A -> not-A
1. A -> not-A
2. A
Therefore,
3. not-A.
Is this argument valid? Why or why not?
According to this source the argument is valid -- https://www.umsu.de/trees/#((A~5~3A)~1A)~5~3A.
I think it is not a valid argument because there is no way that the conclusion can follow from the premises.
2. A
Therefore,
3. not-A.
Is this argument valid? Why or why not?
According to this source the argument is valid -- https://www.umsu.de/trees/#((A~5~3A)~1A)~5~3A.
I think it is not a valid argument because there is no way that the conclusion can follow from the premises.
Comments (992)
1. is a conditional contradiction.
2. fulfils the condition'
And from a contradiction, anything and everything follows. This is the principle of explosion.
That is to say, "1. A -> not-A" is impossible; when the impossible can happen, anything can happen.
1. That Sue is sitting implies that Sue is not sitting.
2.Sue is sitting.
Therefore, Sue is not sitting.
A truth table will tell you this (the whole statement) is true if Sue is sitting or if she isn't sitting.
I just quoted Priest so I have him on hand:
Lewis wrote a lot about this too.
I mean, you can always just laugh at these and ignore them too, there is always a judgement call element in logic anyhow.
Define "follows from".
In ordinary logic, a conclusion follows from a set of premises if and only if there is no interpretation in which all the premises are true and the conclusion is false; and P -> Q is true if and only if at least one of these: (1) P is false, (2) Q is true; and ~P is true if and only if P is true. Thereby:
There is no interpretation in which
A -> ~A
and
A
are both true
and
~A is false
Indeed
A -> ~A
A
therefore ~A
is an instance of modus ponens:
P -> Q
P
therefore Q
The fact that {A -> ~A, A} is inconsistent doesn't contradict that there are no interpretations in which both A -> ~ A and A are true but ~A is false, as indeed there are no such interpretations since there are no interpretations in which A -> ~A and A are both true, even without consulting modus ponens.
If you propose a context with different definitions of "follows from" or different definitions for the truth or falsehood of '->' or '~', then you're welcome to state your definitions.
I can't parse that.
That is correct, but it is not necessary to appeal to explosion, since the argument is valid as it is an instance of modus ponens.
In this case we don't need to appeal to the fact that the premises are inconsistent. If the logic includes modus ponens, then the example is valid, even if the logic does not include explosion.
#1 is a contradiction, reducible to ~ A or ~A. Since it concludes A cannot be true, the antecedent (if A) is always false.
#2 is false and contradicts #1 that establishes ~A.
#3 is not a conclusion, but is a restatement of #1.
As given, 3 is a conclusion.
3 follows from 1 and 2 by modus ponens.
[EDIT:
A -> ~A is not a contradiction.
A is false or not depending on an interpretation.]
The argument seems a bit less problematic if the second premise were changed to: "Sue is not sitting" because then it seems to me that the argument can at least be true in some sense.
Understandable, there is a typo there. I mean the conclusion column is true regardless of the truth value of A.
Indeed.
1 means "If A is true, A is false." This means A can never be true, despite it being true. It's a walking contradiction. This in itself can be taken to mean A is false because, as noted A -> ~A is logically equivalent to ~A or ~A as a disjunction of the conditional ( A --> B = ~A or B). 1 therefore means ~A.
This can be reduced to:
1. ~ A
2. A
Therefore ~A.
The conclusion is a restatement of #1. 2 is a contradiction of 1..
If so, can you say which premise is false and why?
1 is false. "If A is true, then A is false" is a necessarily false statement.
"If A is true, then A is false" is logically equivalent to "A is false or A is false." This means that A is false.
A --> ~A = ~A v ~A
~A v ~ A = ~A
You missed my point.
Of course, there are different ways to show the validity of the argument. But my point is that one of the ways doesn't require appealing to explosion or even contradiction since the argument is in the form of modus ponens.
That's correct. [EDIT: except for use of '=' instead of '<->']
So is:
A -> B
A
therefore B
Well, in the intuitive natural language context I think people would simply want to reject the entailment. E.g., "But that my dog is alive doesn't entail that he is dead."
It could be more interesting in an instance of self-reference. E.g. A is the proposition "this proposition is false." This would, as far as I can see, be a case where intuition would actually tell us that if A is true it entails that A is false.
I cannot think of any concrete examples where we wouldn't simply dismiss it as gibberish though.
A -> ~A is true when A is false and it is false when A is true.
I'd argue A --> ~ A is not of the form A --> B as required as a first premise of modus ponens.
The generic modus ponens syntax requires that the antecedent and consequent be different, meaning that A --> A is not logically equivalent to A -->B because the latter is not reducible to a contradiction.
There is no interpretation in which both premises are true.
If the interpretation has A as true, then A -> ~A is false.
If the interpretation has A -> ~A as true, then A is false.
And no interpretation has both A and A -> ~A as false.
Quoting NotAristotle
The argument is valid, and there is no interpretation in which the argument is sound.
An argument is valid if and only if there is no interpretation in which all the premises are true and the conclusion is false.
An argument is not "inconsistent". What are inconsistent are sets of formulas. What is inconsistent here is the set of formulas that is the set of premises.
It's a valid argument, so the conclusion is true in any interpretation in which all the premises are true. There are no interpretations in which all the premises are true. The conclusion is true in some interpretations and false in other interpretations.
It's a valid argument only if you allow that A --> ~A is of the form A-->~B.
I don't think it follows proper modus ponens syntax. The antecdent and consequent cannot be the same because if they are then it is reducible to simply ~A.
Then you'd argue incorrectly
Quoting Hanover
Modus ponens is any argument of this form:
P -> Q
P
therefore Q
There is no restriction on what P and Q can be.
That includes taking P to be A, and taking Q to be ~A.
A -> ~A
A
therefore ~A
is most certainly an instance of modus ponens.
That is incorrect.
If A is false then "If A is true then A is false" is true.
Quoting Hanover
That is correct.
If my dog does not have have fleas, then "if my dog has fleas, then my dog does not have fleas" is false.
This is where we disgree.
A --> ~A <> A --> ~ B because A-->~A = ~A, yet A-->~B <> ~A.
That is incorrect. Validity is semantic.
A -> ~A
A
therefore ~A
is valid since there are no interpretations in which the premises are true and the conclusion is false.
As to form, we prove that anything in the form of modus ponens is valid.
Quoting Hanover
(1) It is valid whether viewed as modus ponens or viewed by consideration of the fact that the premises are not satisfiable.
(2) You have a serious misconception of modus ponens,
Modus ponens is any argument of this form:
P -> Q
P
therefore Q
There is no restriction on what P and Q can be.
That includes taking P to be A, and taking Q to be ~A.
A -> ~A
A
therefore ~A
is most certainly an instance of modus ponens.
You are welcome to state an alternative logic, but in ordinary truth-functional logic:
If P is false, then P -> ~P is true.
It's where you disagree with the definition of 'modus ponens'.
If P is false then if P is true then it is true that P is true is a contradiction pretty plain and simple.
Nope, we're in agreement with MP. We're in disagreement that P--> Q = P --> P. The former is a conditional, the latter a tautology.
You're confused. I'm not "equating" A -> ~A to A -> B.
Let P and Q be metavariables over formulas. Then modus ponens is any argument of the form:'
P -> Q
P
therefore Q
Instantiate P to A. Instantiate Q to ~A. There is no restriction against such an instatiation.
So
A -> ~A
A
therefore ~A
is an instance of modus ponens.
You're confused.
Look at the truth table by which you will see that if P is false, then P -> ~P is true.
It's ridiculous to argue about it. Just look at it.
Again, this is incorrect. You cannot substitute P and Q to be a statement with the exact same truth value and maintain logical equivalence because once P and Q are the same, you have a different logical statement.
A -> ~A = ~A. That is, it is reducible to that.
A->~B is not reducible to ~A.
Therefore: A-->~A is not logically equivalent to A --~B.
It's like saying A+A = 4 and since it's generic, I can also say A+B=4. In the first case, A=2. In the second, we don't know what A or B equals.
You're confused.
I did not say "P--> Q = P --> P". I said that
A -> ~A is an instance of P -> Q.
Nothing about equality, only instancehood.
And both P -> Q and P -> P are conditionals. The fact that P -> P is a tautology doesn't make it not a conditional.
You can instantiate P and Q to whatever formulas you want.
You somehow got in your head a wrong notion.
Quoting Hanover
I said nothing about logical equivalence.
Read what I wrote.
Quoting Hanover
Actually, the notation is:
(A -> ~A) <-> ~A
(A -> ~A) is not equal to ~A. They are not the same formula. Rather they are materially equivalent. Equality and material equivalence are not the same. The distinction is important.
Quoting Hanover
I said nothing about "reducible".
Again, read what I wrote.
Quoting Hanover
I didn't say they are. Read what I wrote.
Quoting Hanover
No, it's nothing like that. I made no such argument.
Well, one of us does.
Sure. I am just referring to the truth table
You need to reevaluate your mistaken notion about substitutions. [EDIT: replace 'substitutions' with 'instantiations', which is more strictly correct.]
Let P and Q be meta-variables (read as 'phi' and 'psi' if you like) ranging over sentences.
Modus ponens is any argument:
P -> Q
P
therefore Q.
Let A be a sentence letter or any sentence.
Instantiate P to A. Instantiate Q to ~A. Those are perfectly legal instantiations. There is no rule that disallows them. Look in any logic book or ask any logician.
So
A -> ~A
A
therefore ~A
is an instance of modus ponens.
/
And look at the truth table for
A -> (A -> ~A)
to see that it is true when A is false.
/
And, you said, "If A is true, then A is false" is a necessarily false statement.
That's incorrect.
If "A is false" is true, then "If A is true, then A is false" is true.
The antecedent is "If A is true" and the consequent is "A is false".
If "A is false" is true, then the antecedent is false and the consequent is true, so the conditional is true.
So "If A is true, then A is false" is not necessarily false, since there is an interpretation (viz. when ""A is false" is true" in which it is true.
The correct statements are:
(If A is true then A is false) then A is false.
If A is false then (If A is true then A is false).
A is false if and only if (If A is true then A is false).
Yes, that is the truth table.
"is this modus ponens:
A-> ~A
A
~A"
ChatGPT said:
"No, this is not an example of modus ponens. Modus ponens has the form:
?
A?B (If A, then B)
A (A is true)
Therefore,
B (B is true)
In your example, you have
? ¬
A?¬A (If A, then not A), which leads to a contradiction when assuming
So it's not a valid application of modus ponens. Instead, it illustrates a logical inconsistency."
Amazing that someone would take the word of a bot on such a question. One could get bots to generate misinformation over and over again.
But not amazing that bots regularly get things quite wrong.
The bot does not understand substitution. And the bot can't even write formulas or English correctly: [EDIT: replace 'substitution' with 'instantiation', which is more strictly correct.]
"In your example, you have
? ¬
A?¬A (If A, then not A), which leads to a contradiction when assuming"
(1) -> ~
is not well formed
(2) "assuming" doesn't end a sentence there.
(3) What the bot must mean is:
A -> ~A and A lead to a contradiction.
That is correct. But it doesn't show that
A -> ~A
A
therefore, ~A
is not an instance of modus ponens.
The bot incorrectly reasons that the inconsistency of the premises disallows the argument from being modus ponens. But it is not disallowed the premises of a modus ponens argument may be inconsistent
Here is correct information:
Let P and Q be meta-variables (read as 'phi' and 'psi' if you like) ranging over sentences.
Modus ponens is any argument:
P -> Q
P
therefore Q.
Let A be a sentence letter or any sentence.
Instantiate P to A. Instantiate Q to ~A. Those are perfectly legal instantiations. There is no rule that disallows them. Look in any logic book or ask any logician.
So
A -> ~A
A
therefore ~A
is an instance of modus ponens.
AGAIN: There is no rule of logic that prohibits substituting A for P and ~A for Q. [EDIT: replace 'substituting' with 'instantiating', which is more strictly correct.]
ChatGPT is just slamming text together that tends to flow together. Something with P and ~P is going to cause it to talk about contradictions because that's where the text usually appears.
It is bad at logic and uncommon programing languages like Prolog. It is actually surprisingly good at Java and Python, but still sometimes comically bad. I am waiting for the first financial meltdown of a sizeable business based on people using ChatGPT to code spreadsheets or dashboards.
Although I hear they are working on some that will do math well. I'll believe it when I see it.
That bit from the bot is atrocious confusion and misinformation. And it is intellectually shameful for a poster to post a confused and misinformational bot quote as if it is correct and settles a discussion about the logical matter. And risible that a poster quoting a bot doesn't understand that such bots compose text that seems like something that might be said, without fact checking itself that it is actually correct.
Get outta here with that bot garbage!
It should not have to be said more than this:
The rule of modus ponens is:
If P and Q are ANY statements, then the following is modus ponens:
P -> Q
P
therefore Q
Since P and Q may be ANY statements, and A and ~A are statements we have:
A -> ~A
A
therefore ~A
is an instance of modus ponens.
Period.
And these also are instances of modus ponens, even though odd:
A -> A
A
therefore A
~A -> ~A
~A
therefore ~A
Period.
This is NOT the rule of modus ponens:
If P and Q are any statements except Q is not the negation of P, then the following is modus ponens:
P -> Q
P
therefore Q
And this is NOT the rule of modus ponens:
If P and Q are any statements and {P, P -> Q} is consistent then the following is modus ponens:
P -> Q
P
therefore Q
/
One more time, since the poster is presenting as seriously obtuse:
The rule of modus ponens is:
If P and Q are ANY statements, then the following is modus ponens:
P -> Q
P
therefore Q
Since P and Q may be ANY statements, and A and ~A are statements we have:
A -> ~A
A
therefore ~A
is an instance of modus ponens.
I don't usually say, "Please, let's move on" but I'm saying it this time.
Modus ponens "is the rule of logic stating that if a conditional statement (if p then q ) is accepted, and the antecedent ( p ) holds, then the consequent ( q ) may be inferred."
That is, it is the logical basis one asserts in support of the conclusion. If your conclusion is not true, you can't offer MP as the basis of it being true because it's not.
No, it's the DEFINITION of 'modus ponens'.
Quoting Hanover
Modus ponens doesn't require that a conditional is not contradictory, nor that the "major" premise (which must be a conditional) is not contradictory, nor that the "minor" premise (which might or might not itself be a conditional) is not contradictory, nor that the premises together are not contradictory.
Quoting Hanover
What is your source of that quote?
Quoting Hanover
You don't understand basic ordinary academic logic. You need the first chapter of a good textbook in print or online.
Meanwhile, you need to not litter a philosophy forum with confused, misinformational, and malformed bot garbage.
I already cited you the definition, which isn't as you're arguing.
Quoting TonesInDeepFreeze
What is your cite for this definition?
Mine is from Google, which comes from Oxford Languages.
https://www.google.com/search?q=definition+of+modus+ponens&oq=definition+of+modus+po&gs_lcrp=EgZjaHJvbWUqDQgBEAAYkQIYgAQYigUyBggAEEUYOTINCAEQABiRAhiABBiKBTIICAIQABgWGB4yCggDEAAYDxgWGB4yCAgEEAAYFhgeMggIBRAAGBYYHjIKCAYQABgPGBYYHjIKCAcQABgPGBYYHjIICAgQABgWGB4yCAgJEAAYFhgeMggIChAAGBYYHjIKCAsQABgPGBYYHjIKCAwQABgPGBYYHjIHCA0QIRiPAjIHCA4QIRiPAtIBCTExODQ0ajBqOagCAbACAQ&client=ms-android-tmus-us-revc&sourceid=chrome-mobile&ie=UTF-8#ebo=0
Quoting TonesInDeepFreeze
Quoting TonesInDeepFreeze
Calm down, please. You're making this emotive.
is clearly mistaken. As is ChatGPT.
I have no comment on that. Thank you for keeping things polite on your end anyhow.
All definitions I have located say otherwise, as do all Google and AI engines.
Provide to me your cite to close out this incredibly irrelevant question.
Find one that does so, and you will have support for your claim.
Otherwise, the rule is that any formula can be substituted for A and B, including ~A.
And this is quite basic stuff. So from Open Logic:
Nothing says that we may not substitute A for ? and ~A for ?. Hence, we may. Indeed, that's kinda the point.
But this is trivial stuff! Why don't you already know this?
(1) That definition does not contradict that
A -> ~A
A
therefore ~A
is an instance of modus ponens
(2) Here are definitions of 'modus ponens':
"if a conditional holds and also its antecedent, then the consequent holds." (Beginning Logic - Lemmon)
"C is a direct consequence of B and B -> C." (Introduction To Mathematical Logic - Mendelson)
"From the formulas Alpha and Alpha -> Beta, we may infer Beta" (A Mathematical Introduction To Logic - Enderton)
"from P and P -> Q we may infer Q" (as the rule corresponding to the tautology (P & (P -> Q)) -> Q) (Introduction To Logic - Suppes)
"Psi is obtained from Phi and Phi -> Psi" (Mathematical Logic - Monk)
"A, A -> B |= B" (A Concise Introduction To Mathematical Logic - Rautenberg)
"the inference from A and A -> B to B" (Computability And Logic - Boolos, Burgess and Jeffrey)
"Gamma, Phi -> Psi and Gamma, Phi; therefore Gamma, Psi" (Mathematical Logic - Ebbinghaus, Flum and Thomas)
"passing from two formulas Alpha and Alpha -> Beta to the formula Beta" (A course in Mathematical Logic - Bell and Machover)
"Phi -> Psi, Phi; therefore Psi" (Logic: Techniques Of Formal Reasoning - Kalish, Montague and Mar)
"If P and P -> Q are proved, then one is entitled to infer that Q is proved" (Logic For Mathematicians - Rosser)
"A, A -> B |- B" (Introduction To Metamathematics - Kleene)
"p, p -> q |- q" (Foundations Of Mathematical Logic - Curry)
"from the premisses Phi -> Psi and Phi to Psi" (Mathematical Logic - Quine)
"From A -> B and A, to infer B" (Introduction To Mathematical Logic - Church)
"Psi may be entered on a line if Phi and Phi -> Psi appear on earlier lines" (Elementary Logic - Mates)
"From Psi and Psi -> Phi infer Phi" (Model Theory - Chang and Keisler)
"If p then q, p, conclude q" (Symbolic Logic - Copi)
And on and on in as many books on basic formal logic that you may look at.
All those definitions have in common that there is NO requirement that we may not instantiate the variables to A and ~A.
All those definitions have in common that there is NO requirement that the premises are not contradictory
Quoting Hanover
It's not a definition! It's a comment about definitions. It is not itself a definition.
Meanwhile, you will find NO cite of a definition that requires that P can't be instantiated to A while Q is instantiated to ~A. And you will find NO cite of a definition that requires that the premises are not contradictory.
Justifiably.
I thought this forum was going to warn against citing bot misinformation.
It's not a matter of opinion that
A -> ~A
A
therefore ~A
is an instance of modus ponens.
It is a plain fact.
It is quite impolite to continue to ignorantly insist on bad misinformation and to cite wildly erroneous and incoherent bot messages as if they are information.
(1) There is no "self-reference".
(2) The conditional A -> ~A is not contradictory.
(3) Nowhere in the definition of 'modus ponens' is it disallowed to instantiate to P to A and Q to ~A.
(4) Where "pray tell" do you find a definition that says "except Q cannot be instantiated to the negation of what P is instantiated to"? Hint: You don't.
There is no cite, no source, no reference that says such a thing.
You just somehow got it stuck in your head that such a thing is implied by the definition. But it's not.
AGAIN you need to read and comprehend.
P and Q range over formulas.
From P and P -> Q, infer Q by the rule modus ponens
Since A and ~A are formulas, we have:
From A and A -> ~A infer ~A by the rule modus ponens
You cannot show any definition, explanation or argument in any logic book or reliable article that says, implies or insinuates that the definition of 'modus ponens' disallows:
From A and A -> ~A infer ~A by the rule modus ponens
But you many look up arbitrarily many logic books that do imply that
From A and A -> ~A infer ~A by the rule modus ponens
from the plain definition of 'modus ponens' such as:
From formulas Phi and Phi -> Psi, infer Psi
where 'Phi' and 'Psi' are variables ranging over formulas.
Indeed, at a certain point in discussions where a poster is flat out wrong about a matter that is not even a matter of opinion, and persists to insist despite copious explanations given him, then the pertinent question turns from the simple fact of the matter about the subject to what is wrong in the head of the stubbornly clueless poster.
Calm down! You're making this emotive!
Nothing says we can, which is kind of the point.
The absurd question of whether MP includes instances of A causing not A while A is the case doesn't seem to have gained much interest in the world outside the 3 or 4 of us debating it here. Thus the lack of an explicit statement supporting your position anywhere.
But yes, profoundly trivial and entirely irrelevant from a logic perspective. But, if you're asking me to read and define terms, your definition of MP is not logically entailed. It makes as much sense to define MP as excluding instances where A and not A coexist.
The rule DOES imply we can since the rule quantifies over ALL formulas.
For that matter the rule doesn't explicitly mention any particular substitutions. [EDIT: replace 'substitutions' with 'instantiations', which is more strictly correct.] For example, the rule doesn't explicitly mention that:
(A & B) -> C
A & B
therefore C
is an instance of modus ponens. But it is an instance of modus ponens.
And
A -> ~A
A
therefore ~A
is another instance of modus ponens though it too is not explicitly mentioned in particular in the rule.
It is part of the POINT of being a rule that it can be applied to ANY formulas.
Quoting Hanover
It's a DEFINITION. It's not supposed to be "entailed".
It's not the job of the person who is giving correct information to provide a face saving escape hatch for the stubbornly irresponsible person who continues to spew misinformation no matter how many times he or she has been provided ample explanations and citations. If the person doesn't have the intellectual honesty to admit a glaring mistake then that's on the person entirely, especially after having been given copious explanation and citations. Also, one could be as conciliatory as pie to such a poster, and still he or she would not admit his or her error but rather on the contrary, he or she would persist even longer. That is the nature of Internet forums.
Logic is generally handled very badly here - as if it were a question of opinion as to what is valid and what is not, rather than of structure. That a third of folk think the argument in the OP is invalid... that's cause for concern.
Prime real estate was offered for free from the beginning.
Or put another way, the horse was offered the freshest, coolest, cleanest mountain spring water. He won't drink is his choice.
Quoting Banno
It is deplorable the number of people who come into a philosophy forum without having read even page one of a book in logic or mathematics while spewing hyper-opinionated misinformation and nonsense on those subjects. It is utterly reasonable that one would become exasperated by that. Meanwhile, a moderator comes into scold the expression of exasperation while not a word that it is at least seriously frowned upon to cite bot misinformation and confusion, despite that (at least last I happened to read) the forum has said in general that that is not acceptable.
As Tones explained, it's not MP you have misunderstood, but substitution. MP is a rule of inference, saying that if you have ? and ? ??, then you also have ?, where ? and ? are whatever formulae or propositions or sentences you are discussing. That includes substituting the same formula for both, and the negation of ? for ?.
You are mistaken. Sorry.
I've used ChatGPT occasionally to check things, usually nomenclature, sources, who first proposed an idea, or such. This case is a reminder to be aware of confirmation bias. ChatGPT gave @Hanover too great a confidence in his error.
An example of Modus Ponen failure is presented in the Wiki article as the Vann Mcgee case:
https://en.m.wikipedia.org/wiki/Modus_ponens#:~:text=Philosophers%20and%20linguists%20have%20identified,The%20following%20is%20an%20example:
Something I came across in tonight's research.
The antecedent directly contradicting the consequent isn't an example given of MP failure, as far as I can tell, anywhere except here.
So, you're either you're the first to notice it, or it's not really an example of MP failure because it's not MP.
I'm making efforts to clarify the sourcing issue in the guidelines and the mod forum. I'd ask for some patience with us and with other posters on this issue while we sort it out.
Of course, fair enough that it a tough and complicated matter for moderators.
But, in the meantime, I think it is appropriate for a poster to express exasperation when a poster plasters bot misinformation. Indeed, more appropriate to express it than to be quiet about it.
My own view is to not enforce censorship but on, the other hand, to be clear that it is not welcome.
So, while you ask for patience with the moderation, I suggest that the moderation have patience with justified exasperation in reaction to poster abuse of bot quoting, especially not to scold the poster who at least is providing correct info.
If I recall, the Van McGee paper was the subject of a thread. And, if I recall, his argument hinged on adopting a different notion of the conditional.
Anyway, just to be clear, dissent from modus ponens doesn't change what the definition of 'modus ponens' is.
https://thephilosophyforum.com/discussion/11395/a-counterexample-to-modus-ponens/p1
Thanks.
What I wrote eventually:
Quoting TonesInDeepFreeze
If contradiction, then anything.
contradiction.
therefore anything.
The horse has been beaten to death here, but do at least understand I don't struggle with understanding your position, but I simply include within my definition of MP a requirement that it not self contradict.
As I've noted, this is a definitional debate, and we might as well be arguing if a cup with a hole in it entirely incapable of use is still a cup.
That is to say:
If I don't agree with you, I agree with you, and since I don't agree with you, I do. mp.
So says Alice when she's ten feet tall.
How I avoid this logical absurdity is to deny mp permits it, but you may insist that it is as it is. Sometimes cups just don't hold water you say.
I submit p can't be q for a valid mp, except among the speakers in Wonderland.
But at any rate, as always, I do appreciate the passion for such a crazy conversation though. An odd lot we are.
The argument in the OP is:
¬A ? ¬A
A
? ¬A
It's valid, but of course the premises cannot both be true. Necessarily one is false and so the argument is necessarily unsound.
https://thephilosophyforum.com/discussion/comment/943645
https://thephilosophyforum.com/discussion/comment/943647
Not exactly. You were saying that A ? ¬A is necessarily false, which is saying that ¬A ? ¬A is necessarily false. But ¬A ? ¬A is true if ¬A is true, and so A ? ¬A is true if ¬A is true.
Artistotle and Euclid use contradiction in reductio demonstrations all the time. If we have a valid argument with a conclusion we know to be false then we have warrant to reject a premise or assume that at least one is false.
However, I do agree that the common analysis that, if ~A is true, then A?~A doesn't sit well with common sense intuitions about consequence. The truth table also is liable to look confusing because it varies from how the premises are laid out, but I think that it isn't once properly understood.
Nor is my first premise. I think my argument is valid and sound, and expresses the principle of explosion. Similar nonsense results when you divide by zero, and you can disguise that division in a similar way with a bit of algebra. Obviously the universe is the result of God accidentally contradicting Himself by making a mistake, which He cannot do, being infallible. Explains everything!
Impossible cannot happen, therefore nothing can happen?
"Yes."
Fair point. Ok, I concede; "obviously" was a misspoken epithet on my part.
Allowing substitution of any well-formed formula is not a personal foible. It is how propositional logic works. (?, ? ?? ? ? ) for any well-formed formula ? and ?. Nothing says they must be different.
Were debating whether to call certain formulations "modus ponens."
There is no governing body in what to call it. My basis for excluding self contradictory versions has been stated.
As noted, there are exactly zero citations so far found where someone other than us has analyzed whether the OP case belongs in mp. Where we have found debate over invalid mp formulations on the web, exactly zero deal with the OP case.
The point being, should we guage term usage for meaning, I see no evidence supporting your usage.
As I've also repeatedly said, this is a definitions question, not a logic one. We both agree upon what entails what and what can be substituted in for what.
The OP is not a problematic example of mp. It's not mp at all.
.pdf, 32 pages including footnotes.
The OP uses propositional logic. In propositional logic, the argument is valid.
What violates LNC?
Of course, anyone can stipulate their own definition. But your definition is not the one used in ordinary formal logic, or, as far as I know, in any treatment of logic.
And you haven't even stated a definition. You've incorrectly stated that modus ponens disallows certain premises such as those in the example. That's not a definition.
Here are correct definitions (along with the many from standard textbooks I listed a few posts ago, where P and Q range over formulas):
(as an argument form:)
An argument is an instance of modus ponens
if and only if
the premises of the argument are P and P -> Q and the conclusion of the argument is Q
(as a rule of inference:)
The rule of modus ponens
is
from P and P -> Q, infer Q
symbolically:
{P, P -> Q} |- Q
(as a kind of entailment:)
modus ponens
is
the entailment of Q from P and P -> Q
symbolically:
{P, P -> Q} |= Q
Quoting Hanover
There's no reasonable debate. That you fancy that you have your own definition has no bearing on the fact that in the field of study of formal logic, modus ponens is defined and your claims about it are inconsistent with the definition.
Quoting Hanover
That would be true if you constituted a lot.
Here is the truth table for the contradiction A and not-A:
Notice that the column under A and not-A is false for every assignment to A. That's why it is a contradiction: it is always false.
Here is the truth table for A? ~A
Notice that the column for A? ~A is not false if A is false. A ? ~A is not a contradiction. Rather, it says that A is false.
If A is true, then A is false. Therefore A cannot be true.
, 's error, perhaps.
In that post you wrote:
Quoting Hanover
That is wrong, as has been explained to you over and over and over.
The OP is a factual question, not an issues of opinion. The one-third of folk who think that the argument is invalid are wrong. As wrong as if they had asserted that 2+2=5.
One-third of folk who have at least enough interest in logic to respond to the OP do not have a basic understanding of validity.
That's pretty sad. On a philosophy forum, it's pathetic. That is, it arouses pity.
Let me test it.
If the OP uses propositional logic, it doesn't use propositional logic.
It uses propositional logic
Therefore it doesn't use propositional logic.
MP has spoken. It doesn't use propositional logic
This is false. It corresponds to line two of the truth table given above.
Only line 1 is not, ~A. It's A?~A.
It's
1. A ? ~A (assumption)
2. A (assumption)
3. ~A (1,2,MPP)
Not
1. ~A (assumption)
2. A (assumption)
3. ~A (1)
(1) Here is that definition in full:
"Modus ponendo ponens is the principle that, if a conditional holds and also its antecedent, then its consequent holds." (Beginning Logic - Lemmon)
Perhaps your argument is based on taking that to mean this?:
If a conditional holds and also its antecedent, then modus ponedo ponens is the principle that then its consequent holds.
That is wrong.
Modus ponens doesn't require that the premises hold. Modus ponens say nothing about the standalone truth or falsehood of the premises or the standalone truth or falsehood of the conclusion. Modus ponens only says that IF the premises are true then the conclusion is true. That is, there are no interpretations in which all the premises are true but the conclusion is false. That is:
For any formulas P and Q, there are no interpretations in which P is true and P -> Q is true and Q is false.
And that is verified by a truth table.
And it does not disallow P from being instantiated to A and Q from being instantiated to ~A:
If A is true and A -> ~A is true, then ~A is true.
That is an instance of modus ponens.
Again, there is nothing in the principle or rule that says P cannot be instantiated to A while Q is instantiated to ~A.
A is a formula.
~A is a formula.
Modus ponens is the principle that for any formulas P and Q, if P and P-> Q, then Q.
So, one instance of modus ponens is: if A and A -> ~A, then ~A.
(2) Not that it matters for the point above, but as a matter of fact, A -> ~A does hold when A is false.
But (A->~A) & A is a contradiction.
If you assert A->~A, and then go on to assert A, then you have contradicted yourself.
The set {A->~A, A} is not a contradiction because it is not a formula, but a set. It is, however, inconsistent.
Would there be any harm in requiring that the conditional in a modus ponens have fresh variables on the right hand side? We would lose, in effect, only this one and A->A, which is either useless or the LEM, and thus innocent or tendentious, depending on how you look at it.
I mean, mathematicians always prefer the greatest generality, at the minor, to them, cost of letting in the degenerate case. If your project is automated reasoning, you'll go with the usual. But for doing philosophy, we don't have to let the mathematicians have the last word.
If the only question here is "How does formal logic work?" we know the answer to that. But around here we're more interested in the practical use of logic, and it seems to me letting mathematical logic have the last word is the tail wagging the dog.
Quoting Banno
I mean, I get that MP requires a "->", but (A->~A)<->~A, so I'm puzzled by insisting on this nicety. In classical logic it's materially equivalent to the disjunctive syllogism, isn't it?
2. Hanover is correct
3. Hanover is not correct (1,2 mp)
4. Hanover is not correct or 3 is an invalid conclusion derived from mp.(3, introduction)
5. 3 is an invalid conclusion derived from mp (3,4)
Yep.
Quoting Srap Tasmaner
Well for a start you would no longer be dealing with a complete version of propositional calculus...
Quoting Srap Tasmaner
Too often this is an excuse for poor logic.
Back to the question: Do you, Srap, agree that the argument in the OP is valid?
Again, 1 is false, and your argument (1-3) valid but unsound.
So what? Modus ponens is well understood and defined in thousands of books and articles and your remarks about it are not consistent with the common definition. You can define it any way you like (though you haven't defined it but only given certain qualifications about it), but then you will be talking about something very different from other people - such as logicians, philosophers, mathematicians, and students of logic, philosophy or mathematics - are talking about. What is the point of that?
You only engender misinformation and confusion on this point. No one is stoppiing you from giving a different from giving it a name ('hodus honens' would be good) and defining it.
Quoting Hanover
So what? I gave you over a dozen defintions where the variables range over formulas or statements, and such that it is trivial for us to deduce that the example is an instance of modus ponens.
You might as well say, "We haven't seen a citation in which is found an analysis that
54322995731999373272287 + 229699797833575592 is an instance of summation"
We haven't seen it since anyone can carry out the analysis for themself:
For ANY natural numbers x and y, x+y is a summation.
instantiate x to 54322995731999373272287
instantiate y to 229699797833575592
so 54322995731999373272287 + 229699797833575592 is an instance of summation.
For ANY statements P and Q, the inference of Q from P and P -> Q is modus ponens.
instantiate P to A
instantiate Q to ~A
so the inference of ~A from A and A -> ~A is an instance of modus ponens.
Quoting Hanover
You mean McGee?
(1) McGee says himself that the invalidity does not concern the material conditional.
(2) AGAIN, since you SKIP this point, that people may dissent from modus ponens doesn't affect what the definition of 'modus ponens' is.
Quoting Hanover
I've given you nearly two dozen textbook definitions from which anyone who has read the first chapter could deduce that the example is an instance of modus ponens.
Quoting Hanover
You know nothing about it. Nothing at all.
Thanks, that's an interesting one.
The first premise is false though. We are only affirming a contradiction if we affirm A and ~A.
In this one 2 is false. It is possible to have a valid argument that has some true premises and a true conclusion without all the premises necessarily being true.
It is indeed contrary to intuition that A ?~A should be true if A is false however, that affirming that "if you are incorrect" then "your being correct implies that you are incorrect," is pretty tortured, I'd agree.
:up:
I cannot think of a way to frame this as a real example outside of self reference and removing A?~A would solve that.
No. Because the premise "If the OP uses propositional logic, it doesn't use propositional logic" is false.
At least read the first chapter of a book on the subject.
Yes, and you are wrong that
A -> ~A
is a contradictory.
Right. But there is a person here who claims that A -> ~A is necessarily false. That is what is being addressed.
Also, the person claims that if the premises are contradictory, then it is not an instance of modus ponens. That also is being addressed.
Quoting Srap Tasmaner
As I recall, that is a needed theorem along the path to proving the important sentential theorems from certain axiomatizations. Indeed, it must be a theorem for the sentential calculus to be complete.
Quoting Srap Tasmaner
Whatever the merits of that view, the most recent discussion is not how logic should be set up but as to what is the case with the way ordinary propositional logic is set up.
True, but if, pace Frege, we assume assertoric force, then to claim "A ? ~A" along with "A" is to contradict oneself, and therein I think lies the confusion.
1. Meaning what exactly?
2. Is the answer to (1) something I should care about?
Quoting Banno
I don't really care. It's abusive.
Quoting Count Timothy von Icarus
I'll come up with one. I think you see it around the forum and elsewhere in the wild pretty regularly. Informally, we're looking for a case where you try to disagree with me, but that attempt misfires because in stating your position you have to tacitly agree with me.
So you claim A, but I show that still leads, by some chain of reasoning or argument to my claim ~A. Introducing A?~A is just icing on the cake, because it's still just an extravagant way of saying ~A.
I don't know what question you are raising.
In ordinary formal logic:
(A -> ~A) <-> ~A
(A -> ~A) <-> ~A v ~A
(~A v ~A) <-> ~A
etc.
A -> ~A
A
therefore, ~A
is an instance of modus ponens.
Whether that should be the logic for certain reasoning is another question. But one can't properly address that question if one doesn't at least understand how that logic does operate and not have misconceptions about it.
No, you need to know the difference between truth and validity.
That difference has been explained in this forum at least a hundred times. It is fundamental to formal logic.
Or read a logic book.
Meaning that it wouldn't be the case that all tautologies are theorems.
Quoting Srap Tasmaner
If you are interested in the basics of ordinary formal logic, then it would be a question that would naturally occur to you. But I don't see why you couldn't study other branches of philosophy without understanding the completeness of the propositional calculus.
It's incorrect. But what do you mean by 'abusive'?
Yeah, I actually thought of a more concrete one we see on this forum: "it is true that nothing is true."
Normally this is just the claim "nothing is true," made with assertoric force. Same for "it is true (I know) that knowledge is impossible."
This is still self-reference though I guess.
Exactly that. If you modify the substitution rule to remove substitution of the same variable on both sides of a function, can you demonstrate that the resulting calculus will be complete? Can you prove A?A, for example?
Yep.
Quoting Srap Tasmaner
That's entirely up to you. But you are on this thread, so forgive my presumption. Failing to see that the argument in the OP is valid is an indication of a lack of understanding of basic logic. Refusing to give an opinion says something else.
Quoting TonesInDeepFreeze
Some folk think that pointing out an error os abusive. Odd, sad, but true
Yes, A -> ~A with A is contradictory.
I don't know anyone who has said otherwise.
But I can't say what is the source of the mental block in people who don't understand that "A -> ~A with A is contradictory" doesn't entail "A -> ~A is contradictory", other than that those people have never read the first page of a book or article on the subject and are stuck with misconceptions that they adopted blindly.
Meanwhile, such ignoramuses don't understand even the most basic distinction between truth and validity while they promulgate that ignorance on a supposed philosophy forum.
1.Life therefore death
2.Life
Therefore
3.Death.
Both valid and sound it seems.
Also famously:
Life then Death and Taxes.
Life.
therefore Death and Taxes.
The first states modus ponens as a principle not an argument form. But as an argument form it is such that the premises are a conditional and its antecedent, and the conclusion is the consequent of the conditional, so that if both the conditional and its antecedent are true then the consequent of the conditional is true.
The second says that if a conditional and its antecedent are true then the argument is modus ponens if the premises are the conditional and its antecedent, and the conclusion is the consequent of the conditional.
The first is a correct definition of 'modus ponens'. The second is not a correct definition of 'modus ponens'.
The first does not require that the conditional and its antecedent are true; only that IF they are true then the consequent of the conditional is true.
The second allows us to take an argument as an instance of modus ponens if and only if the premises are a conditional and its antecedent, and the conclusion is the consequent of the conditional, and the conditional and its antecedent are true.
These also are not correct definitions of modus ponens:
A modus ponens argument is one in which the premises are a true conditional and its true antecedent, and in which the conclusion is the consequent of the conditional.
That's wrong, since modus ponens does not require that the premises are true.
A modus ponens argument is one in which the premises are a conditional and its antecedent, and they are not together contradictory, and the conclusion is the consequent of the conditional.
That's wrong, since modus ponens does not require that the premises are consistent.
Yeah, but if you affirm that "death" is equivalent with "not-life," you'll be stuck affirming Plato's argument for the immortality of the soul in the Phaedo, which in turn implies that you may be reincarnated for innumerable lifetimes where you have to debate these same topics before finally achieving henosis and completing the process of exitus and reditus. That's a pretty rough commitment to have to make.
It doesn't seem that hard to understand to me. If people don't use formal logic often, then the most common thing to do is to translate into natural language. In natural language, we don't say that a falsehood implies anything. There is a relevance condition to consequence. Thus, to say "if my dog is alive then my dog is not-alive," is to say something that seems necessarily falsefalse regardless of if the dog is alive, dead, or even if the dog never existed. And the "necessarily false" is in the same mental bucket as "contradiction."
Or, if something like the OP is framed in terms of self-reference, e.g. "nothing is true," with assertoric force, it would be a self-refuting statement, which also is in the same vein. A Catch-22 would be similar, and people might even call it contradictory in common parlance.
That cuts a crucial part of the sentence:
Quoting TonesInDeepFreeze
Of course I understand that many people refer to everyday senses. My point though is that after it is made clear that the context is formal logic, those people persist to incorrectly declare what is the case in formal logic.
Also, I don't say "from a falsehood anything follows". It is better to say, "from a contradiction anything follows". And we don't need that principle in this case anyway, but only the more special instance: from a conditional whose consequent is the negation of the antecedent, that consequent follows.
Well, if you've been taught that a contradiction has a truth table that is always false and you think you have identified something that is necessarily/always false, it seems possible to conflate the two.
I mean, obviously people do confuse this quite often, I can recall several threads, so I figure it's something like that, similar mental buckets.
In his first post:
Quoting Hanover
That misconception is not explained merely as a contrast with everyday reasoning. I have never heard everyday reasoning say ""~ A or ~A" is a contradiction".
Moreover the poster is using symbolizations that at least suggest a formal, not everyday, context.
Only because the poster refuses not to conflate.
Quoting Count Timothy von Icarus
'buckets' is a good word choice.
Quoting TonesInDeepFreeze
Quite. I worked through some of the usual metatheorems years ago when I was studying formal logic. If you're interested in the properties of these formal systems, such results are just what you're interested in. And I'm sure there are issues that come up in philosophy that depend substantively on such results.
But for the everyday use of logic just to schematize and clarify arguments, you get a lot more mileage out of de Morgan's laws, contrapositives, a solid understanding of quantifiers, and such. The cash value of completeness for such applications is nil.
Quoting Banno
As Tones suggested, it might be necessary for proving certain metatheorems, but of course in real applications of logic ? such as on TPF ? "A?A" usually only appears as an accusation of question-begging. It's not something anyone would have any reason to argue for, and it's not a premise anyone would intentionally rely on. ? Hence my suggestion that we could usually get along without it.
Quoting TonesInDeepFreeze
The basic idea is "formally correct but misleading". Akin to sophistry. Or to non-cooperative implicature, like saying "Everyone on the boat is okay" when it's only true because no one is left on the boat and all the dead and injured are in the water.
In this case, for instance, it is suggested that we conclude ~A by modus ponens. The form is indeed instantiated ? I'm not contesting that ? but the first premise is materially equivalent to ~A. People worry over the sense in which the conclusion of a deductive argument is "contained" in the premises ? here it is one of the premises. Who needed modus ponens?
(Besides, you are effectively arguing from the set {A, ~A}. You could as well conclude A from that ? or any B you like ? so in what sense should this count as a "demonstration" that ~A? In what sense is the relationship of A and ~A revealed or clarified? It may be modus ponens in form, but hardly in spirit.)
And if we step back and look at the offending premise, we get to ~A by noting that A?~A is materially equivalent to ~A v ~A. Now what kind of disjunction is that? It's a well-formed-formula ? no one can deny that ? but it's hardly what we usually have in mind as a disjunction. It's "heads I win, tails you lose." That's abusive.
There is, in this case, a veneer of logic over what could scarcely be considered rational argumentation. If this appearance of rationality serves any purpose, it must be to mislead, hence abusive, eristic, sophistical, non-cooperative. ? Again, I am only talking about how logic is used as an aid to ordinary philosophizing, not what people get up to in a logic lab.
There the poster is (in horrible confusion) discussing truth values per formulas in a context of formal logic (a response to my own mention of modus ponens). Whatever is in the slop bucket of his mind about everyday reasoning, he's just plainly confused and wrong about the formal logic at hand.
Heh, describes how these threads normally go.
FTFY
Actually, I have personally gained a lot from study of mathematical logic, in use outside of mathematics, in practical applications, in organizing ideas, and in appreciation of rigor and clarity in practical applications.
If one happens to be interested in the subject of symbolic logic, then it is an eminently natural and wise question to ask: Does this symbolic calculus prove all and only the valid formulas? I wondered about that and had no idea that mathematicians had answered the question with proofs about it. When I saw the proofs, I found the intellectual curiosity acted on, the intellectual honesty and the intellectual creativity to be tremendously inspiring. If you don't, then so be it, but also, so what.
/
Some mathematicians have a solid understanding of quantification naturally without studying symbolic logic. But personally, I found that studying symbolic logic brought me up to speed with what is natural for others.
And, for example, consider the most economically important question in mathematics: P v NP with its economic implications thus its million dollar prize. Work on that presupposes mathematical logic. If one is not interested, then fine, but what is the point then in carping about it?
Quoting Srap Tasmaner
You used the pronoun 'it' right after the argument of the first post was mentioned. So I didn't know whether you meant 'it' to refer to indeed its antecedent. Now I see that you didn't.
Anyway, the counterargument to the original argument is not misleading or like sophistry.
Quoting Srap Tasmaner
It is equivalent with one of the premises.
The original post asked about a symbolic argument. The matter was not whether such things are useful in your life. At least speaking for myself, I have no interest in convincing you that you should regard such matters as useful to you. But that does not vitiate my interest in technical matters in formal logic, even if for the mere mathematical/intellectual pleasure, and especially as that mathematical/intellectual pleasure is attached to a field of study that provides core context for the mathematics for the sciences, engineering, and computing, and especially since I also enjoy philosophical aspects of the subject and having a solid technical founding greatly contributes to both appreciation and understanding of the philosophical discussions.
Quoting Srap Tasmaner
The argument was:
A -> ~A
A
therefore ~A
There was discussion that this is an example of aspects of negation and contradiction. My point was that we can see that the argument is valid in another, quite immediate way, viz. that it is staring us right in the face as modus ponens.
I am not declaring any profundity. It's just a correct observation, with a nice reminder, and pertinent to the context of the discussion.
Quoting Srap Tasmaner
Another poster took that path. It's correct and pertinent too to the formal matter at hand. To call it "abusive" is to abuse the word 'abusive'.
Quoting Srap Tasmaner
The purpose is to be clear about the formal logic. That is a good purpose. One my wish not to talk about formal logic, or to disagree with it, but it is a good purpose to at least be correct and clear about what does happen in formal logic, especially when the first post regards a symbolic argument.
And "logic lab". The original post regarded a symbolic argument. So what, you gonna sue a bunch of people for taking it in that very context? And, by the way, "the logic lab" is part of philosophy too, as formal logic and symbolic logic are regular undergraduate philosophy courses, and sometimes even set theory and mathematical logic too.
And it is not a "veneer".
You might choose to heap on more opprobrium against posters who are interested in the subject enough to strive to be correct about it, but you've not at all successfully impeached them, not even touched them.
Exactly. As cranks and people ignorant of the subject inject their confusions and ignorance, harbored in such forums on the Internet.
Quoting Hanover
All purely symbolic. Get outta here with this [paraphrase:] "Oh but the poor boy was just trying to get in a bit of common sense everyday logic; not about formal logic" stuff.
Folk who understand that the argument is valid yet not sound will make no such conclusion.
The argument presented in the OP is valid, and has the form MPP. Pointing out the error of thinking otherwise has more of disabuse than abuse.
When questions about formal logic come up, it is appropriate to check that the claims about formal logic are correct and coherent. And if one wishes to regard the questions in a context other than formal logic or other than the ordinary versions of formal logic, then of course that is fine, but it helps if the person makes clear that they are not mixing contexts, or if they are comparing contexts or even critiquing the ordinary versions of formal logic, then at least they should not misstate, misrepresent or mangle what actually happens in the formal logic, which is abusive.
I'd have thought it is the opposite. If death is equivalent to not-life it means no afterlife.
I'll just say that no opprobrium was intended. I too have gained, I believe, from my study of logic and mathematics, and I have found formal methods intensely interesting, so neither am I disparaging that as an interest.
Some people are interested in formal logic full-stop. Some people are only interested in formal logic as a help meet to argumentation and analysis. For the latter, results about logic ? the completeness theorems and such ? are not only of less interest, but less everyday use. All I was saying.
Nor was I accusing you or the original poster, or anyone in this thread, of engaging in sophistry, if that's what you thought. Maybe it seemed to you I was engaging in the current controversy on the other side, but I was not. You are, of course, correct about the formal question.
I was looking at the argument schema presented in the OP. If you imagine this as the formal representation of a substantive argument, you would have to have serious doubts about what was going on in that argument. This was the "veneer" of logic I was talking about. Any argument that could be formalized in the schema presented would instantiate an accepted form in a deeply questionable way. Hence "sophistry". That wasn't intended to refer to you, to your explanations, to anyone in this thread, but to a hypothetical argument that would fit the schema under discussion. ? You'll note that @TonesInDeepFreeze and I were trying to think of a genuine example of such an argument, and I'll now be getting back to that.
I hope my "position" is clearer now.
I figured this would be an interesting thread. This is the standard set piece where Banno and Tones think logic is arbitrary symbol manipulation and others think it has to do with correct reasoning, but this thread brings it out quickly.
For my money the question here is whether modus ponens is arbitrary or non-arbitrary. (Whether what is at stake is a mere matter of definition.)
Quoting Srap Tasmaner
Yep. :up:
Thanks.
No thanks.
Common sense wise, yes, but Plato has Socrates make an argument that relies on notions contrariety and us accepting death as the polar opposite of life (e.g. as darkness is the absence of light).
A lot of scholars think this argument is meant to be bad and to have this hole in it (i.e. that death is not a straightforward negation of life). I tend to agree. Socrates chains several arguments and only one is really good (more or less the same argument used against forms of reductionism to this day), and then suggestively breaks into an interlude where he tells his interlocutors that they shouldn't give up on reason if they happen to discover that arguments they once embraced turn out to be bad ones. I honestly don't love the philosophy of the dialogue as far as Plato goes, but the execution is brilliant.
"Argument is valid if and only if it would be contradictory for the conclusion to be false if all of the premises are true.[3] Validity does not require the truth of the premises, instead it merely necessitates that conclusion follows from the premises without violating the correctness of the logical form. If also the premises of a valid argument are proven true, this is said to be sound.[3]
https://en.m.wikipedia.org/wiki/Validity_(logic)#:~:text=An%20argument%20is%20valid%20if%20and%20only%20if%20it%20would,correctness%20of%20the%20logical%20form.
From the same wiki article:
"A Formula of a formal language is a valid formula if and only if it is true under every possible interpretation of the language. In propositional logic, they are tautologies."
So:
1.
A -> ~A
~ A
Therefore A (1,2 mp)
But
2.
A->~A
~A
Therefore ~ A (2)
Test 1 for validity: It is valid if it would "be contradictory for the conclusion to be false if all of the premises are true."
So, #1, could A be false if the premises true? Yes, see #2. Same premises, yet in #1 A is true, but in #2 A is false.
Test #2 for validity (which is really just a clearer restatement of #1): "A formula of a formal language is a valid formula if and only if it is true under every possible interpretation of the language."
Note "every possible interpretation of langauge."
Premise #1 is logically equivalent to ~A. That is, a possible interpretation of this syllogism:
~A
A
Therefore A.
Therefore ~A is also true.
This is not a valid argument.
not (P->Q)
Therefore, not-P
is not an obviously bad argument (bad argument though it is).
I would also note that the argument "A->not-A, Therefore not-A", though it is apparently a valid argument, does not make much sense in natural language; it would be like saying "if it is raining then it is not raining." Maybe someone could infer from that statement that it is not raining, but the statement seems more like a contradiction then a "valid" logical statement.
I think you're treating A -> ~A as if it's hypothetically true. They're just declaring it to be necessarily false.
I have never believed that logic is arbitrary symbol manipulation. I have never posted that logic is arbitrary symbol manipulation. I have never posted anything that implies that I believe that logic is arbitrary symbol manipulation.
1
No, I get the distinction between a deductive conditional premise, and a linguustic counterfactual. I'm just engaging in the pedantry of determining whether the OP satisfies a hyper analyzed definition of "valid."
As @Banno notes, validity is determined by asking if the conclusion flows from the premises, and so he argues under mp, it does, so it is valid.
The wiki cite adds criteria, namely (1) that the negation of the conclusion cannot also flow from the premises for validity and (2) the premises under any formulation must also reach the same conclusion.
The OP falls under those criteria because: (1) both A and ~A can be derived from the premises, and (2) when Premise 1 is changed from a conditional format to a disjunctive one, it reduces simply to ~A, clearly contradicting the second premise A, and further violating criterion 1 that prohibits the negation and assertion to consistently flow from the same premises.
This is to say, if I were reviewing a contract, and it said "you get $1,000,000 if the OP is a valid syllogism," I'm saying no if I'm the guy who has to pay. Does the other side have a colorable argument? Maybe, but it must argue validity despite contradiction and accept the absudity that follows.
I do think @Benkei's comment regarding the necessity of acknowledging the LNC as foundational is correct.
How are you getting A as a conclusion?
Quoting Hanover
That is not a valid argument and it is not modus ponens.
Quoting Hanover
Quoting Hanover
The arguments themselves don't declare the truth or falsehood of A.
Quoting Hanover
#2 is not a restatement of #1.
Quoting Hanover
A -> ~A is equivalent with ~A.
Quoting Hanover
Strictly speaking an argument has only one conclusion. You have two conclusions there. But we can use conjunction:
~A
A
therefore A & ~A
That is a valid argument.
Can you provide a citation for that criterion of validity? I did not find it in the wiki article.
Also, Hanover, thanks for articulating an argument against validity as I was not sure how to do so.
But what if you say the first premise is necessarily false? It can't be true. Then what do you get?
I appreciate that you say that now.
Yet:
Quoting Srap Tasmaner
I was the one who remarked that the argument whose conclusion is ~A is modus ponens. I'll take your word for it that you didn't mean that my remark was non-cooperative and abusive, but I don't see how it would not be natural to take you as first claiming that my remarks were non-cooperative and abusive.
Instead you could have first said what you say now: That some hypothetical argument, one not given in this thread, is abusive.
Also:
Quoting Srap Tasmaner
The disjunction argument was given by another poster in this thread. I'll take your word for it that you didn't mean that his remarks were abusive, eristic, sophistical and non-cooperative, but I don't know how it would not be natural to first take you as claiming that his argument is abusive, eristic, sophistical and non-cooperative.
If a premise is necessarily false, then the argument is valid.
But with validity, aren't we looking at what happens when all the premises are true? If a premise is necessarily false, can we still look at the argument in terms of validity?
'valid' has three senses:
(1) an argument is valid if and only if there are no interpretations in which all of the premises are true and the conclusion is false
(2) a formula is valid if and only if there is no interpretation and assignment for the free variables in which all of the premises are satisfied and the conclusion is not satisfied
(3) a sentence is valid if and only if there is no interpretation in which all of the premises are true and the conclusion is false
(3) reduces to a special case of (2).
~A
A
therefore A & ~A
valid
~A
A
therefore ~(A & ~A)
Of course, we recognize that that is problematic to many people regarding everyday reasoning and, more pertinently here, in different philosophical points of view.
We are not restricted to looking only at the interpretations in which all of the premises are true.
If there is no interpretation in which all of the premises are true and the conclusion is false, then the argument is valid. If there is an interpretation in which all of the premises are true and the conclusion is false, then the argument is invalid.
So we know the first premise is necessarily false. That means the conclusion has to be false for validity. Is the conclusion false?
No, he's claiming that A -> ~A is necessarily false, and we are pointing out that it is true when A is false, so it is not necessarily false.
I think the first premise is necessarily false in propositional logic.
(1) The first premise in that argument is not necessarily false.
(2) I don't know what 'conclusion is false for validity' means.
(3) The conclusion is true in some interpretations and false in others.
You seem not to grasp the meanings, in the context of ordinary formal logic, of 'true', 'false', 'valid' and 'invalid'.
The first premise is:
~A
That is not necessarily false.
Is it not? It's expressing a contradiction. Contradictions are necessarily false, right?
~A is a negation but it is not a contradiction.
But wait, which argument are we talking about?
(1)
~A
A
therefore A & ~A
(2)
~A
A
therefore ~(A & ~A)
(3)
A -> ~A
A
therefore ~A
In all three case, no premise is itself a contradiction. But in all three cases, the set of both premises together is contradictory (is inconsistent).
Isn't the first premise: If A, then not-A? That's what it looks like
I see, the earlier argument.
See my edit that I composed while you were posting.
A -> ~A is not contradictory.
A formula is contradictory if and only if the formula proves a contradiction. A contradiction is a formula of the form P & ~ P (or, sometimes we say, a pair of formulas of the form {P ~P}).
Using the propositional calculus, you cannot derive a contradiction from A -> ~A.
I think Hanover was talking about the argument in the OP. It can't be valid because the first premise is necessarily false, right?
Agreed, a natural reading, but my target was really someone who might present an argument in the OP's schema, as a perfectly respectable modus ponens. It's MP alright, but it's a degenerate case.
Similarly for the disjunction. My point was that disjunctions that amount to "heads I win, tails you lose" are disjunctive in form only, and we expect something more substantive.
In short, no attack was intended on you or any other poster, but only on the illicit use someone might make of legitimate argument form. To the extent that I was offering criticism, it was to say that we are not helpless when confronted with correct inference in form only, and can choose to block such deviant uses if we like.
And we can leave formal logic alone, as a study in its own right, but not import it wholesale when all we really need is the convenience of schematizing arguments.
A related example would be various attempts to deal with what many people find counterintuitive about the material conditional. There are several ways to block troublesome cases.
Hence my casual suggestion that we have very little practical use for "If grass is green then grass is green" or "If grass is green then grass is not green."
No, not right. The first premise is not necessarily false.
It's been correctly pointed out over and over and over, by different posters in this thread, that
A -> ~A
is true when A is false.
Truth tables have even been adduced.
Please look at those truth tables.
That's okay for me, as long as I take 'degenerate' in a non-pejorative sense as often in mathematics.
Quoting Srap Tasmaner
Of course, formal logic, or at least a particular formal logic, does not always apply in everyday and even in all philosophical contexts.
Quoting Srap Tasmaner
I would need to dig up documentation, but, I tend to think that P -> P does have importance in Boolean logic used along the way in switching theory, computation, etc. Just for starters, we use the Boolean 1-place function whose value is always 'true' (or '1') and it is definable propositionally as P -> P. Logic is a vast field of study, including all kinds of formal and informal contexts. I would not so sweepingly declare certain formulations otiose merely because one is not personally aware of its uses.
"A conditional statement is false if hypothesis is true and the conclusion is false.".
here
And if A is true, we can't have not-A as the conclusion, so the conditional in premise 1 is false.
How would you be warranted to examine what happens when A is false?
I might have mistyped at some point.
The OP:
1. A->~A
2. A
3. Therefore ~A (1,2 mp)
A cab also be concluded from the second premise.
A (2)
I can also continue from the conclusion:
4 ~A v A (3, disjunctive introduction)
5. ~ (~A) (2, double negation)
6. Therfore A.
All grass is green
All grass is not green
Cows can bark.
You need to remedy your misunderstandings of this. I suggest starting with the first chapter of a good textbook in formal logic.
Yes, a conditional is false in all and only those interpretations in which the antecedent is true and the consequent is false.
And, yes , if in an interpretation, A is true, then A -> ~A is false.
But if, in an interpretation, A is false, then A -> ~A is true.
And a sentence is necessarily false if and only if it is false in all interpretations.
But A -> ~A is not false in all interpretations. So A -> ~A is not necessarily false.
Please read each of those lines again now carefully. And look at the truth table.
Yea, you're right.
Thank you.
Yes, so?
In other words, you can interpret that cows can bark if you want to.
Wait a minute. If A is false, then the first premise is:
If not-A, then not (not-A)
You can't change one of the A's to false and not the other one. If A is false, they both have to be false.
I'm not swapping any premises, and I'm not making a reinterpretation.
Let G stand for "grass is green".
Let C stand for "cows bark".
G
~G
therefore C
There are no interpretations in which both the premises G and ~G are true. Perforce, there are no interpretations in which both the premises are true and the conclusion is false. So the argument is valid.
I think you did swap out the first premise when you made the first A false, but not the second one. Is that wrong?
I didn't change any premises. And I didn't make anything true or false. And there is no "first A" and "second A". There is only one A. I merely pointed out that A -> ~A is true in the interpretation in which A is false.
I suspect you don't know what is meant by 'interpretation'.
Ok. So with a false premise, the conditional is true by default.
That means the first premise is actually not-A, right?
Wait, no, the first premise doesn't say anything at all if A is false. It's trivially true.
So the conclusion to the argument should be the 2nd premise. It should be A.
I get that you're frustrated. Thanks for hanging in there. If the hypothetical in the first premise is false, isn't the first premise trivially true? It doesn't say anything in that case.
You're mixing up 'premise' and 'antecedent'.
If the antecedent is false then the conditional is true.
As to premises, let's not mixup two things, and which argument are you talking about now?
(1)
A -> ~A.
That's not an argument and it has no premises. It is a formula that is true in an interpretation in which A is false, and it is false in an interpretation in which A is true.
(2)
~A
A
therefore A & ~A
That's an argument. It is valid since there are no interpretations in which all the premises are true and the conclusion is false.
(3)
A -> ~A
A
therefore ~A
That's an argument. It is valid since there are no interpretations in which all the premises are true and the conclusion is false.
It's not a matter of frustration. Rather, since you want to know about this, my sincere helpful suggestion is for you to get a book that explains this stuff methodically, step by step, starting at page 1.
If the hypothetical of a conditional is false, the conditional is trivially true. Is this correct?
Which argument?
A conditional sometimes called a 'hypothetical'. Sometimes the antecedent is called 'the hypothesis'.
To avoid confusion between 'hypothetical' and 'hypothesis' let's stick with this terminology:
A conditional has an antecedent and a consequent.
For example:
P -> Q
is a conditional in which P is the antecedent and Q is the consequent.
If Jack is good then Jack reads Faulkner
is a conditional in which "Jack is good" is the antecedent and "Jack reads Faulker" is the consequent.
Ok. If the antecedent of a conditional is false, the conditional is vacuously true. Right?
If, in an interpretation, the antecedent is false, then, in that interpretation, the conditional is true.
In more lax formulation:
If the antecedent is true then the conditional is false.
But with that lax formulation, do not forget that it is still implicit that truth and falsehood are relative to interpretations. That is, look at the truth table.
Vacuously true. Trivially true. Correct?
The term 'vacuously true' is used that way.
If the antecedent is false, the conditional is trivially true, right?
If I gave you a quote from a respected authority advising that if the antecedent of a conditional is false, the conditional is trivially true, would you believe it?
Do you intend for this to be a Socratic interview?
No. It's that if A is false, the first premise is trivially true.
So the argument is one in which the first premise doesn't say anything. The argument would be:
1. Trivial truth
2. A.
Conclusion: not-A.
That's not valid.
That is wrong. You are plainly misusing the terminology.
Considering different interpretations doesn't change formulas.
And I'm not making "one A" false and not "the other A".
There is no sense of "another A" and thus no interpretation in which A is false but "another A" is true.
Again, you need to know what 'interpretation' means.
I think you need to know what "trivially true" means.
Per the definition of "valid":
Quoting Hanover
Assuming all premises in the OP true, the conclusion of not A is shown to be false because a valid conclusion of A was shown.
Yes. In the only interpretation where both premises are true, there's no way to conclude not-A
A -> ~A
says
If A then it is not the case that A
That is equivalent with
it is not the case that A
In an interpretation in which A is false, "it is not the case that A" is true; in an interpretation in which A is true, "it is not the case that A" is false.
Quoting frank
Yes, that that is not a valid argument. But when you replace A -> ~A by "trivial truth" the new argument is not equivalent with the original.
(1) A -> ~A is not a truth simpliciter nor a falsehood simpliciter. As has been explained to you about 15 times today, it is true in an interpretation in which A is false, and it is false in an interpretation in which A is true.
(2) What you may say is trivial is not the truth of A -> ~A, since it is true or false depending on the interpretation of A, but rather that A -> ~A is true in any interpretation in which A is false, which, if you like, you may choose to call trivial.
(3) 'trivially true' is a term logicians and mathematicians usually use to describe a statement such that the truth of the statement is very obvious. It's not a technical term in a context such as this. So the judgement that a particular statement is trivially true does not count toward the validity of an argument. Validity is determined by the set of all interpretations, no matter whether one considered certain of the statements to be trivially true or not.
(4) A -> ~A has a deductive role in the argument. It's not the same argument if you take it out and replace it with "trivial truth".
(5) We actually do have a defined* constant called 't' such that t is interpreted as true in all interpretations (though 'trivial' is not part of this). In that case there is this argument:
t
A
therefore ~A
That argument is invalid.
Indeed, adding t as a premise to any argument has no effect on validity:
Let G be any set of premises. Let P be a statement. Let the argument be
(5a) From G, infer P.
Now add t. So we have:
(b) From Gu{t}, infer P.
Then (5a) is valid if and only if (5b) is valid.
* Usually, the definition is of the form, for some formula P, P -> P.
(6) So, even if we supposed that the premise A -> ~A were true in every interpretation, we can't just replace it with "trivial truth". That is, if we suppose that A -> ~A has as some property (such as being true, or trivially true, or having four symbols, or being a conditional, or having only one sentence letter, etc.) you can't just replace the formula with a mention of an adjective that applies to it. That would be blatantly fallacious.
Sure. I would encourage you to write out in English the only case where both premises are true, and see if you think not-A makes sense as the conclusion. If it does, great. Bon Voyage.
You think incorrectly.
I well know how logicians and mathematicians use the verbiage 'trivially true' and 'follows trivially' for statements and arguments respectively. Usually it means that the statement is very obviously true or the inference is very obviously valid. In logic and mathematics it doesn't mean that the statement or the argument is otiose (though, sometimes it does mean that the vacuous case needn't be considered since the generalization being proved is not claimed to include the vacuous case).
But 'true' has formal import while 'trivial' does not. There's nothing wrong with saying 'trivially true' or 'follows trivially' but it doesn't count toward evaluation of an argument.
That is, there is no definition:
P is trivially true if and only if [definens]
It's a personal choice for any author whether to use 'trivial' or not as part of the informal prose in which mathematics is often written.
And if we choose to say that "condition is true if the antecedent as false" is trivial, then we can say that about all the connectives:
It's trivially true that a conjunction is true if both conjuncts are true.
It's trivially true that a disjunction is true if at least one of the disjuncts is true.
It's trivially true that a negation is true if it is the negation of a false statement.
It's trivially true that a biconditional is true if either the left and right are both true or both false.
/
Moreover, see my previous post that explains in detail the fallaciousness and irrelevance of your argument.
Yes, that was my meaning, as with the boat example. I think, though, we can allow a somewhat negative connotation because reliance in argumentation on degenerate cases is often inadvertent or deceptive. "There are a number of people voting for me for President on Tuesday [and that number happens to be 0]."
Quoting TonesInDeepFreeze
Absolutely. I mentioned automated reasoning projects when you need classical logic in full generality with P ? P and all. Of course.
But that is not the only sort of application of logic, and as I noted reducing someone's argument to P ? P is pointing out that they are "begging the question," generally considered a fatal problem for an argument. That conditional is legitimate in form, and is generally a theorem, but it is fatal if relied on to make a substantive point or demonstrate a claim. It will only happen inadvertently ? in which case, a good-faith discussant will admit their error ? or with an intent to mislead by sophistry.
Quoting TonesInDeepFreeze
A fair point. Let's say, I'm only pointing out forms, or uses of particular schemata, that might raise our suspicions. "Heads I win, tails you lose" may in some cases demonstrate that I'm necessarily right and you necessarily wrong. Hurray! But in some cases it might amount to me stacking the deck against you.
Fundamentally, all we're talking about in this case is arguing from a set a premises which are inconsistent ? in fact, here, necessarily inconsistent. As I keep saying, that's either inadvertent or deceptive. Very often on this forum people attempt to show that a set of premises is inconsistent precisely by making correct inferences that make the inconsistency obvious. But people arguing from inconsistent premises often make inferences that, while in themselves correct, continue to hide their inconsistency. (Sometimes this is because not all premises are explicitly stated, and the inconsistency is in what is "assumed".) In such cases, it is not uncommon for people to insist on the correctness of their inferences. But it's not validity we usually disagree over, but soundness, and inconsistent premises make valid inferences unsound.
None of this news to you, I'm sure.
The argument:
A -> ~A
A
therefore ~A
valid
Another argument:
A -> ~A
A
therefore A
valid
The fact that the premises are inconsistent doesn't vitiate that the argument is valid. Actually the fact that the premises are inconsistent entails that the argument is valid.
(2) A conclusion itself is valid if and only if it is true in all interpretations. An argument is valid if and only if there are no interpretations in which the premises are all true and the conclusion is false.
A -> ~A
A
therefore ~A
There is no interpretation in which both the premises are true.
I don't know why you continue to ignore the definitions and explanations given you.
PS. 'vacuous' is more informative for the conditional than 'trivial', since 'vacuous' is specfic while 'trivial' is not. That is, vacuousness is a certain kind of triviality.
Another twist. A conditional may turn out to be vacuously true, but it might be quite nontrivial to prove that the antecedent is false.
If the antecedent in the conditional is false, then the first premise is true. Now say the second premise is true. Then the conclusion does not follow.
If you insult me one more time, we're done. I'm satisfied with ending this discussion.
The premises are consistent and the conclusions are not.
The conclusion is not true under all interpretations. Sometimes it's A and sometimes it's not A.
Correct
Quoting Srap Tasmaner
In cases of inconsistent premises what happens is that the person arguing arbitrarily makes use of some premises while conveniently ignoring others. For example:
Or a reductio, which has been shown elsewhere to falsify one side of a contradiction rather than the other side for no necessary reason. Is the argument above or a reductio valid? Are they sound? Neither answer is obvious. We can't just say, "Ah, it's cut and dry. The argument is valid but unsound."
Similarly, the arguments over the OP turn on the nature of modus ponens, which is not a simple question. If modus ponens is just a matter of symbol manipulation then the OP is valid. If modus ponens is more than that then the OP is probably not even valid.
"Trivial" has a clear meaning in analytical philosophy.
Still hunting for a solid example, but in the meantime there's
The sequence from here is most likely
And then the argument shifts from whether I got more points than you (or whatever) to whether the rules were followed.
But "the rules" can be surprising. Casinos universally have a rule against "card counting," which amounts to a rule against being too good at playing cards.
Sometimes among children using greater skill or knowledge is treated as cheating. It's clear they have an intuition about what fair competition is, but they mistakenly treat every advantage as unfair, or every unfair advantage as cheating.
What's happening here, broadly, is that the competition continues "by other means." The losing party, in one sense, grants that they lost, but continues in the competitive spirit, which means they have to shift ground from whether they "officially" or "technically" lost to whether that was a "real" loss, or whether there had a been a "real" competition in the first place.
So we have two versions here:
Right: the conclusion must flow from the premises. The premises must provide the aitia for the conclusion. A contradiction is not an aitia.
As I argued at length in Flannel's thread, contradictions and inconsistencies are not meaningful. To pretend they are meaningful is to become lost in the logical abyss. If you feed the "argument" of the OP into the propositional logic machine, the answer is neither "invalid" or "unsound." It is, "Does not compute."
If under #1, I assume A (the negation of the conclusion) and I prove A from that (as is shown under #2), then I've proven invalidity by negation because I've shown my negation is true.
Feynman had a party trick he used to do, I think in grad school. He could tell whether any mathematical conjecture was true.
What he would do is imagine the conditions concretely, in his mind. Like start with a tennis ball to represent some object; then a condition would be added, and he'd need some explanation of what it means, to know whether to paint the entire ball purple, or half, or maybe add spots or something. He would follow the explanations making changes to his imaginary object and then when asked, is it X?, he could check and see.
But the trick is this: when he got one wrong and the math students explained why, he would say, "Oh, then it's trivial!" which to the mathematicians was always completely satisfying.
I'm restating in my own manner some of the points you've made.
Quoting Srap Tasmaner
I agree that a pedantically correct application of notions in formal logic could be abusive sophistry in a context in which they are not understood But, at least at the moment, an actual example from public discourse doesn't leap to my mind. On the contrary, for example, if you were interviewed by a news outlet about the sunken boat and you tried to pull the stunt you mentioned, you would be pilloried. Or if you tried to make a vacuous argument among your non-logician friends, the best you'd get would be "Huh?" Meanwhile, of course, contentious, especially polemical, public discourse is rife with hideous, obnoxious, downright sneaky and pernicious use of all kinds of informal fallacies.
Quoting Srap Tasmaner
Question begging happens a lot. But, again, I can't think of an instance in public discourse in which the speaker appealed to P -> P as a tautology in order to convince anyone about anything.
As to complaints about formal logic, people who don't know about the subject often miss the point about such things as axioms proving themselves. In formal logic it is completely open that that one doesn't assert that a non-logical P holds in and of itself without respect to being either an axiom or derivable from the axioms. That is, when we say "Put P on the line as P is an axiom", we don't hide what we are doing. It's a kind of "benign" question begging. On the other hand, informal question begging is malign when, as is usual, it tries to hide the nature of the inference.
Quoting Srap Tasmaner
It is such a case. It is also a case of modus ponens. I've mentioned why I pointed that out.
Quoting Srap Tasmaner
I don't know what that means.
Quoting Srap Tasmaner
That is very very common in public discourse. Maybe it's the norm! It's maddening, frightening and ultimately depressing.
I'm not sure what post you are responding to, but there is of course a substantive issue here. It is the difference between rules-as-arbitrary and rules-as-substantive, and logic-as-arbitrary and logic-as-substantive. There are true charges of cheating and false charges of cheating, and it's not always easy to disentangle the two.
The move is always to a meta-level. What is the game? What is the competition? What is logic? Our world has a remarkable tendency to try to avoid those questions altogether, usually for despair of finding an answer.
I realize a lot of people like this claim, but I don't think it is right. You are confusing consequence or inference with identity.
Even on a very formal reading, this is invalid. "A?A" and "A, ?A" are not the same statement. Even so, there is a dispute here about what '?' and '?' mean. In that way it is the same problem of trying to hold to truth-functionality (turtles) all "the way down."
Only arguments are valid, and "A, therefore A," is not an argument. Argument, at the very least, involves rational movement.
-
The core of truth in @Hanover's variegated posts is that "A?A" is not a conditional and "A, ?A" is not an argument. If you admit such things to the bar of conditionals and arguments, you are fudging the meaning of "conditional" and "argument." You are prioritizing truth-functional process over logical telos.
It's hardly an insult to say that I don't understand why you are not minding the definitions and explanations. I've patiently given you information and explanations, repeated as has been needed. I am sincere when I wonder why you, in one post, seem to understand, but then blow right past again. As to whether you choose to reply to me, of course, that is entirely your choice
Here we go again:
If, in an interpretation, the antecedent is false, then, in that interpretation, the first is premise true, and the second premise is false not true. In that interpretation, the second premise is false.
You need to learn what an interpretation is. That's not an insult. It is good advice, given lagniappe in addition to all the information and explanations I'm giving you.
To deny A flows from the premises makes the curious argument that a premise has been eliminated by other premises.
In any event, premise 1 is reducible to ~A, so when you couple that with the second premise of A, you then can claim "A and ~A," allowing you to prove whatever you want.
Premise A & ~ A
Inferences:
:
A
~ A
A v C (cows bark)
~A
Therefore C
and so on and on
Quoting Leontiskos
-
Ergo:
These are all based on the same error, that of a non-inference inference.
There is too little knowledge of Aristotle on these forums, and that is why we don't seem to understand what arguments are. :grin:
I'm interested in what that definition is.
Meanwhile, the context here is examination of a particular formal argument. In that context, you fallaciously used the notion of triviality, as I detailed for you.
There are no interpretations in which both premises are true.
Sure. If a statement is trivially true, it's not informative. For instance, a tautology is trivially true. The T-sentence rule is trivially true.
Under what definition of "valid" is the argument in the OP valid? I'm not being Socratic, I'm just asking.
I agree with all that. The toy examples we're dealing with here are too transparent for anyone to get away with much.
1 A -> ~A ... premise
2 A ... premise
3 ~A ... {1 2}
4 A & ~A {2 3}
So the premises are not consistent.
A -> ~A
A
therefore ~A
Let A be false. That is an interpretation in which the conclusion is true. So the conclusion is a consistent statement, or put as a set: {~A} is consistent.
For example:
P1. A -> ~A
P2. . A
1. ~A (1,2 m.p.)
2. ~ A v A ( 1 and disjunctive introduction)
Therefore:. A (P2, 2 negation of ~ A)
Note I've not just reasserted P2 in my conclusion, but I've logically deduced that since not A could not be true based upon A being a given premise, by elimination, A must be true.
I'm sure there are more convoluted ways to go about it, but does that satisfy your objection?
And it's a good thing that it is valid because we often can reason from necessarily false conclusions in valid arguments to identifying false premises. This example is simple, but sometimes self-refutation is not simple.
None, or .
Just trying to think of real world examples of a formula like "A ? ~A", likely dressed up enough to be hard to spot. Excluding reductio, where the intent is to derive this form. What I want is an example where this conditional is actually false, but is relied upon as a sneaky way of just asserting ~A.
I suppose accusations of hypocrisy are nearby. "Your anti-racism is itself a form of racism." "Your anti-capitalism materially benefits you." "Your piety is actually vanity." Generalize those and instead of saying, hey here's a case where the claim is A but it's really ~A, you say, every A turns out to be ~A. Now it's a rule.
Still thinking.
Quoting Leontiskos
With good reason, as you well know.
I said nothing about identity.
Certain posters are disputing the validity of an argument.
I am only remarking about what happens to be the case in ordinary formal logic.
It is fine if posters wish to provide a logic and definition of 'validity' in which the argument is invalid. And fine also to point out that, in many contexts, ordinary formal logic is not used or could be misleading. That is not in dispute.
But when critiquing ordinary formal logic, one should at least not be confused and self-misinformed as to how it does go.
In ordinary formal logic:
A -> ~A
A
therefore ~A
is valid
Again, that does not dispute that that may be quite counter-intuitive to many people, nor that there are many other formal and informal logics, notions of validity, understanding of the conditional, and all kinds of other everyday and academic contexts.
The key word there is "at first glance". Upon consideration, it is seen that the first premise is not contradictory.
You can't be serious. I've given the definition probably at least fifteen times already.
An argument is valid if and only if there are no interpretations in which all the premises are true and the conclusion is false.
Back to my truth by negation maneuver then.
The opposite of (A & ~ A) is (A v ~ A), which is a tautology
So, if I can prove from the OP that (A v ~ A) flows, then the argument is invalid because I would have shown F is T.
P1. A -> ~ A
P2. A
1. A&~A (1,2)
2. ~A (1)
3. ~A v A ( 2 disjunctive introduction)
Still not valid, considering the contradiction allows me to prove anything I want, even that T is F.
That argument has been addressed extensively in another thread.
In ordinary formal logic, the argument forms mentioned are valid. "ignoring premises" has nothing to do with it. Indeed, the logic is monotonic. And this pertains even to certain natural language situations.
And my saying that does not at all entail the decidedly and outrageously false representation that I take logic to be just symbol manipulation or that my view of logic is confined to truth-functionality.
No, I am not doing that.
The original argument is one thing. I don't substitute anything in it.
But you claimed that the premises are consistent.
So I gave a proof that they are inconsistent.
Quoting Hanover
(1) Whatever you mean by "flows", I say "is entailed by", or "follows from" or is "implied by".
(2) In classical logic, A v ~A follows from any set of premises whatsoever. Proving A v ~A from a set of premises does not in and of itself tell us anything about the premises nor does it show that a particular argument is invalid.
I'll do it for you:
(3) A -> ~A
A
therefore A v ~A
valid
Indeed, we can explain its validity in at least two different ways: Valid since there is no interpretation in which all the premises are true. Valid since there is no interpretation in which the conclusion is false.
meanwhile, still
(4) A -> ~A
A
therefore ~A
valid
Valid in at least two ways: Valid since there is no interpretation in which all the premises are true. Valid since it is modus ponens, which is proven to be a valid form.
I'll need some more explanation.
I can better comment on that if you provide the specific arguments you have in mind.
The way to prove the invalidity of an argument is to show that there is an interpretation in which all the premises are true but the conclusion is false.
The premises are
A -> ~A
A
The conclusion is
~A
So provide an interpretation in which both A -> ~A and A are true, and ~A is false.
Hint: There is no such interpretation, since there is no interpretation in which both A -> ~ A and A are true. I'll do it for you:
There are only two interpretations:
(M1) A maps to true (i.e."A is true")
(M2) A maps to false (i.e. "A is false")
In M1, A -> ~A is false
In M2, A is false
So there is no interpretation in which both A -> ~A and A are true.
Your "disjunctive syllogism" is different than my A?A, so in that sense, sure. You are effectively saying that A flows or follows from the contradiction, not from itself.
So a second objection would be that nothing flows or follows from a contradiction (which is the flip side of saying that everything flows or follows from a contradiction).
It would be nice to have a specification of your logic so that other people could determine for themselves what obtains and does not obtain in it, without having to take you as the arbiter in each particular case.
Oh. So then any argument that has no true premises is valid. That's weird.
1. A -> ~A ... premise
2. A ... premise
3. A & ~A {1 2}
4. ~A {1}
5. ~A v A {}
That is correct. Each inference is valid.
No invalidity has been shown.
Isn't that reductio?
I would say that, like argument, contradiction also requires a kind of middle term, and is therefore never direct. For example:
Quoting Leontiskos
People can only make this inference because they do not see that they are being inconsistent. When there is neglect we hold them responsible for the mistake.
So A?~A is never a self-conscious premise.
You can't be serious. For the umpteenth time:
An argument is valid if and only if there is no interpretation in which all the premises are true and the conclusion is false.
In this case, there is no interpretation in which all the premises are true. Perforce, there is no interpretation in which all the premises are true and the conclusion is false. So the argument is valid.
I can't put it more starkly than that.
Yes. I edited that post. It's just weird that any argument that can't have all true premises is going to be valid.
I very much appreciate that it may be quite counter-intuitive to many people.
What you've done is imported the artificial truth-functionality of the material conditional into the consequence relation itself. You have contradicted 's "flows from." You are effectively saying,
I imagine you finally had to retire to the insane asylum. Enjoy the rocking chair.
You can retire to the blazes.
Sounds uncomfortable.
As I said, in this particular regard, I'm merely applying the definitions of ordinary formal logic. As I said, I don't claim that those definitions have dominion over all other contexts.
We could say with that if the conclusion flows from the premises then the argument is valid.
1. P?Q
2. P
3. ? Q
4. A?~A
5. A
6. ? B
Now one could say that (3) flows from (1) and (2); and that (6) flows from (4) and (5). But this latter use of "flows from" is very different from the former. 's contention that they are the same use is not "merely applying the definitions of ordinary formal logic."
Quoting TonesInDeepFreeze
Ordinary formal logic does not define the consequence relation as identical to the material conditional.
What is the definition of 'flows'?
Quoting Leontiskos
I didn't say they are the same. They are very different. This the second time, in this thread alone, that you've put words in my mouth.
Validity in propositional logic involves a relativization of truth-values with respect to inference-relations. Inference-relations are held steady, and if the truth-values cash out given the stable inference-relations, then we call it "valid." The inference relations are conceived as meaning-stable, and the variables are conceived as meaning-variable (i.e. truth-variable). But in this case what is at stake is the meaning and stability of the inference-relations themselves. The contentious move is to claim that the consequence-relation involved in the OP is the stable, familiar consequence relation of modus ponens. It isn't. That's that place to start.
To claim that it is involves:
Quoting Leontiskos
Put differently, the notion of validity assumes a truth-functional context where truth and form are entirely separable. Yet when we think deeply about inferences themselves, such as modus ponens, truth and form turn out to be less separable than we initially thought. When we stop merely stipulating our inferences and ask whether they actually hold in truth, things become more complicated.
Probably unfortunately relevant to we Americans' immanent election (and the last one, and 2000... a pattern emerges). However, I think this would be an issue of unclear/disputed termsequivocity re "winning"not a case of A?~A.
Interestingly, it's another example involving stipulated rules, just as with conventional self-reference. In the thread on logical nihilism I was thinking out loud about the older distinction between formal and material logic. This goes back to Aristotle, who discusses pure form in the Prior Analytics and the "matter" of discourse in the Posterior Analytics. The latter discusses the ways in which subject matter shapes discourse.
I think it is at least possible that one might be able to ground the selection of different consequence relationships in the relevant subject matter in a way that is rigorous, preserving the intuitions of both pluralists and monists.
John Poinsot (John of St. Thomas) and later CS Peirce (who took a lot from the Scholastics) could lay the groundwork for this with their well developed theory of signs (including attention to the unique aspects of stipulated signs and sign systems). The sign relation is irreducibly triadic. There is always the object signified joined to an interpretant by a sign vehicle. Yet in cases of self-reference in stipulated signs systems the object [I]is[/I] the sign vehicle.
Obviously, this isn't true in every sense. When we read "this sentence is false," there is a sense in which the paper or screen is the object, light acts as the sign vehicle, and we are the interpretant. Yet in the universe of the stipulated system, taken by itself, we have collapsed the necessarily triadic relationship into a dyadic one. The result, apparent "true contradictions."
And this would also bear out some of the intuitions of the post-modern semiotics that grew out of Sausser, which collapses the triadic relationship, while at the same time allowing us to at least plausibly overcome their more radical and destructive (destructive to notions of truth and meaning) theses by demonstrating how these are limited to a specific area of discourse and not all sign relations. Maybe.
Surely a hard sell, since "freeing the sign" has been bound up in notions of human freedom and flourishing in that tradition.
.
No, with a false antecedent the conditional is true, sometimes described as 'vacuously true'. It's the conditional that is deemed true when the antecedent is false.
Quoting Benkei
Yes, A -> ~A and A together are contradictory.
All he had to do is say that there aren't any cases where both premises are true, therefore it's valid.
I said it over and over and over for you.
All you had to do is read the replies given you. And that's hardly the only point I explained for you.
Yea, well...
Well, while I think Srap has a good point about our being able to live without A?~A in most situations, I think it is important that statements like "nothing is true," are able to entail their own negationthat logic captures how these claims refute themselves.
:up: :100:
What about my earlier example:
Quoting Janus
Not quite propositional, though, using modus ponens on a mass noun.
Still sad that it remains that high.
I am not sure about this one. The person is not arguing that A is actually ~A. Presumably, they believe real piety exists, just that this person doesn't possess it. They are arguing that what the person claims is A is actually just B. Perhaps, "anti-racism is racist," is closer to the mark, but again, I think this is still more of the same, a claim that what is presented under the term A is actually B. Presumably the person who earnestly makes such claims normally believes that one can be actually anti-racist without being racist (normally by being "colorblind.")
And on second thought, about the first use case, I think that often, when people argue that the other party is accidentally implying the falsehood of their own position, the issue will also be unclear terms. Not always, sometimes people do refute themselves. But such arguments might not settle the issue even if both sides are acting in good faith, not because one party rejects the form, but rather the content.
On a related note, although not the case here, I think a lot of the "gotcha" puzzles that involve presenting good formal arguments alongside what appear to be faithful natural language translations of them, which are nonetheless either clearly wrong or at least not obviously right, involve equivocation. I think these are particularly disarming because, at least in my experience, the basics of form is taught while ignoring the possibility of vagueness, which is a problem because arguments can fail in three ways, invalid form, false premises, or unclear terms.
I didn't really get a straightforward introduction to this risk until being led to the ol' "three acts of the mind," in historical treatments of logic curricula in the past. Although perhaps my experience is not typical.
Edit: Just for an obvious example:
Everything that runs is an animal.
My refrigerator is running.
Therefore, my refrigerator is an animal.
Works great formally if you're allowed to us "R" for "that which runs" in both premises.
Perhaps I misunderstood you. I had taken "it" and "the inference" to be the argument in the OP. Hence it appeared you were saying the argument in the OP was invalid.
A thought I have is sarcasm, but in the context of asserting a falsehood mistakenly.
So I can sarcastically say "George is going to open the store tomorrow" to mean that George is usually late and we are the ones who open the store on the regular. But if George opens the store tomorrow then the conditional was false because I asserted A to imply not-A, but in fact A is true so the conditional is false.
But does logic really capture how these claims refute themselves? I don't think so. It merely defines a formal notion of contradiction and shows that a contradiction has occurred. The how/why question is beyond the logic (as is the reductio-remedy), and I believe you yourself pointed earlier to the logical simplification of 'contradiction' (i.e. an all-false truth table).
Quoting Count Timothy von Icarus
Why would it be a good thing? It is good that we can reason from non-necessarily false conclusions in valid arguments to identifying false premises. An argument from a necessarily false conclusion is a reductio, and the question of whether an absurdity is valid is part of the very question at hand.
What is the conditional?
/
Maybe there could be an intensional logic with a defined irony operator 'i'.
s is a speaker
P is a statement
B(s P) iff s believes P
T(s P) iff s states P
Df. i(s P) <-> (T(s P) & B(s ~P))
If I understand your point, I agree with it, and I think it is incisive and apropos.
Let Rx mean x runs (where 'runs' includes both 'moving quickly on feet' and 'operates')
Let Mx mean x is an animal
Let t mean the refrigerator
1 Ax(Rx -> Mx) ... false premise
2 Rt ... true premise
therefore Mt ... {1 2} ... false conclusion
valid but unsound
A = "George is going to open the store tomorrow"
So, by substitution:
George is going to open the store tomorrow implies George is not going to open the store tomorrow.
If it turns out, extensionally at least, that George opens the store tomorrow then the implication is false -- and I don't think that sarcasm means to invoke material implication, but this seems an example of everyday communication which material implication seems to capture. George opens the store tomorrow, so tho I state one and believe another it turns out that my belief is false and the assertion true (attempting to use your intensional definition here) -- so the implication turns out to be false. I'm thinking more baby logic here:
A -> ~A
Put it in a truth table and if A is true then the implication is false.
I like the idea of an irony operator :D
I mean sure, if you want to collapse "moving quickly on feet" and "operates" into a single term. You could cover other equivocations of "run" as well and have a single term cover "flows," "seeks elected office," "is a candidate for winning," etc.
But if you want to use your terms in any sort of a broad fashion, or if you want to make things simple, you can simply demand that the terms be disambiguated.
Plus, these are just obvious examples, relying on equivocity. When it comes to analogous predication it will not be so simple to use such a solution.
And at any rate, prior to recent advances in robotics, it was true that only animals run in the proper (proper here) sense.
I edited my post to mention that, if I understand your main point, I agree with it.
And, if I understand, I agree with your disjunction:
Either make the predicate encompassing, in which case the first premise is false.
Or have two separate predicates, in which case the argument is invalid.
/
Quoting Count Timothy von Icarus
I don't know what you mean.
"George is opening tomorrow, and we all know what that means."
"George isn't opening tomorrow."
The conditional here is actually true, because George never opens.
If George is George Atkins then "George is going to open the store tomorrow" is true iff "Atkins is going to open the store tomorrow".
You mean substitute "George will open the store" with "If George will open the store then George will not open the store"?
Why make that substitution? I don't see how that is what the ironic speaker is saying.
It occured to me, when I thought of the irony operator, that we have the problem that we can't substitute salva veritae within a belief operator. That even makes negations tough, since we can't assume the ironic speaker knows double negation. I glossed over it. So we ould have to formulate further:
Let P be not a negation:
Df. i(s P) <-> (T(s P) & B(s ~P))
Df. i(s ~P) <-> (T(s ~P) & B(s P))
But I'm just fooling around. And probably someone has already worked out an irony logic.
[EDIT: I should have seen right away that my "irony operator" is not adequate since it does not distinguish between speaking ironically and lying in general.]
In natural language, predication is often not totally univocal, but is also not totally equivocal. There is a vagueness problem. For example, we might say that "lentils are healthy," or "running is healthy." These are true statements. And we might also say "Tones is healthy." Yet you would not be "healthy" in the same way that lentils are. However, neither is the usage totally equivocal. We call lentils "healthy" precisely because (normally) they promote the health of human beings, i.e. the same "health" we refer to in "Tones is healthy."
Perhaps we could dismiss this as just a case of equivocation in disguise, but I don't think so.
I know that people have tried to formalize this sort of thing; I am not particularly well-versed in how though. My understanding is that no attempt has proven particularly popular because they do not seem to fully capture how analogous predication is used.
It's sort of like how, as far as I am aware, there is no popular formalization of the distinction between quia vs. propter quid demonstrations (i.e. demonstrating "that something is the case," vs. demonstration "why it is the case.") I don't think most people would deny that they're different (although some would), but rather it seems that the difference should be entirely reliant on the arguments' content, not their form (i.e. an issue of material logic).
:D
This seems the easier approach to making sense of A -> ~A in a commonsense setting.
Quoting Moliere
I can give you a story that comes to mind in which I'd assert something like that -- say I'm commiserating with a coworkers frustration about George not being as reliable as we'd like, even though he's a good enough fellow.
The substitution is there only because the OP starts with A -> ~A and asks for validity, so substitution seems to work as a model for the sarcastic talking. I agree that the person speaking sarcastically does not in any way mean these logical implications, though -- it's only an interpretation of everyday speech to try and give some sense to the original question that's not purely formal.
I don't claim to have academic definitions of 'univocal' and 'equivocal', but at a naive level, as I'm merely winging it here, it seems to me that:
'totally univocal' is redundant. An expression is univocal if and only if it has one meaning. That's total.
'totally equivocal' is hard to conceive. An expression is equivocal if and only if it has more than one meaning. What would it mean to say it is totally equivocal?
Quoting Count Timothy von Icarus
Right.
Quoting Count Timothy von Icarus
Hmm, I'm not sold on that. That "Tones is healthy" and "This apple is healthy" are true two in different senses doesn't suggest to me that there's any matter of totality to consider.
/
By 'analogous predication' you mean as with the Tones/Apple example?
Do you mean that
This apple is healthy
is analogous predication with
Tones is healthy
I do understand that.
So
The animal runs
is analogous predication with
The refrigerator runs
Quoting Count Timothy von Icarus
I would think
'that it is the case' is a matter of giving an argument
but
'why it is the case' is a matter of exposition, not argument
It's interesting that in mathematics, some people demand to know "Why is that theorem true?" And I can't think of an answer other than "Because there is a valid argument for it from true premises, and here it is ..." That is, I can show you the proof, which, at least for me, does answer "why?". I may be able to give real world examples, and abstract analogies, and point to coherency. But those don't fully answer "why" in the same definitive way that proof does.
Quoting TonesInDeepFreeze
Aristotle calls such a thing a "pros hen" homonym.
Very good. Thank you.
Of course, I understand the basis of the sarcasm.
Quoting Moliere
The original post challenges the validity of
A -> ~A
A
therefore ~A
I don't see your substitution capturing irony.
When I say A sarcastically, I mean ~A, of course. And that is equivalent with A -> ~A. But I don't present it like that at all. I just say A and there is an implicit premise that when I say it, I mean its negation. I don't know how even modal logic could capture that. Or maybe, I am saying that A is true in an alternative world and false in the actual world, but even that seems far-flung.
Getting back to @Srap Tasmaner, he's looking for a use of A -> ~A in everyday discourse. I don't think your proposal works, since people don't acutually say things of the form A -> ~A to convey sarcasm. It seems to me that you followed an interesting idea, but it doesn't do the job here.
Though, related to a different kind of formula, people do say things like:
If 'Fear Factor' is great television then I'm the Queen of Roumania.
That is:
If P then Q (where Q is false)
/
And reductio ad absurdum may occur too (aside from mathematics where it is prevalent):
If Jack robbed the store, then has the loot in his car.
He does not have the loot in his car.
So, the claim that Jack robbed the store leads to a contradiction.
So Jack did not rob the store.
Of course, it can also be cast, more tersely, as modus tollens:
If Jack robbed the store, then he has the loot in his car.
He does not have the loot in his car.
So Jack did not rob the store.
I think that is likely often true, but it also seems possible in some cases to construct a syllogism that addresses the "why" (as well as syllogisms that do not seems to address it.) Since I just shared some of the relevant sources in another thread I have them on hand:
Aquinas relates this to causes (although his concept of "cause" is Aristotle's four causes, so they might still be invoked in mathematics)
Now, I do think this is probably something that has to stay to one side of form. It was long considered part of "logic," but this is logic interpreted broadly as the study of "good reasoning" (even rhetoric was sometimes lumped in with logic on curricula). However, when it comes to more amorphous debates like pluralism and the "correct logic" vis-a-vis some subject matter, it seems possible that an argument could be advanced that states that a certain sort of logic is "correct" because of the nature of the subject matter, in which case content (matter) would inform form (which I guess it always does, just not in a way everyone can agree upon).
So would it be fair to say that 'why' is more amenable to being answered when causality is involved? That goes to the point that in mathematics it is difficult to answer 'why' without finally resorting to showing the proof, which some people might consider to not be an answer to 'why'.
Yes, I think that's fair to say. We particularly care about this sort of thing in the sciences, the large focus on "correlation versus causation" for instance, or as respects states of affairs.
Interestingly, Aristotle and St. Thomas do make some recourse to causes in discussing mathematics. Even though mathematics is the understanding of form abstracted from matter, they include a form/matter distinction within mathematical entities. The essence (or "what-it-is-to-be") of a triangle is its form and the form determines what is true of all triangles. But particular triangles vary according to their dimensions and this variance is attributable to their "matter," which is termed "intellectual matter" due to these abstractions existing in the mind (ens rationis). We might say the lines are the "material" that compose a triangle. And we might say that triangle is a genus with different species, e.g. "isosceles," with there being things that are true for all isosceles by virtue of their species form.
For example, the Pythagorean theorem can be explained in terms of formal cause, whereas the values for a, b, and c will be explained in terms of material cause.
Whether this distinction is useful is another matter. In the mathematics they had available, it seems like it could be helpfully explanatory, but whether one wants to try to bother reforming the concepts for modern mathematics will probably largely depend on if one thinks the rest of the metaphysics backing it is worth developing.
They do have very interesting philosophies of mathematics though, particularly the potential/actual distinction as Aristotle applies it to the notion of the infinite in physics.
* Because of Aristotle's metaphysics, everything except God has both act and potency, and so some analagous form(actuality)/matter(potentiality) distinction.
A premise is defined as an analytic truth. It cannot be false, regardless of its synthetic falsity. If C means "Cows bark," it is irrelevant if they don't for the purposes of formal logic.
My point is simply that if you have an analytically false premise (meaning it cannot be true in any world), it fails to meet the definition of "premise."
An argument without premises is not a syllogism.
That is to say, accepting what you've argued as true, the OP is not a valid argument because it's not an argument at all.
A premise is an assumed truth.
An analytic truth is true by definition, e.g. "bachelors are unmarried men." Premises need not be analytic or considered so.
Aren't truth and form mixed together in any tautology or contradiction? We wouldn't want to exclude those though, right?
It seems you could do without it too. I hadn't really given it much thought.
No, it's not.
Quoting Hanover
Yes. So?
Quoting Hanover
A rigorous definition:
An argument is an ordered pair such that the first coordinate is a set of statements* and the second coordinate is a statement*.
The members of the first coordinate are the premises. The second coordinate is the conclusion.
If purely symbolic, the premises and the conclusion are symbolic formulas. If in natural language, the premises and conclusion are natural language declarative sentences.
So, written in that form we have this argument:
<{A -> ~A, A} ~A>
The set of premises is {A -> ~A, A}, and the conclusion is ~A.
Written informally (where everything above 'therefore' is a premise and what follows the word 'therefore' is the conclusion.
A -> ~A
A
therefore ~A
* Or more generally, formulas.
A natural language example:
If the Great Pumpkin is orange, then Great Pumpkin is not orange.
The Great Pumpkin is orange.
therefore, the Great Pumpkin is not orange.
/
With your requirement, even the following would not be an argument (not just one of the premises is not analytic, but neither of the premises are analytic):
If Bob has poor eyesight, then Bob wears glasses.
Bob has poor eyesight.
Therefore Bob wears glasses.
Really, you want to disqualify that from being an argument because the premises are not analytic?
/
You've tried a few incorrect arguments, based on misconceptions, that the argument is not valid. And now another one.
Just to note: tautology is semantic and contradiction is syntactic.
Quoting Leontiskos
Quoting TonesInDeepFreeze
Quoting Leontiskos
Here is Gensler speaking about validity in his introductory chapter:
Here is Enderton:
Here is SEP:
Quoting Logical Consequence | SEP
Here is Wikipedia:
Quoting Validity | Wikipedia
@TonesInDeepFreeze wants to say that an argument is definitionally/trivially valid if it its premises cannot all be true (i.e. if it is inconsistent). He says that he is "merely applying the definitions of ordinary formal logic." Except the reputable sources and logicians simply do not define validity in such a way.
(@Hanover)
That is the third time in this thread that you've put words in my mouth.
Quoting TonesInDeepFreeze
Indeed. Equivalent to the definition I've been stating.
Quoting Validity | Wikipedia
Indeed. Yet another way of saying the definition.
Quoting Leontiskos
Indeed. It is a point I've made many times. It is the completeness and soundness of first order logic.
You're wrong.
The definition of 'valid argument' there is standard.
And with the argument mentioned in the original post, it is the case that there is no interpretation in which all the premises are true. Perforce, there is no interpretation in which all the premises are true and the conclusion is false. So the argument is valid.
And that does not make the argument valid for Gensler, Enderton, SEP, or Wikipedia.
But it does for you.
Because you are leveraging an idiosyncratic notion of validity.
Quite not idiosyncratic.
And the argument is valid by Gensler, Enderton, SEP and Wikipedia.
Gensler:
"An argument is valid if it would be contradictory (impossible) to have the premises all true and conclusion false."
It is impossible to have both A -> ~A and A true. Perforce, it is impossible to have the premises all true and the conclusion false.
Wikipedia:
"an argument is valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false."
It is impossible to have both the premises true. Perforce, it is impossible for the premises to both be true and the conclusion nevertheless false.
SEP:
Quoting Logical Consequence | SEP
That is true, though usually it is addressed as a theorem not a definition. It is the soundness theorem. Still:
The conclusion follows deductively from the premises:
1. A -> ~A ... premise
2. A ... premise
3. ~A {1 2 by modus ponens}
Enderton:
"the concept of validity turns out to be equivalent to another concept (deducibility)"
Again, that's not a definition of 'valid' but rather it mentions an equivalence with deducibility (as it applies to first order logic). Still
The conclusion is deducible from the premises:
1. A -> ~A ... premise
2. A ... premise
3. ~A {1 2 by modus ponens}
Fucksake.
Not just consistent, but equivalent with.
Let's take the first:
Quoting TonesInDeepFreeze
The question is whether we should read Gensler as presupposing that the premises are consistent. You want to say, "The premises are inconsistent, therefore the argument is valid," and you want Gensler to agree with you. But the quotes I gave from Gensler (and everyone else) do not support your interpretation:
Your interpretation flies in the face of the bolded sentence. Gensler is talking about a consequence relation between premises and conclusion. A consequence relation is not established by your, "The premises are inconsistent..."
(As I've pointed out, you are turning the consequence relation into a material conditional, and claiming that inconsistent premises trivially show an argument to be valid in the same way that the false antecedent of a material conditional trivially shows the conditional to be true.)
Three equivalent variations:
"Valid: an argument is valid if and only if it is necessary that if all of the premises are true, then the conclusion is true; if all the premises are true, then the conclusion must be true; it is impossible that all the premises are true and the conclusion is false." [bold added]
https://web.stanford.edu/~bobonich/terms.concepts/valid.sound.html
There is no question. He does not presuppose it.
Quoting Leontiskos
(1) The consequence relation is this:
{
(2) "The premises are inconsistent" is not what I wrote.
(3) I did not claim that validity requires that there is no interpretation in which the premises are all true. Rather, I applied the definition of validity to the case in which there is no interpretation in which all the premises are all true.
I'll spell it out for you again:
(4) Df: An argument is valid if and only if there is no interpretation in which the premises are all true and the conclusion is false.
(5) Now, consider this simple thing:
If there is no interpretation in which all the premises are true, then there is no interpretation in which the premises are all true and the conclusion is false.
(6) So, if there is no interpretation in which all the premises are true, then the argument is valid.
(7) There is no interpretation in which both A -> ~A and A are true.
(8) Therefore, the argument is valid.
Quoting TonesInDeepFreeze
Well done.
(1) I did not say it is "trivial". That was another poster. I already pointed out to you that I did not say it is "trivial". So this is the fourth time in this thread that you put words in my mouth.
(2) Of course, we can state two equivalent ways:
there is no interpretation in which all the premises are true and the conclusion is false
every interpretation in which all the premises are true is an interpretation in which the conclusion is true.
And, yes, the equivalence is per the material conditional.
As in this quote:
"an argument is valid if and only if it is necessary that if all of the premises are true, then the conclusion is true; if all the premises are true, then the conclusion must be true; it is impossible that all the premises are true and the conclusion is false." [bold added]
Ordinary formal logic adopts the material conditional, not just in the object theory but in the meta-theory too.
Quoting frank
Spelled out here:
Can you see it now?
No, that's not correct.
If there is no assignment in which all the premises are true, then the argument is valid.
That is very different from what you mentioned.
There is no question that he would reject your tendentious interpretation, which fully ignores the bolded sentence of Gensler's.
Suppose you are on the jury. @Hanover presents his defense. It is a garbled mess of incoherent and contradictory gibberish. He concludes, "...Therefore, the defendant is innocent." The jury goes into deliberation. You say, "Well, we must first recognize that Hanover's defense was a piece of valid reasoning." The rest of the jury looks at you with blank stares. You continue, "His premises were inconsistent, and any argument with inconsistent premises is necessarily valid." The blank stares only become more protracted.
Now it would not help you in any way if Gensler and Enderton were fellow jurors. Even more than the other jurors, they would think you were confused. Gensler might say, "Did you read past the first sentence of my explanation of validity? Very few people would construe it in the bizarre way you have, but even so, I went on to clarify the concept in the following sentences."
Quoting TonesInDeepFreeze
And I never said you did (you are falling into the fallacy of affirming the consequent). You think that any argument with inconsistent premises is automatically valid, not that every valid argument has inconsistent premises. Here is Gensler:
Validity has to do with the conclusion following from the premises. Your claim is, "Whenever the premises are inconsistent, the argument is valid." But inconsistent premises do not show that the conclusion follows from them. Hanover's defense is not valid reasoning just because it is confused.
(Now you can hold to your tendentious position if you like, but it is not the position of Gensler, or Enderton, or SEP, or Wikipedia.)
Any argument with inconsistent premises is valid, according to Tones. Weird indeed. It requires a strained reading of the fine print of portions of definitions of validity, taken out of context. Earlier posters usefully leveraged the word "sophistry."
(Note that this is different from the modus ponens reading of the OP and it is different from the explosion reading of the OP. The effect of explosion requires explicit argumentation. The OP, for example, is susceptible to explosion, but it is not wielding explosion. Tones is just doing a weird, tendentious, definitional thing.)
I ignored nothing. The bolded part is another way of saying the unbolded part:
"An argument is valid if it would be contradictory (impossible) to have the premises all true and conclusion false. In calling an argument valid, we arent saying whether the premises are true. Were just saying that the conclusion follows from the premises that if the premises were all true, then the conclusion also would have to be true."
So, flip the bolding:
"An argument is valid if it would be contradictory (impossible) to have the premises all true and conclusion false. In calling an argument valid, we arent saying whether the premises are true. Were just saying that the conclusion follows from the premises that if the premises were all true, then the conclusion also would have to be true."
They are two ways of saying the same thing. And:
Quoting TonesInDeepFreeze
Which you ignored.
And you ignored these:
"A sentence Phi is a consequence of a set of sentences Gamma if and only if threre are no interpretations in which all the sentences in Gamma are true and Phi is false." (Elementary Logic - Mates)
"An argument is deductively valid if and only if it is not possible for the premises to be true and the conclusion false." (The Logic Book - Bergmann, Moor and Nelson).
With bolding for your bolding pleasure:
"A sentence Phi is a consequence of a set of sentences Gamma if and only if threre are no interpretations in which all the sentences in Gamma are true and Phi is false." (Elementary Logic - Mates)
"An argument is deductively valid if and only if it is not possible for the premises to be true and the conclusion false." (The Logic Book - Bergmann, Moor and Nelson).
/
You've mentioned a hypothetical trial. I don't claim that the formal notion is suitable in all situations.
My point has never been that the formal notion should have dominion over all contexts. Rather, it has been shown that the original argument is valid per ordinary formal logic.
Quoting Leontiskos
I affirmed no consequent. To be clear, I said I did not say the thing, whether you said that I said it or not.
Meanwhile, I did not say that the inference is "trivial" though you've twice now claimed I did.
Quoting Leontiskos
That is the second time you put quotes around words I didn't say. The fifth time in this thread you've put words in my mouth.
"If the premises are inconsistent, then the argument is valid" is equivalent with my wording of the definition, but for the purpose of the definition, I don't mix consistency (syntactical) with satisfiabilty (semantical). So you are incorrect when you put those words in quotes and ascribe them to me.
It's called paraphrase, and we both know you hold to the paraphrased proposition. You should be a lawyer given the way you constantly complain, nitpick, and manage bizarre readings interpreted via a form of legalese.
Quoting TonesInDeepFreeze
These are not conclusive in favor of your reading, and you would need to quote the context around these sentences given the way you have shown yourself willing to ignore context.
"It is not possible for the premises to be true and the conclusion false" is not uncontroversially fulfilled by a set of inconsistent premises. You are literally interpreting English conditionality via the idiosyncrasies of the material conditional, which is ironic given the way you protest labels which reduce your thinking to truth-functional categories. It is curious to me that you do not recognize the way your argument rests on a mere technicality.
I'm really not convinced this is going anywhere given how many times you have now repeated yourself, but the issue here has to do with consequence or inference vs. the material conditional. I gave examples of sources which agree that a valid argument requires that the conclusion follows from the premises, and everyone knows that the idiosyncratic/trivial case of the material conditional, where a false antecedent automatically makes the conditional true, is not a case of "follows from." This is why logicians refused to admit the material conditional for many decades after Frege had attempted to introduce it.
"Hanover's defense was logically inconsistent, therefore his conclusion follows from his defense," is not correct. B does not automatically follow from A whenever A is incoherent.
"a major topic in the study of deductive logic is validity. This is a
relationship between a set of sentences and another sentence; this relationship holds whenever it
is logically [b]impossible for there to be a situation in which all the sentences in the first set are true
and the other sentence false[/b]." [bold added]
https://logiclx.humnet.ucla.edu/Logic/Documents/CORE/LogicText%20Chap%200%20Aug%202013.pdf
From that same source:
"An "argument", in its technical sense, consists of two parts: a set of sentences, called the premises, and a sentence called the conclusion. The term "argument" may suggest a dispute, but in logic something is called an argument whether or not any people ever have or ever will disagree about it. Likewise, the "premises" of such an argument may or may not have been believed or asserted by somebody, and it is sometimes useful to examine arguments whose "premises" would never be believed by any rational person. Likewise, by calling something a "conclusion" we do not suggest that anyone ever has or even should "conclude" this thing on the basis of the premises given."
The idea that it is a relationship already excludes your reading. If a relationship between A and B must be established, then one must know something about both A and B. Yet you think that merely knowing something about Athat it is inconsistentproves validity. If an isolated fact about A proved validity then validity would not be a relationship between A (premises) and B (conclusion). This is another source that excludes your view. The other (single-sentence) sources you presented favor my view but do not exclude your tendentious view.
In this instance, the use of quote marks made it look like a quote, and not just a paraphrase.
And even as a paraphrase, it would be incorrect.
Quoting Leontiskos
Oh, please. (1) I do not ignore context. (2) They are simple definitions. (3) How much context would I have to type for you? (4) The cites you gave are equivalent with my wording and the wording of the two recent cites I gave.
A reasonable person would see those quotes and say, "Okay, I do see that your definition is used too and that yours and the others are equivalent." Instead, you can't stand to concede even the simplest point.
Quoting Leontiskos
It's not at all controversial in ordinary formal logic. It is easy to see that if a set of sentences is inconsistent then there is no interpretation in which all the members are true.
And look at your own cites:
"An argument is valid if it would be contradictory (impossible) to have the premises all true and conclusion false. In calling an argument valid, we arent saying whether the premises are true. Were just saying that the conclusion follows from the premises that if the premises were all true, then the conclusion also would have to be true." [bold added]
"In logic, specifically in deductive reasoning, an argument is valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false. It is not required for a valid argument to have premises that are actually true, but to have premises that, if they were true, would guarantee the truth of the argument's conclusion." [bold added]
It's remarkable that you can't stand to be wrong - to the degree that you don't heed even your own cites!
/
"it is impossible that all the premises are true and the conclusion is false"
"it would be contradictory (impossible) to have the premises all true and conclusion false"
"there are no interpretations in which all the sentences in Gamma are true and Phi is false"
"impossible for there to be a situation in which all the sentences in the first set are true
and the other sentence false."
"not possible for the premises to be true and the conclusion false"
"there is no interpretation in which the premises are all true and the conclusion is false"
Those are all ways of saying the same thing.
But you can't recognize that which is blazingly clear.
The level of dishonesty here is extraordinary.
The validity relation is:
{
So, easily we have that an argument is valid if and only if it is a member of that relation.
The validity relation is a relation in the ordinary formal sense of a set of ordered pairs. That is distinct from any of the ordered pairs themself.
Thanks for that post. It is helpful and I need to look more into the subject.
However, the reductio shows that the first premise is unsound but why is it unsound? It's unsound because it's logically contradictory. If A then not-A necessarily implies A and not-A, which tells me the argument must be invalid.
But that is not so. It doesn't. And I don't have Tone's patience. Have a think about what happens when A is false.
Yes, I understand.
Inconsistent? If you look at the argument in the OP, there can never be a case where both premises are true. According to the definition of validity in the SEP article on propositional logic, the argument in the OP is valid. It's odd at first glance.
Yep.
Quoting Leontiskos
Compare (A & ~A) ? B. Of course Tones is right, because anything follows from such a contradiction.
Leon has no idea.
1. "Sound" in this context means "the premises are true and the conclusion follows". It doesn't make sense to say that premises are sound or unsound; it is arguments that are sound or unsound.
2. "Valid" in this context means "if the premises are true then the conclusion follows". An argument can be valid even if one or more of its premises are false (and even if one or more of its premises are necessarily false).
3. A ? ¬A does not mean A ? ¬A. It means ¬A ? ¬A. The argument in the OP is equivalent to:
¬A ? ¬A
A
? ¬A
To offer a more meaningful example:
I am not 36 years old or I am not 36 years old
I am 36 years old
Therefore, I am not 36 years old
It's possible for the first premise to be true (it's true if I am not 36 years old) and it's possible for the second premise to be true (it's true if I am 36 years old), but it's not possible for both premises to be true.
The argument is an example of the principle of explosion; from a contradiction anything follows.
Sure, tautologies might be semantic, e.g. "bachelor's are unmarried men," or "triangles are three-sided." Yet aren't these really just expressing that the two terms are actually the same term, i.e. A = A, something taken to be true by virtue of its form?
Anyhow, it seems like we also have cases the formula itself is always true, regardless of which valuation is used for the propositional variables. Wouldn't these be tautologies?
For example:
(p?q) ? (~q?~p) (contraposition)
Or:
Or, in the context of this thread A?~A and ~A or ~A
At least in propositional logic, my understanding is that tautologies are defined in terms of form, hence Wittgenstein's claim that the propositions provable in logic are all in some sense tautological.
How do you figure that?
https://en.wikipedia.org/wiki/Material_implication_(rule_of_inference)
P ? Q ? ¬P ? Q
In this case, P = A and Q = ¬A.
A ? ¬A ? ¬A ? ¬A
See: "implications are disjunctions."
https://discrete.openmathbooks.org/dmoi3/sec_propositional.html#:~:text=Implications%20are%20Disjunctions.&text=P%20%E2%86%92%20Q%20is%20logically%20equivalent%20to%20%C2%AC%20P%20%E2%88%A8%20Q%20.&text=Example%3A%20%E2%80%9CIf%20a%20number%20is,else)%20it%20is%20even.%E2%80%9D
This plus De Morgan's Laws end up being important because we can do transformations into disjunctive normal form, which is easy for computers to check.
That's cool. But if the antecedent is negated, why wouldn't it be:
¬A ? ¬(¬A)
In other words, why wouldn't you negate both A's?
Fair enough. I agree it doesn't fit the form -- people don't actually say the implication, it's only equivalent to the implication (but so are so many other formulas...).
1. P ? Q
2. ¬Q ? ¬P
These do not mean the same thing:
1. P ? Q
3. P ? (P ? Q)
The misunderstanding is that many here are misinterpreting (1) as (3). "if ... then ..." in propositional logic does not mean what it means in ordinary English.
I haven't given it much thought but I am pretty sure this holds for strict implication as well.
If: "if p is true then q is necessarily true (true in all possible worlds)" then if follows that we cannot have both not-q and p. So, I guess the xor.
And my guess is that this would be true in other attempts to capture relevance vis-á-vis entailment too, since it intuitively makes sense.
I read it, thanks. It just looks like that if the A in the antecedent is false, the A in the consequent should be false too. I think you were only making the antecedent A false.
No. It doesn't say that Q being true depends on P being true. Q can be true whether P is true or false.
It just can't be that P is true and Q is false.
But in this case, they're the same variable. They're both A.
No, P is A. Q is ¬A.
Ok. I see. But then, what about the second premise? If A is false, wouldn't the second premise actually be not-A?
I'll rephrase it into English for you.
1. If Socrates is mortal then Socrates is not mortal
2. Socrates is mortal
3. Therefore, Socrates is not mortal
Given that P ? Q ? ¬P ? Q, this can be rephrased as:
1. Socrates is not mortal or Socrates is not mortal
2. Socrates is mortal
3. Therefore, Socrates is not mortal
This can be simplified to:
1. Socrates is not mortal
2. Socrates is mortal
3. Therefore, Socrates is not mortal
The argument is valid (as per the principle of explosion) but is unsound because (1) and (2) cannot both be true.
If we exclude necessarily false premises can we still demonstrate explosion? Or does keeping contradictory premises out of valid arguments remove explosion?
I see. I don't think that's what Tones was saying though. He was saying that since there are no cases where both premises are true, the argument is valid.
Yes, the argument is valid as I said. But it isn't sound because one of the premises is false.
You're giving a different reason for why it's valid versus Tones.
We both agree that the argument is valid because the conclusion deductively follows from the premises, i.e. that if the premises are both true then the conclusion is true.
Quoting Michael
You are. He's just using the definition of validity:
Quoting TonesInDeepFreeze
There is no interpretation in which all the premises are true. Therefore, the argument is valid.
And as previously mentioned, P ? Q ? ¬P ? Q. So the above can be rephrased as:
a. One of the premises is false or the conclusion is true.
And (a) is true because one of the premises is false.
That's not what he's saying.
Quoting TonesInDeepFreeze
He's not saying what you think he's saying. These are two different claims:
1. An argument is valid if there is no interpretation in which all the premises are true
2. An argument is valid if there is no interpretation in which all the premises are true and the conclusion is false.
You are claiming that he is asserting (1), when in fact he is asserting (2), as am I.
Notice that 1 and 2 are saying the same thing: The argument is valid if there is no interpretation in which
All the premises are true AND the conclusion is false.
There aren't any interpretations where all the premises are true. So it's valid.
That's not what he's saying. I don't know how to explain this to you in an even simpler way.
You may be right. Let's double check with him. @TonesInDeepFreeze
It depends on the length to which we "interpret" an argument and how you interpret "interpret."
P1. P->~P
P2. P
Conclusion: ~P
can be interpreted as:
1. ~P (P1, which is equivalent to ~P v ~P)
2. P (P2)
3. ~P v P (1, 2, this is correct as either v or &)
4. P - > P (3)
Conclusion: P (2,4 )
These two arguments are interpretations of each other because they maintain truth throughout based upon the premises provided.
Interpreting the same argument, we arrive at contradictory conclusions, which violates the definition of "valid."
This is the explosion issue. Everything follows from a contradiction. The question of validity versus soundness doesn't typically contemplate the contradiction, but it instead contemplates synthetic falsity of contingent premises yet valid logical structure (e.g. All cats can fly, I have a cat, my cat flies, valid but unsound because cats don't fly versus If all cats can fly then all cats can't fly, I have a cat, my cat can't fly.).
I'll put this to rest if someone can find an article outside our blabbing that actually considers the issue of the "validity" of the OP.
I don't quite understand what you're trying to say here. I'm just explaining very basic terminology.
If the conclusion follows from the premises then the argument is valid. If the argument is valid and the premises are true then the argument is sound.
The argument in the OP is valid because the conclusion follows from the premises, but it's unsound because one of its premises is false.
I'm saying that if you can interpret the same argument and obtain contradictory conclusions, then the argument is not "valid" under this definition of "valid":
"An argument is valid if there is no interpretation in which all the premises are true and the conclusion is false."
If we interpret it under my first iteration, we receive the conclusion ~P.
If we interpret under my second iteration, we receive the conclusion P.
We therefore have an "interpretation" in which all the premises of #1 are true and the conclusion is shown to be false via interpretation #2.
What reductio?
A premise is not sound or unsound. An argument is sound or unsound.
Df. An argument U is sound if and only if U is valid and all the premises of U are true.
But, to be more rigorous, note that 'true' is relative to an interpretation. So, to be more rigorous:
Df. An argument U is sound per an interpretation M if and only if U is valid and all the premises of U are true per M.
Note that if the set of premises is inconsistent, then there is no interpretation in which all the premises are true, so the argument is unsound per every interpretation.
/
It is not correct that A -> ~A implies both A and ~A.
Rather, (A -> ~A) along with A implies both A and ~A.
So the set of premises is inconsistent. So there is no interpretation in which all the premises are true. EDIT: That is going along with you mentioning inconsistency, though my earlier arguments have not mentioned consistency.
/
Df. An argument is valid if and only if there is no interpretation in which all the premises are true and the conclusion is false.
Note that if there is no interpretation in which all the premises are true, then there is no interpretation in which all the premises are true and the conclusion is false.
A -> ~A
A
therefore ~A
There is no interpretation in which all the premises are true. So the argument is valid and unsound.
I really don't understand what you're trying to say. Have a look at this.
The following argument is valid:
It is raining
It is not raining
George Washington is made of rakes
And the following argument is valid:
It is raining
It is not raining
George Washington is not made of rakes
And the following argument is valid:
It is raining
It is not raining
It is raining
And the following argument is valid:
It is raining
It is not raining
It is not raining
As the article says, "this arises from the principle of explosion, a law of classical logic stating that inconsistent premises always make an argument valid; that is, inconsistent premises imply any conclusion at all. This seems paradoxical because although the above is a logically valid argument, it is not sound (not all of its premises are true)."
This is just what the word "valid" means. I think you think it means something else.
And also an example of modus ponens.
An example where we might want to argue that both premises are true might be instructive.
Suppose A = "This sentence is false."
We might suppose that "A ? ~A." And, because, if the sentence is false, it will thereby be true, we might also want to assert "A" as a second premise.
The argument preserves the truth of our two (assumedly) true premises. If A is true it implies that it is also false, but if A is false it is true. Our problem here is a "truth glut," we have too much truth for A on account of both A and its negation being (arguably) true.
The inferences in OP's argument are right in line with the idea that if A is true then it is also false. And so the argument is valid in that, if we want to maintain the truth of both premises, the conclusion will follow.
However, it seems possible to construct an argument where you have a necessarily false premise that is nonetheless invalid.
For example:
All cats are mammals.
Samuel Clemens is not Mark Twain (necessarily false because they are the same person)
Therefore, Mark Twain is a mammal.
We have no middle term (and none implied, i.e, this is not an enthymeme) in something that looks like a syllogism. My first thought is that we could also probably construct an undistributed middle fallacy with a necessarily false premise.
This is the misunderstanding.
A ? ¬A does not mean "if A is true then A is also false".
As I said above, these mean two different things:
1. A ? ¬A
2. A ? (A ? ¬A)
"if ... then ..." in propositional logic does not mean what it means in English.
I don't mean that sense of 'tautology'. I mean the sense: a tautology is a sentence that is true on every row of the truth table. (If we are confined to just propositional logic, then that is equivalent with: a tautology is a sentence that is true in every interpretation, i.e. a tautology is a valid sentence).
Quoting Count Timothy von Icarus
Of course, (P -> Q) <-> (~Q -> ~P) is a tautology.
Of course, (A -> ~A) <-> (~A v A) is a tautology.
Quoting Count Timothy von Icarus
That is a common notion and quite fine. But there is a different, though compatible, notion: a tautology is a sentence that is true on every row of the truth table.
You are correct. I was speaking to our intuition about: "This sentence is false." If it is true it is false, yet we say also because it is apparent that if it is false it is also true.
A choice of three ways to figure it:
(1) Prove it in the sentential calculus.
(2) Show it as an instance of an already proven theorem schema (as @Michael did).
(3) Show it on a truth table.
Would you please at least learn the most basic thing, which is to write truth tables?
It was an interesting idea, though.
Ah, gotcha. That makes sense.
In an interpretation, a sentence is either true or false and not both, and has the same truth value no matter where it occurs in the formulas.
With an interpretation in which A is true:
A is true
~A is false
A -> ~A is false
With an interpretation in which A is false:
A is false
~A is true
A -> ~A is true
/
This is so basic that reading even just an easy Internet article on truth tables would allow you to understand sentential formulas in general.
Thanks.
This is what "valid" means: "An argument is valid if there is no interpretation in which all the premises are true and the conclusion is false."
It is raining
It is not raining
George Washington is made of rakes
Per our definition, this argument is not valid becasue all the premises are true and that conclusion is false because you also indicated:
It is raining
It is not raining
George Washington is not made of rakes
I fully understand that the conclusion is also true, so there's that, but that's the nonsense of contradictions. That is, these arguments both meet and do not meet the definition of "valid."
They're not all true. One of them is false. Either it is raining or it is not raining.
But if it were the case that both "it is raining" and "it is not raining" were true then it would be the case that "George Washington is made of rakes" is true (and that "George Washington is not made of rakes" is true).
It is what I'm saying.
The above is not the definition of 'valid argument' but it is a consequence of the definition.
(1) Two equivalent definitions:
(1a) Df. An argument is valid if and only if every interpretation in which all of the premises are true is an interpretation in which the conclusion is true.
(1b) Df. An argument is valid if and only if there is no interpretation in which all of the premises are true and the conclusion is false.
Therefore:
(2) Th. If there is no interpretation in which all of the premises are true, then the argument is valid.
Right, so you're talking about the principle of explosion?
Given that frank and I were talking about the definition of "valid", I (mis)understood him as claiming that you were saying "an argument is valid if and only if there is no interpretation in which all the premises are true".
He claimed that you and I were giving different reasons for why the argument in the OP is valid.
You say that because you're not linking your first argument to your second. That is, I consider Argument 1 to be "an interpretation" of Argument 2, not as two seperate arguments. This is one argument with 2 conclusions, both Q and ~Q. The premises must be true because they are taken as givens. Given P1 and P2, both Q and not Q are implied. The conclusion can be shown to be false by analysis of the same premises.
I didn't say "if and only if." I just said that since there are no cases where both premises are true, the argument is valid.
They are different, but (1) follows from (2).
Df. An argument is valid if and only if here is no interpretation in which all the premises are true and the conclusion is false.
Th. If there is no interpretation in which all the premises are true, then the argument is valid. (Proof: see Df.)
Why would I? Every argument is its own thing. If the conclusion deductively follows from the premises then the argument is valid.
The fact that two contradictory premises entail two contradictory conclusions does not mean that neither argument is valid. It says it right there in the Wikipedia article:
I agree, but this was the specific exchange:
Quoting frank
What he says certainly follows from what you said, but it isn't what you (literally) said (at least not in the quote he posted), and isn't the definition of validity.
You and I don't have different definitions of validity.
That's what I was trying to clarify.
Ok. What I was trying clarify is that he's not talking about explosion. It's simply that if there is no interpretation in which all the premises are true, the argument is valid.
That is explosion.
We interpret by assigning a truth value to each sentence letter. In sentential logic, that's all there is to it.
Each row of a truth table represents an interpretation and a determination of the sentence based on that interpretation. For example:
Suppose there are two sentence letters, P and Q. Then there are four interpreataion. (In general, if there are n number of sentence letters, then there are 2^n interpretations.)
interpretation 1: P is true and Q is true
interpretation 2: P is true and Q is false
interpretation 3: P is false and Q is true
interpretation 4: P is false and Q is false
Each row of the last four rows here represents one of the four interpretations with two sentence letters:
P Q
T T
T F
F T
F F
Now determine the truth value of a sentence (such as P -> Q) per each interpretation:
P Q ... P->Q
T T .......T
T F .......F
F T .......T
F F .......T
Explosion is that any proposition can be proven from a contradiction. What Tones is explaining is that if you have an argument in which there is never a case where both premises are true, the argument is valid.
That's the same thing.
If you have an argument in which there is an interpretation where both premises are false, but there are no cases where both premises are true, then the argument is valid. That wouldn't be a case of explosion.
The reason that there is no interpretation where both premises are true is because the premises are inconsistent, i.e. that their conjunction is a contradiction. As such the argument is valid whatever the conclusion (i.e. anything follows).
You may be right. Nevertheless, what Tones is pointing out is that anytime there are no cases where both premises are true, the argument will be valid. The premises don't have to be inconsistent for that. They're just never both true.
If they are consistent then they can both be true. If they can never both be true then they are inconsistent.
Checking the validity of one argument using another is done all the time.
@TonesInDeepFreeze is this true?
Couldn't it be:
1. The present King of France is bald.
2. The present King of France is wise.
Therefore: Cows bark.
It's valid, right?
Checking the soundness of one argument using another is done all the time.
Here are two arguments:
P1. If my name is Michael then I am 36 years old
P2. My name is Michael
C1. Therefore I am 36 years old
P1. If my name is Michael then I am not 36 years old
P2. My name is Michael
C1. Therefore I am not 36 years old
Both arguments are valid, but only one is sound.
No, they are not. But (1) is a consequence of (2).
They are different but related.
Anyway, yes, my point was about question begging in everyday polemical discourse. But I also contrasted it with the situation in formal logic in which axioms are also theorems.
Quoting Leontiskos
Lots of people are not paying attention to the differentiation of arguments for why the OP might be valid. Three options have been given: modus ponens, explosion, and the definition of validity. @TonesInDeepFreeze's is the latter, and it is tendentious but also probably just sophistic. It is very close to this argument:
Tones' argument:
This is what Srap usefully called "reliance in argumentation on degenerate cases":
Quoting Srap Tasmaner
(And I would be willing to explain why this sort of thing deserves a negative connotation even apart from inadvertence or deception.)
What's interesting here is that Tones is literally applying the material conditional as an interpretation of English language conditionals, and he is relying on the degenerate case of the material conditional to try to make a substantive point. He has trapped himself within a truth-functional paradigm, and has convinced himself that his "reliance in argumentation on degenerate cases" is a normative reliance, such that he is, "merely applying the definitions of ordinary formal logic." This is an especially clear case of the deep confusion that results from the excessive formalism of folks like Tones or Banno. They cannot interpret real English; they cannot distinguish absence from privation; they cannot discern rocks from corpses; they cannot recognize that validity involves a relationship between premises and conclusion.
(Cf. , , )
-
Edit:
Quoting Leontiskos
This is a matter of different modal levels, so to speak, or different domains or levels of impossibility. Tones is committing a metabasis eis allo genos. He is committing a category error where the genus of discourse is not being respected. Contingent falsity, necessary falsity, and contradictoriness are three different forms of denial or impossibility. The definition of validity that Tones favors is dealing in the first category, not the second or third. The domain of discourse for such a definition assumes that the premises are consistent. It does not envision itself as including the degenerate case where an argument is made valid by an absurd combination of premises. An "argument" is not made valid by being nonsense.
That doesn't make sense and it is not how interpretations and validity work.
An interpretation assigns one and only one truth value to each sentence letter. (If there are n number of sentence letters, then there are 2^n number of interpretations.)
An interpretation then determines the truth value of any formula that uses only those sentence letters.
Then for an argument, per a given interpretation, all of the premises and the conclusion have a determined truth value.
An argument is valid if and only if there is no interpretation in which all the premises are determined to be true but the conclusion is determined to be false.
A -> ~A
A
therefore ~A
There is no interpretation in which all the premises are true. So there is no interpretation in which all the premises are true and the conclusion is false. So the argument is valid.
An argument is sound per an interpretation if and only if the argument is valid and every premise is true per the interpretation.
A -> ~A
A
therefore A
Ther is no interpretation in which every premise is true. So the argument is unsound per every interpretation.
I mentioned it several posts back, but it seems possible to have an invalid argument with necessarily false premises.
You could construct a syllogism with an illicit negative, exclusive premises, undistributed middle, etc. and an inconsistent premise.
All triangles are not three-sided shapes.
F is not a three sided shape.
Therefore F is a triangle.
Aside from the first premise being necessarily false, this is not a valid syllogism. Even if we assume true premises and a true conclusion we get something like:
All dogs are not reptiles.
Chloe is not a reptile.
Therefore, Chloe is a dog.
You might want to double-check that.
Quoting Leontiskos
Actually, he isn't. The OP's question was not about ordinary English at all:
Quoting NotAristotle
I mainly use formal logic for analysing ordinary language arguments, so that's what I've been thinking about, but the original question was not about that.
This shouldn't be about choosing sides.
English as a meta-language regarding formal logic. In that meta-language, 'if then' is taken in the sense of the material conditional.
Indeed, we could even formalize the meta-language, and the formal conditionals would be the material conditional.
In ordinary contexts, including a natural language meta-language, ordinarily, when logicians (since at least the advent of 20th century logic) use 'if then', they use it as the material conditional.
Tones is interpreting English-language definitions of validity according to the material conditional, not merely the OP. He himself now recognizes this:
Quoting TonesInDeepFreeze
Edit:
And now explicitly:
Quoting TonesInDeepFreeze
He thinks the consequence relation of logic (?) is the material conditional, such that a contradictory set of premises automatically makes an argument valid, irrespective of any explosive argumentation within the argument.
I mentioned it lately only because the matter was raised. It is taken for granted that in such contexts, the material conditional is used. But since the matter was raised, I responded.
What in the world? There is no interpretation in which both a statement and its negation are true.
I agree, but Tones is talking about assignment or inconsistency, not necessary falseness. A (formal-propositional) contradiction is necessarily false, but not everything that is necessarily false is a (formal-propositional) contradiction.
"All triangles are not three-sided shapes," is necessarily false, it is contradictory, but it is not contradictory in the formal-propositional sense. I think this goes somewhat to my edit about levels of modality. Your earlier post about the relevance of matter and form within abstract fields like mathematics also gets at this point. See:
Quoting Leontiskos
Well, we also have cases like:
Quoting Leontiskos
Quoting TonesInDeepFreeze
Validity is a relationship between premises and conclusion. This is what I say is the common interpretation of your sources on validity:
1. Assume all the premises are true
2. See if it is inferentially possible to make the conclusion false, given the true premises
3. If it is not possible, then the argument is valid
Your interpretation changes the ordering of the conjunction and condition, and probably also the nature of the condition. You want to say that if we cannot assume that all the premises are true (on pain of contradiction), then the argument is valid by default. There is no need to look at the inferential structure.
Your interpretation is mistaken because validity is an inferential relationship between premises and conclusion. You would establish an inferential relationship without examining the inferential structure and relations. To say, "The premises are contradictory, therefore an inferential relationship between premises and conclusion holds," is to establish an inferential relationship without recourse to inferential relations.
This whole thing is an unwieldy topic in general. For example, can premise (1) of the OP be assigned a true value? And can both premises of the OP be assigned a true value? I suspect that the answers to these questions go beyond the purview of standard propositional logic, and creep into the space of Frege's judgment stroke. So it's not even obvious that Tones is right when he says that the premises of the OP cannot both be assigned a true value, although I have no real dog in that fight.
Good catch. Trying to translate English into proportional logic is hard.
Explosion is related, but I didn't mention it or need to mention it for the purpose at hand.
There are both semantical and syntactical versions of principles. These are definitions I use. Different authors have variations among them, but they are basically equivalent, except certain authors use 'valid' to mean 'true in a given interpretation', which is an outlier usage. I mention only sentences here for purpose of sentential logic; for predicate logic we have to also consider formulas in general and some of the definitions are a bit more involved.
Semantical:
Valid sentence: A sentence is valid if and only if it is true in all interpretations. A sentence is invalid if and only if it is not valid.
Logically false sentence: A sentence is logically false if and only if it is false in all interpretations.
Contingent sentence: A sentence is contingent if and only if it is neither a validity nor a logical falsehood.
Satisfiable: A set of sentences is satisfiable if and only if there is an interpretation in which all the members are true.
Validity of an argument: An argument is valid if and only if there is no interpretation in which all the premises are true and the conclusion is false.
Entailment: A set of sentences G entails a sentence P if and only if there is no interpretation in which all the members of G are true and P is false.
Sound argument (per an interpretation): An argument is sound (per an interpretation) if and only if it is valid and all the premises are true (per the interpretation). Note: When a certain interpretation is fixed in a certain context, we can drop 'per an interpretation' in that context. For example, if the interpretation is the standard interpretation of arithmetic. For example, informally, when the interpretation is a general agreement about common facts (such as that Kansas is a U.S state).
Explosion: For a set of sentences G, if there is no interpretation in which all the members of G are true, then G entails every sentence.
Syntactical:
Proof: A proof from a set of axioms per a set of inference rules is a finite sequence of sentences such that every entry is either an axiom or comes from previous entries by application of an inference rule. (And there are other equivalent ways to formulate the notion of proof, including natural deduction, but this definition keeps it simple.)
Theorem from a set of axioms: A sentence is a theorem from a set of axioms if and only if there is a proof of the sentence from the axioms.
Contradiction: A sentence is a contradiction if and only if it is the conjunction of a sentence and its negation. (Sometimes we also say that a sentence is a contradiction when it proves a contradiction even if it is not itself a conjunction of a sentence and its negation.)
Inconsistent: A set of sentences is inconsistent if and only if it proves a contradiction. (Sometimes we say the set of sentences is contradictory)
Explosion as a sentence schema: For any sentences P and Q, (P & ~P) -> Q.
Explosion as an inference rule: For any sentences P and Q. From P & ~P infer Q.
/
So explosion and "any argument with an inconsistent set of premises is valid" are similar.
Is this what you mean:
'Validity' is being defined as a concept that applies to arguments which have the form
[math]P_0 \land P_1 \land ... \land P_n \to C[/math]
when it should be defined for some other relation than ?, because ? does not properly capture the root intuition of logical consequence, or "... follows from ...", or whatever.
There are a couple issues here, I think.
One is at least somewhat technical, and I hope @TonesInDeepFreeze can figure out what I'm trying to remember. There is a reason we don't need an additional implication operator ? that is, one that might appear in a premise, say, and another for when we make an inference. In natural deduction systems, if you assume A and then eventually derive B, you may discharge the assumption by writing 'A ? B'; this is just the introduction rule for ?, and it is exactly the same as the '?' that might appear in a premise.
Thus the form for an argument above is, I believe, exactly the same as writing this:
[math]\{P_0, P_1, ... P_n\}\vdash C[/math]
That is, we lose nothing by treating an argument as a single material implication, the premises all and-ed together on the LHS and the conclusion on the RHS. (And I could swear there's an important theorem to this effect.)
Quoting Leontiskos
Okay, so yeah, this is what you were saying, but in formal logic identifying the consequence relation with material implication is not an assumption or a mistake but a result. I believe. Hoping @TonesInDeepFreeze knows what I'm talking about.
Yep. :up:
Quoting Leontiskos
I mean your post does use two different operators?
In fact there are a few that come to mind:
1. A ? ¬A
2. A ? ¬A
3. A ? ¬A
4. A ? ¬A
As a specific example:
1. I am a penguin ? My name is Michael
2. I am a penguin ? My name is Michael
(1) is true and (2) is false.
It is a relation. It is the relation whose members are all and only those arguments that are such that there is no interpretation in which all the premises are true and the conclusion is false.
Quoting Leontiskos
You didn't even read among some of the first posts I made in this thread about modus ponens, and I went on about it. I was the one who pointed out that it is an instance of modus ponens; and there was even an extended discussion about that, as at least one poster disputes that it is an instance of modus ponens. And I even mentioned that it is modus pones just a few posts before yours. For Pete's sake!
I affirm that it is valid by any of these considerations:
(1) Apply the definition of 'valid argument'.
(2) See that it is an instance of modus ponens and note that modus ponens is a valid argument form.
(3) See that the set of premises is not satisfiable, so, by explosion, the argument is valid.
(4) Prove the conclusion from the premises and note that the soundness theorem: "If a sentence is provable from a set of sentences, then the sentence is entailed by the set of sentences."
Quoting Srap Tasmaner
This is a source of the disagreement. I don't disagree that you can "discharge" the consequence in that way, but it avoids the crucial matter of the degenerative case of the material conditional, and this is precisely what Tones wants to rely upon. It seems to me that the only reason people tend to substitute consequence with ? is because arguments de facto exclude the degenerative case that Tones wants to re-introduce. An argument is a teleological act that aims at legitimate validity, not degenerative validity. Validity in logic is desirable, not undesirable.
And that is the option we are talking about, nitpicker.
From the post you sidestepped:
Quoting Leontiskos
What a stupid thing to say.
The original argument was symbolic. Of course, that could be taken as symbols meant to stand for natural language sentences. But in any case I made clear that my explanation is per ordinary formal logic and that other natural language contexts may differ.
Yes that's probably necessary, but something I overlooked.
Here's the sort of thing I was trying to remember. It's Gentzen's stuff.
Quoting wiki
And similarly
Quoting same
What I forgot is that you move the turnstile ? to the left of the whole formula, with an empty LHS.
So the result I was trying to remember was probably just cut-elimination. I never got very far in my study of Gentzen, so the best I can usually do is gesture over-confidently in his direction.
Quoting Srap Tasmaner
That is one way of looking at it. But we don't need to refer to inconsistency (which is syntactical) as we can also just note that semantically, there is no interpretation in which both premises are true.
Either is okay, but I note that in fact, I kept it all semantical.
Quoting same
Actually I expected the footnote just to be a reference to Gentzen, but it was glossed!
I mean, I don't think you can turn it into an argument that doesn't sound very stupid at any rate.
My thoughts were just that an argument isn't considered valid just because there is no way for the premises to be true and the conclusion false.
Consider:
A is not A
B is not A
B is A
This cannot have true premises and a false conclusion because one premise is necessarily false. But surely we don't want to claim that the fallacy of exclusive premises is true just in cases where it is possible for its premises to be true.
To be sure, one might use disjunctive syllogism to prove that B is A from the contradiction, but that doesn't make the form of the above valid.
Correct that I didn't mention inconsistency.
But "never both true" implies inconsistency.
It is a theorem: If as set of sentences is not satisfiable then it is inconsistent.
I'm taking this out of context, for the sake of a comment.
I'm a little rusty on natural deduction but I think reductio is usually like this:
Not sure how to handle the introduction of ? but it's obviously right, and then our assumption A is discharged in the next line, which happens to be the definition of "~" or the introduction rule for "~" as you like.
Point being A is gone by the time we get to ~A. It might look like the next step could very well be A ? ~A by ?-introduction, but it can't be because the A is no longer available.
What you do have is a construction of ~A with no undischarged assumptions.
#
We've talked regularly in this thread about how A ? ~A can be reduced to ~A; they are materially equivalent. We haven't talked much about going the other way.
That is, if you believe that ~A, then you ought to believe that A ? ~A.
In fact, you ought to believe that B ? ~A for any B, and that A ? C for any C.
And in particular, you ought to believe that
and you ought to believe that
If you combine the first two, you have
while, if you combine the second two, you have
These are all just other ways of saying ~A.
#
Why should it work this way? Why should we allow ourselves to make claims about the implication that holds between a given proposition, which we take to be true or take to be false, and any arbitrary proposition, and even the pair of a proposition and its negation?
An intuitive defense of the material conditional, and then not.
"If ... then ..." is a terrible reading of "?", everyone knows that. "... only if ..." is a little better. But I don't read "?" anything like this. In my head, when I see
I think
The relation here is really ?, the subset relation, "... is contained in ...", which is why it is particularly mysterious that another symbol for ? is '?'.
The space of a false proposition is nil, and ? is a subset of every set, so ? ? ... is true for everything.
The complement of ? is the whole universe, unfortunately, and that's what true propositions are coextensive with. When you take up the whole universe, everything is a subset of you, which is why ... ? P holds for everything, if P is true.
Most things are somewhere between ? and ?, though, which is why I have 'probability' in parentheses up there.
Quoting Moliere
Which is the interesting point here.
Ask yourself this: would "George will not open tomorrow" be a good inference? And we all know the answer: deductively, no, not at all; inductively, maybe, maybe not. But it's still a good bet, and you'll make more money than you lose if you always bet against George showing up, if you can find anyone to take the other side.
"George shows up" may be a non-empty set, but it is a negligible subset of "George is scheduled to open", so the complement of "George shows up" within "George is scheduled", is nearly coextensive with "George is scheduled". That is, the probability that any given instance of "George is scheduled" falls within "George does not show up" is very high.
TL;DR. If you think of the material conditional as a containment relation, its behavior makes sense.
((Where it is counterintuitive, especially in the propositional calculus, it's because it seems the only sets are ? and ?. Even without considering the whole world of probabilities in fly-over country between 0 and 1 ? which I think is the smart thing to do ? this is less of a temptation with the predicate calculus. In either case, the solution is to think of the universe as being continually trimmed down to one side of a partition, conditional-probability style.))
I don't know what you mean. Example?
Yes.
Quoting frank
Wrong. (Even considering the difficulty with the definite description 'the present King of France'.)
Yes, that's my point.
Tones thinks it is valid by definition, because any argument with inconsistent premises is (trivially) valid.
Now the question arises: is it invalid? I don't claim that.
Quoting Count Timothy von Icarus
Not sure what you mean by this.
Why is it wrong? There is no interpretation where both premises are true.
Quoting Srap Tasmaner
That was a really interesting post, and it presents an interesting attempt to bridge propositional logic and real-world reasoning. I am reading Burnyeat on Aristotle's Enthymeme, which is closely related to your discussion of George. Unfortunately I've already spent too much time on TPF today, so I am not going to say a whole lot more.
My take on material implication:
Quoting Leontiskos
The purpose of material implication is inferences like modus ponens and modus tollens. Degenerative uses are improper. The consequence relation can appropriate the material conditional without any risk of degenerative use (at least until you do the weird stuff Tones is doing, in which case the risks are re-introduced).
See also:
Quoting Leontiskos
When a formalist takes up logic, they neglect its teleological character, and when logic has no teleological character there can be no degenerative or non-degenerative uses. That is the problem, methinks.
I sympathize. I think a lot of our judgments rely on what I believe @Count Timothy von Icarus mentioned earlier under the (now somewhat unfortunate) heading "material logic", distinguished from formal logic.
A classic example is color exclusion.
When you judge that if the ball is red then it's not white ? well, to most people that feels a little more like a logical point than, say, something you learn empirically, as if you might find one day that things can be two different colors. (Insert whatever ceteris paribus you need to.)
Wittgenstein would no doubt say this comes down to understanding the grammar of color terms. (He talked about color on and off for decades, right up until the end of his life.)
Well, what do we say here ? leaving aside whether color exclusion is a tenable example? What you're after is a more robust relationship between premises and conclusions, something more like grasping why it being the case that P, in the real world, brings about Q being the case, in the real world, and then just representing that as 'P ? Q' or whatever. Not just a matter of truth-values, but of an intimate connection between the conditions that 'P' and 'Q' are used to represent. Yes?
These are interesting topics that Aristotle also takes up, but I don't think I'm being overly greedy in what I desire. I am not requiring a special kind of aitia/account/explanation. Here is what I said above:
Quoting Leontiskos
As Enderton notes, validity is about deducibility. It is not merely about truth values. It is about the inferential relationship between premises and conclusion. In order to show that Q follows from P, we have to show how Q is correctly inferred from P, and we need to have evidence that ~Q cannot also be inferred from P.
A key contention of mine is that I am representing the notion of validity in formal logic better than Tones is. I don't even need to advert to real-world cases, like that of color. Even within propositional logic itself, validity has to do with "follows from" and deducibility.
I was hoping this thread would be a discussion investigating deduction, implication, and validity. I am thankful that that is what everyone is discussing and other topics. I wish I had more to add to the discussion, but I am not as well-versed in logic. I have learned what I think is a strong definition of validity, which TonesinDeepFreeze stated earlier in the thread. I encourage respectful discussion of these topics by all parties.
Good lad.
Quoting NotAristotle
Even better.
Similar to what I said earlier about the genus of discourse, some arguments are apparently neither valid nor invalid:
Quoting Leontiskos
Probably they are not "arguments" at all.
To give another example using Srap's color idea:
That is the sort of thing that is occurring when one tries to claim that any argument with inconsistent premises is trivially valid. The domain of discourse when speaking about validity is arguments, and arguments do not contain premises that are known to be inconsistent. Some arguments have premises that are inconsistent but are not known to be inconsistent, and that is where reductio comes in. Are these latter kind truly arguments? Not in any perfect or ideal sense, but they are in the sense that the arguer believes the premises to be consistent.
...And I want to say that an argument is supposed to answer the "why" of a conclusion. Inferential argumentation is an explanation for a proposition/conclusion. Validity is one aspect of the goodness of such an explanation.
Quoting Leontiskos
Well, the thing is, deducibility is for math and not much else. That's the point of my story about George, and my general view that logic is ? kinda anyway ? a special case of the probability calculus.
Quoting Leontiskos
I agree with this in spirit, I absolutely do. I frequently use the analogy of good proofs and bad proofs in mathematics: both show that the conclusion is true, but a good proof shows why.
I'll add another point: when you say something another does not know to be false but that they are disinclined to believe, they will ask, "How do you know?" You are then supposed to provide support or evidence for what you are saying.
The support relation is also notoriously tricky to formalize (given a world full of non-black non-ravens), so there's a lot to say about that. For us, there is logic woven into it though:
It goes without saying that Billy can't be in two places at once. Is that a question of logic or physics (or even biology)? What's more, the story of why Billy isn't at work should cross paths with the story of how I know he isn't. ("What were you doing at the pharmacy?")
As attached as I've become, in a dilettante-ish way, to the centrality of probability, I'm beginning to suspect a good story (or "narrative" as @Isaac would have said) is what we are really looking for.
Yes, that's a common view today. Analogy is a difficulty for logic. The move towards the univocity of being in the late medieval nominalist period (important for theology, but also for how the rest of philosophy developed) was largely born out of a period in scholasticism that was intensely focused on logic (perhaps analogous to early analytic philosophy). Yet if one wants to develop a metaphysics that avoids atomism or nominalism, it might end up being quite important to have analogy as an option.
The most obvious example where this comes out is something like:
"The shot that Lee Harvey Oswald took that killed Kennedy was a good shot."
We can say this is true, because it was something that only a "good marksmen" could regularly accomplish. But, unless we really don't like Kennedy, we would not say this is "good" in a moral sense. And yet, if we are forced to claim that the "goodness" of things like "good food" and "being a good basketball player" or "being a good teacher," have nothing to do with moral goodness, I would argue that we effectively isolate moral goodness. I don't think "castrating the Good" would be too strong a term here. By my reckoning, this change in philosophy seems to be directly responsible for the descent into emotivism in ethics, until we reached a place where Moore is forced to argue that "goodness" is just a "non-natural" quality that "just is."
I guess my first thought was essentially agreement. A syllogism with two negative premises is not valid. Making the premises inconsistent doesn't seem like it should change this.
However, I understand how the claim that "if it is impossible for all the premises to be true while the conclusion is false, then an argument is valid," flows with the idea of validity as truth preservation. And I won't deny that you can find this definition in some logic textbooks. It seems like something akin to the paradoxes of material implication in terms of the "smell test" at first glance though.
That is equivalent with my argument. But my argument did not mention consistency.
There is a conceptual reason for that. Though, it is not incorrect to mix semantical and syntactical considerations, I prefer the clarity of keeping it to only one of them in this definition. (And of course, there are other times when we do want to use both semantical and syntactical considerations, especially as they relate to one another.)
Quoting Leontiskos
Again, I used the notions of interpretation (which involves truth and falsehood), not the notion of inconsistency.
Quoting Leontiskos
Perhaps I've overlooked, but I don't recall any of my cites saying "assume", "see" and "make" - verbs.
You've reinterpreted the definition for your tendentious purpose.
Quoting Leontiskos
My interpretation is literal in an example such as Mates, and virtually literal in certain others, and equivalent with the rest of them.
Your tendentious interpretation is quite a departure as it imposes a routine to be carried out, described with a series of verbs.
The cited definitions don't mention routines to be carried out.
Put my wording next to the cites. Put your interpretation next to the cites. See that yours is nowhere near as close as mine.
Quoting Leontiskos
(1) No, it doesn't. (2) Even if it did, it would be okay as long as the definition were equivalent.
Quoting Leontiskos
I didn't say anything about anybody assuming anything.
No, I do distinguish between what is object-language and what is meta-language.
'->' is in the object language. Or if the object-language is English, then 'if then' is in the object language. And in ordinary formal logic, that is the material conditional.
In the meta-language, we also use 'if then'. I put it here in italics to distinguish from 'if then' in the object-language. But still, also the meta-language 'if then' is the material conditional.
In ordinary formal logic, writers don't stop regarding 'if then' as the material conditional (if then) just because it occurs in the meta-language.
For example:
(2) If the sentence "P -> Q" is false, then P is true and Q is false.
'->'is in the object-language, and 'if then' is in the meta-language. But both of them are the material conditional.
(2) If the sentence "If John went to the store, then John got bread" is false, then "John went to the store" is true and "John got bread" is false.
The outer 'if then' is in the meta-language' and the inner 'if then' is in the object language. But both of them are the material conditional.
Quoting Janus
Propositional logic deals in propositions. Your piece has the form of a modus ponens, but doesn't deal in propositions. That makes it interesting in several ways. But "not-a" is pretty well defined in propositional logic, in various equivalent ways. And by that I mean that the things we can do with negation in propositional logic are set. There are not different senses of "not-A" in propositional calculus.
The argument in the OP is for all intents a propositional argument. It is an instance of the application of modus ponens. And it is valid. The thread should have finished at 's post. The subsequent discussion displays ignorance of basic logic rather than any failing of that logic.
Quoting Janus
is not an example of 'not-A', nor of propositional logic, although it is a striking example of the creativity of language.
Formal logic can set out some of the structures we might wish to find in natural languages.
The confused ignorance on display hereabouts might turn folk off looking at logic on detail, or encourage them to think it useless. The lesson from this thread, if there is one, might be that if folk begin by misunderstanding logic, they cannot conclude that logic tells us nothing about language.
Those who pretend to be defending a supposed common sense logic that is incompatible with formal logic are doing a disservice to rationality.
@TonesInDeepFreeze and to a lesser extent @Michael have presented a patient, consistent and correct account of the validity of the argument in the OP in the face of some extraordinary rubbish from folk we might have expected to know better.
A sad thread, this one. A low point in the history of the forums.
https://en.wikipedia.org/wiki/Proof_by_contradiction
Good list.
(1) P -> Q ... is a sentence that is interpreted as true if and only if either P is interpreted as false or Q is interpreted as true.
Indeed, sometimes '->' is not primitive but is defined:
Df. P -> Q stands for ~P v Q
or
Df. (P -> Q) <-> (~P v Q)
(2) If G is a finite set of formulas and P is a formula:
G |- P means there is a proof of P from G
(3) If G is a set of formulas and P is a formula
G |= P means P is entailed by G. (where entailment is semantic)
(4) G ? P might be the same as (2) or the same as (3) depending on the author.
I don't need to read a whole article that I've read before. I'm just wondering whether you'd give a particular example.
That gives the impression that I opt for the latter more than the others. But that is not the case:
I started in the thread by pointing out that the argument is modus ponens. Then I was challenged about that and more about the question of validity came up. Then I adduced the definition of validity and showed that the argument is valid. [EDIT: That's not correct. I started in the thread by both an appeal to the definition of validity and that the argument is an instance of modus ponens, and the fact that the premises are inconsistent does not disqualify the argument from being valid. In any case, whatever approach, yes, finally to show the validity of an argument boils down to showing that the definiens of the definition holds for the argument. But my point stands that it's not like I just chose one of the three options, as in subsequent post I especially stressed modus ponens.]
It is not "nitpicking" that I now mention that, you putting words in my mouth, distorting, confused and clueless about basic formal logic, side stepping, intellectually dishonest, would be conversation controlling, tendentious distraction.
Quoting Leontiskos
I addressed the matter of a 'relation'. You sidestepped that. And a bunch more that you sidestepped. Including that my definition is virtually the same as Mates and equivalent with the others.
Quoting Leontiskos
Wrong. As been explained to tedium. And you are seriously confused in distorting me:
When A is false, A -> B is true.
And B follows from {A -> B, A}.
I have not at all taken that to provide that B follows from A.
You're ridiculous.
Quoting Leontiskos
He said that validity and deducibility turn out to be equivalent. We easily may define (and many or most authors do) 'valid formula' and 'valid argument' without need to mention deducibility. Indeed, Enderton defines 'valid formula' (if I recall, he doesn't define 'argument' in that book) without mention of deducibility. It is only later that he remarks that validity turns out to be equivalent with deducibility.
I refer only to sentences here, not formulas in general, to keep it simple:
Df. A sentence is valid if and only if it is true per all interpretations and assignments for the variables.
Df. An argument is valid if and only if there is no interpretation in which all the premises are true and the conclusion is false.
Df. A set of sentences G entails a sentence P if and only if there is no interpretation in which all the members of G are true and P is false.
None of those mention derivation (proof, deducibility).
Df. A sentence P is derivable from a set of sentences G, per a set of axioms and a set of inference rules, if and only if there is a sequence of sentences such that each entry is either an axiom or is inferred from previous entries by an inference rule.
Then the key theorems:
Th. If a sentence is derivable from logical axioms alone, then it is valid. (soundness). Equivalently: If a sentence P is derivable from a set of sentences G, then G entails P. (soundness)
Th. If a sentence is valid, then it is derivable from logical axioms alone. (completeness) Equivalently: If a sentence P is entailed from a set of sentences G, then P is derivable from G. (soundness)
So:
Th. A sentence is derivable from logical axioms alone if and only if it is valid. (soundness and completeness). Equivalently: Th. A sentence P is derivable from a set of sentences G if and only if G entails P. (soundness and completeness)
That is what Enderton is referring to.
Quoting Leontiskos
You, totally cluelessly misconstrue a central matter in logic.
Validity is semantic. It is usually defined with regard to truth values and interpretations. Such definitions do not mention deducibility. In sentential logic, truth tables represent the determination of validity.
Again, what Enderton refers to is the fact that validity and deducibility turn out to be equivalent. But still the definition itself of validity does not require mention of deducibility.
[EDIT: Leontiskos displays typical rank sophistry. He has never read Enderton's book, let alone studied it and understood it. He just cavalierly, unthinkingly picked a quote from it out of context to support his false claim. If he had actually read Enderton, he would see that Enderton's definition does NOT mention deducibility, indeed it is entirely semantic, and that Enderson's point is that it "turns out" that validity and deducibility are equivalent. Enderton didn't say that validity is "about" deducibility. Just as Leontiskos puts words in my mouth, he puts words in Enderton's mouth. Moreover, what Enderton mentioned is just a well known and central proven fact. Anyone familiar with the basics of this subject knows that validity is semantical, deducibility is syntactical, and they have separate definitions, but we prove an equivalence.
Leontiskos also says:
Quoting Leontiskos
That is not what Enderton wrote and not implied by anything Enderton wrote.
(1) Above I addressed the misrepresentation that Enderton wrote that validity is about deducibility.
(2) In order to show that Q is entailed by a set of sentences G (Enderton's terminology is 'logical consequence' rather than 'entailed') it is NOT required to show an inference, especially not an "inferential relationship" (whatever that would mean other than that there is a correct inference) and especially not a requirement to show that ~Q cannot be inferred from G. Rather, it suffices to show that there is no interpretation in which all the members of G are true and Q is false.
It is true that if we show that there is a deduction from G to Q, then Q is entailed by G (that is the soundness theorem). But it is not required that we use that method. We still may use the semantical consideration alone: showing there is no interpretation in which all the members of G are true and Q is false.
And it is true that if G proves Q and G is consistent, then G does not prove ~Q. But it is not true, contrary to Leontiskos's ignorance and tendentious mangling, that, to show that G entails Q, we are required to show that G does not prove ~Q.
Leontiskos is so often in really bad faith when talking about logic. It's fine that he has a different notion of logic, and fine even to critique what he doesn't like, but it is bad faith and destructive to reasoned dialogue that he misrepresents what he critiques and blatantly misrepresents other posters too and then blatantly misrepresents a cite he pulled blindly without reading and understanding the basics of logic to which the cite pertains.]
Quoting Leontiskos
Hilarious!
It is not valid since there are interpretations in which the premises are true but the conclusion is false.
I guess you mean there are interpretations where the sentences are uttered in a context where they could be true. Thanks for your help.
Quoting Banno
I don't think so. My experience with logic is with the logic gates that make up a computer's microprocessor. If it's an or-gate, either input goes through, that kind of thing. I never had to worry about validity. :lol:
No, that is not what I mean.
In a post, I spelled out in detail what an interpretation is.
Ok. Thank you!
Respect includes not intentionally or carelessly putting words in the mouth of a poster, especially after the poster has dropped a flag on it and more than once. Respect includes not intentionally or carelessly seriously mischaracterizing a poster's main point, even to the point of reversing it.
Awww. Do you feel bad now @Michael?
I took it to refer to the word count. :smirk:
Not sure how this fits in with the OP. I've used this approach from time to time, but never to the extent of assuming ~P is true and showing P follows. Usually one shows a logical contradiction of sorts short of P being true. But I digress from the conversation, which has long ago become absurd. @Tones clarified the issue way back imo.
This is a meaty post.
Almost too much for me :D -- one thing that's interesting is your reduction of material implication to set theory. I'm not sure how to understand that, really -- if the moon is made of green cheese then 2 + 2 = 4. That's the paradox, and we have to accept that the implication is true. How is it that the empirical falsehood, which seems to rely upon probablity rather than deductive inference, is contained in "2 + 2 = 4"?
I'm intentionally throwing wrenches/spanners here so kindly tell me to 'ef off if it's uninteresting or simply misinformed. I'm starting to feel the tread in this conversation where I'm in too deep over my head.
Of course, reduction ad absurdum. But how is that "checking the validity of one argument using another"?
https://web.stanford.edu/class/cs103/tools/truth-table-tool/
I specified exactly what a sentential logic interpretation is. To add to that, here is what is meant by "true (or false) per an interpretation" or "true (or false) in an interpretation":
First we define 'is a sentence' by induction:
Every sentence letter is a sentence (drop parentheses when not needed):
If P is a sentence, then ~P is a sentence.
If P and Q are sentences, then (P & Q) is a sentence.
If P and Q are sentences, then (P v Q) is a sentence.
If P and Q are sentences, then (P -> Q) is a sentence.
If P and Q are sentences, then (P <-> Q) is a sentence.
It is in "stages" ('P' and 'Q' here range over sentences):
Then, an interpretation assigns a truth value to each sentence letter. So a sentence letter alone has, per that interpretation, the truth value assigned by that interpretation ('P' and 'Q' here range over sentences):
If P is just a sentence letter, then P is true per the interpretation if the interpretation assigns true to P; otherwise P is false per the interpretation.
~P is true per the interpretation if P is false per the interpretation; ~P is false per the interpretation otherwise.
P & Q is true per the interpretation if both P and Q are true per the interpretation; P & Q is false per the interpretation otherwise.
P v Q is true per the interpretation if at least one of P or Q is true per the interpretation; P v Q is false per the interpretation otherwise.
P -> Q is true per the interpretation if either P is false per the interpretation or Q is true per the interpretation; P -> Q is false per the interpretation otherwise.
P -> Q is true per the interpretation if either both P and Q are true per the interpretation or both P and Q are false per the interpretation; P <-> Q is false per the interpretation otherwise.
An example:
(P -> Q) v (R & Q)
Suppose the interpretation is:
P ... true
Q ... true
R ... false.
Then:
P -> Q is true per the interpretation
R & Q is false per the interpretation
so, abracadabra, voila, and drumroll please ...
(P -> Q) v (R & Q) is true per the interpretation
Similarly, in stages like that, for arbitrarily complicated sentences.
That's what is meant by 'true (or false) per an interpretation' or 'true (or false) in an interpretation'.
/
Various definitions we've seen mention things like 'cases', 'circumstances'.
Those can be taken to mean 'interpretations'.
And sometimes 'possible' and 'impossible' are used.
Those can be taken to mean 'true in at least one interpretation' and 'true in no interpretation', respectively.
/
Quoting Leontiskos
Whatever is meant by "See if it is inferentially possible to make the conclusion false, given the true premises", here instead is one* common method for checking for the validity of a sentence in sentential logic. *There are some more efficient ways, but they are harder to specify in a post.
Df. An argument is valid if and only if there is no interpretation in which all the premises are true and the conclusion is false:
To check for the validity of an argument (with finitely many premises):
1. Write the conjunction of the premises. Follow that with '->'. Follow that with the conclusion.
2. Write the truth table for the above formed sentence.
2. If there is a row in which the antecedent is true and the conclusion is false, then the argument is invalid, and it is valid otherwise.
Indeed, this highlights a connection between arguments and conditionals.
No "assuming". No seeing "if it is inferentially possible to make the conclusion false, given the true premises" whatever that means. No messing with the modality of possibility. Indeed, just a simple, utterly clear, step by step mechanical method.
Note: There is no mechanical procedure to check for the validity of arbitrary formulas of predicate logic.
/
I mentioned that I don't mention 'inconsistency' when defining 'valid argument'. There is good reason for that, which is:
The notion of consistency requires the notion of deducibility and deducibility is a whole subject in itself.
Df. A set of sentences is consistent if and only if there is no deduction of a contradiction from the set.
But that requires having a deduction system from which to define 'is a deduction'.
But we may wish to consider validity without having first done all stuff we have to do to set up a deduction system, which we can do later.
Indeed, often textbooks in logic devote early chapters to semantics (truth/falsehood, interpretation, entailment, validity, etc.) and then separate chapters to deduction. And then, chapters in which we prove meta-theorems about the connection between semantics and deduction. Such, as I recently mentioned, the central theorems of soundness and completeness. That is a conceptually elegant approach. Indeed, this engenders two branches of study in logic: model theory (interpretations) and proof theory (deductions).
/
Another definition of 'valid argument' to add to the list:
"it is impossible that all the premises should be true and the conclusion false" (Intermediate Logic - Bostock)
Sorry. Wrong @Hanover post.
Maybe you mean by analogy?
For example:
Check
(1) If Churchill was English then Churchill had a stiff upper lip
Churchill had a stiff upper lip
therefore Churchill was English
Compare with:
(2) If DeGaulle was German then DeGaulle was born in Lille
DeGaulle was born in Lille
therefore DeGaulle was German
(2) has true premises and a false conclusion. therefore (2) is invalid. but (2) is analogous in form with (1). so (1) is invalid
A ? ~A
A
Therefore, B
In a logic with a relevance condition such that not everything follows from a falsehood. And suppose our logic also has removed disjunctive syllogism and disjunction introduction so that it is not explosive.
In this case, wouldn't it be true that:
-A premise is still necessarily false
-B does not follow from the premises by any inference even if both are assumed true?
The point being, the argument would be valid in the sense that it is impossible for the premises to both be true, but even if they [I]were[/I] both true they wouldn't entail the conclusion anyhow.
But definitions of validity (e.g. IEP) very often will define it in both terms, i.e. that the conclusion cannot be false while the premises are all true and that the conclusion must follow from the premises or "be contained in them."
Wouldn't this be a case where the two are seperate?
It's not that complicated.
The whole space is people, say. Some are rulers, some monarchs, some kings, some none of those. A lot of monarchs these days are figureheads, so there's only overlap with rulers. All kings are monarchs, but not all monarchs are kings.
There are some things you can say about the probability of a person being whatever, and the ones we're interested in would be like this:
That is, the probability that x is a monarch, given that x is a king, is 1. The space of "being a king" is entirely contained in the space of "being a monarch".
Similarly we can say
which is the contrapositive.
The complement of Monarchs is contained in the complement of Kings, but the latter also contains Queens and I don't know, Czars and whatnot. Not a king doesn't entail not a monarch, and sure enough Pr(x is a monarch | x is not a king) > 0.
Conceptually, that's it. (There are some complications, one of which we'll get to.)
I find the visualization helpful. We're just doing Venn diagram stuff here.
Quoting Moliere
For this example, there's a couple things we could say.
Say you partition a space so that 0.000001% of it represents (G) the moon being made of green cheese, and the complement ? 99.999999% ? is it not (~G). Cool. Little sliver of a possibility over to one side.
2 + 2 = 4 is true for the entire space, both G and ~G. Both are contained in the space in which 2 + 2 = 4, which will keep happening whatever your empirical proposition because it's, you know, a necessary truth.
What's slightly harder to express is something we take to be necessarily false, like 2 + 2 = 5. The space in which that's true is empty, and the empty set is a subset of every single set, including both G and ~G. It could "be" anywhere, everywhere, or nowhere, doing nothing, not taking up any room at all. It doesn't have a specifiable "location" because Pr(2 + 2 = 5 | E) = 0 for any proposition E at all.
Both necessary truths and necessary falsehoods fail to have informative relations with empirical facts.
Only if you agree to write the preface. And it should be trenchant.
Leon and Hanover are more of an inspiration for Tones. They bring forth his best work.
Yes that's very helpful, thanks. I was getting lost in the idea of a probability space and how that relates to "contains in", but the visualization makes it quite literal and easy to comprehend.
So going back to
Quoting Srap Tasmaner
The probability space here is the set of possible outcomes we've thus far observed and, under the assumption that the distribution over that probability space has not changed -- George hasn't converted to the church of punctuality, giving us a reason to believe the probability space has changed -- the good bet is he'll show up late.
EDIT:
Wrapping that back to the OP, now...
A -> ~A
A
Therefore ~A
The (probability) space of A is entirely contained within the (probability) space of not-A.
Well, of course it is. That's almost a restatement of the probability of P v ~P equals 1.
Not sure where I'm going with this, just thinking out loud more than anything.
?
A and its complement ~A are disjoint. If A is contained in ~A, it must be ?.
Sort of thinking about future events in analogue to the bag of different colored marbles -- George is late has 99 white marbles, George is on time has 15 red marbles, and George doesn't show up is 1 black marble.
Oh, not what I was saying at all.
The impetus for talking about this at all was the material conditional, and my suggestion was that you take P ? Q as another way of saying that P ? Q.
It helps me understand why false antecedents and true consequents behave the way they do.
Having gone that far, you might as well note that there are sets between ? and ?, and you can think of logic as a special case of the probability calculus.
That's how it works in my head. YMMV
Okay. I was twisting things around with probability because of the example, but they're not related.
You mean it is a subset of every set (the empty set is not a member of every set).
Not a correction, but a reminder: We prove that the empty set is a subset of every set by using the material conditional:
Show for all S and x, if x is in 0 then x is in S:
It is not the case that x is in 0, so if x is in 0 then x is in S.
I haven't followed all of your conversation, so this might not be pertinent, but if it is, it is good to keep in mind: So, if '->' is construed as in terms of subsets to make sense of the material conditional and vacuous instances, then that tack would be circular, since the empty set being a subset of every set is based on the material conditional. But of course, we could say the notions are compatible, though that is no surprise.
Yeah that's a funny thing. Mathematics cannot be reduced to logic, it turns out, but it appears to have an irremediable dependency on logic.
Sometimes it suggests to me that mathematics and logic are both aspects or expressions of some common root.
Anyway, much as I would like for probability to swallow logic, I'm resigned to mostly taking the sort of stuff I've been posting as a kind of heuristic, or maybe even a mathematical model of how logic works. (I have some de Finetti to read soon, so we'll see what he has to say.)
By the way, I understand the main focus for unifying math and logic in recent years has been in category theory, which I haven't touched at all. Is that something you've looked into?
This is a correction ? not a member, but a subset.
A nitpick, for sure, but making exactly that distinction took a long time, and there were questions that remained very confusing until those concepts were clearly separated.
Your probability exploration is interesting. I think there's probably (pun intended) been a lot of work on it that you could find.
Indeed, logic and mathematics - chicken and the egg.
I am not up to speed on category theory though I know some of its basics. One problem I've had is finding an axiomatization. However, ZFC+"exists an infinite cardinal" is an axiomatization of category. So, as far as I can tell, category theory does not eschew set theory but rather, and least to the extent of interpretability (different sense of 'interpretation' in this thread) it presupposes it and goes even further.
Indeed. I'd have to check, but I think Ramsey used to suggest that probability should be considered an extension of logic, "rather" (if that matters) than a branch of mathematics. It's an element of the "personalist" interpretation he pioneered and which de Finetti has probably contributed to the most. I'm still learning.
Quoting TonesInDeepFreeze
Yeah not clear to me at all. A glance at the wiki suggests there have been efforts to replace set theory entirely, but I'm a font of ignorance here.
On the other side, it did catch my eye when some years ago Peter Smith added an introduction to category theory to his site, Logic Matters. One of these days I'll have a look.
Then let me nitpick that. You didn't mean 'nitpick' pejoratively, but subset vs member is not a nitpick, and the point about circularity is a good catch.
One other tiny point of unity: I always thought it was interesting that for "and" and "or" probability just directly borrows ? and ? from set theory. These are all the same algebra, in a sense, logic, set theory, probability.
I should also have mentioned that it matters because ? has no members but ? ? ? is still true, in keeping with how the material conditional works.
0 subset of 0 holds by P -> P.
I'm not expert either. But my understanding is that yes, category theory couches mathematics in different terms from set theory, and thus provides a different way of thinking, but category theory is inter-interpretable with ZFC+"exists an inaccessible cardinal"*. And ZFC+"exists an inaccessible cardinal" provides a model of ZFC, so it is a quite strong theory in that sense.
* But I have never been able to find what that really means. I know what interpretability is. And I know what ZFC+"exists and inaccessible cardinal" is. But interpretability is between two theories, but what exactly is the theory category theory that is inter-interpretable with ZFC+"exists and inaccessible cardinal"?
/
Peter Smith offers some nice content. And he used to post at sci.logic, but, if I recall correctly, got disgusted with all the cranks.
The duals run all through logic and mathematics. The main result concerning propositional logic is that there is an isomorphism between propositional logic and the Tarski-Lindenbaum algebra (a particular Boolean algebra). Then Tarski also showed an isomorphism between predicate logic and cylindrical algebra (the details of cylindrical algebra are beyond mere)
I've granted that mathematics is dependent upon logic ? but, for the sake of argument, are you sure this is right?
That is, we need logic in place to prove theorems from axioms in set theory, to demonstrate that ???, for instance, but do we want to say it's because of the proof that it is so?
This close to the bone, I'm not sure how much we can meaningfully say, but something about "holds by" ? rather than, "is proved using" ? looks wrong to me.
Am I missing something obvious?
Quoting TonesInDeepFreeze
I used to enjoy reading his reviews of logic textbooks, because he was very picky about how they presented logic schemas and the process of "translating" natural language into P's and Q's. Unforgiving when authors were too slapdash or handwavy about this, which I thought showed good philosophical sense.
Quoting TonesInDeepFreeze
Just the sort of thing, I understand, that motivates category theory.
#
Honestly, I'm not quite sure why formal logic (mathematical logic) isn't just considered part of mathematics. It would be part of foundations, to be sure, as set theory is, and you need it in place to bootstrap the rest, as you have to have sets (or an equivalent) to do much of anything in the rest of mathematics, but so what? What does mathematics get out of pretending it's importing logic from elsewhere?
That's a solid point. It felt natural and intuitive when talking about "areas", subspaces of a partitioned probability space, and so on. But it's an awful word, as @Moliere proved.
Absolutely sure.
If x in 0 then x in 0.
That's an instance of P -> P.
Quoting Srap Tasmaner
I don't opine as to 'because'.
Quoting Srap Tasmaner
I meant 'is proved by'. I find that the word 'holds' is used in at least two senses in mathematics: (1) is true, (2) is proven. But with set theory, 'true' could be taken as 'true in any model of the set theory axioms'. So "0 subset of 0" is proven by deploying "P -> P" and it is also true by deploying "P -> P" (given the soundness theorem in this version: if a sentence is provable from a set of axioms then the sentence is true in any model of the axioms).
Yet, often censoriously regarded as bad philosophical sense in The Philosophy Forum.
Category theory centers on arrows, and as involving functions, composition of functions and morphisms and things.
Someplace to start writing without having to explain yourself. I honestly think that's it.
I think of mathematical logic sub-subject of formal logic.
I guess because logic and formal logic have under philosophy, and mathematical is a part of formal logic, we have mathematical logic under the wide umbrella too.
Mathematical logic is typically a course in Mathematics. But sometimes such things as set theory are taught as a Philosophy course.
Symbolic logic is often found as a Philosophy course. But it can be a warmup for mathematical logic, and it usually includes translation of natural language arguments; even if one goes on to use symbolic logic mainly for mathematics, it helps to first know how to translate natural language since so much of mathematical prose is in natural language.
And, of course, mathematical logic is extended in formal logics regarding all kinds of philosophical subjects - modal, epistemic, etc. And, of course, in philosophy of language. And, prominently, computing.
And, of course, philosophy of mathematics is steeped in considerations about mathematical logic, set theory and mathematics.
What pretending? Would you mention a specific writing?
Set theory axiomatizes classical mathematics. And the language of set theory is used for much of non-classical mathematics. Those are two answers to "so what?"
OK thanks. It does seem to be a propositional statement in ordinary langauge.
1. If there is life (A) there is death (not-A)
2. There is life
3. Therefore there is death.
But as I've said before my undertsnding of formal logic leaves much to be desired.
I'm okay with that.
The chicken and egg still bothers me, though, so one more point and one more question.
Another issue I have with treating logic as just "given" in toto, such that mathematics can put it to use, is that one of the central concepts of modern logic is nakedly mathematical in nature: quantifiers. If you rely on ? anywhere in constructing set theory (so that you can construct numbers), you're already relying on the concept of "at least one", which expresses both a magnitude and a comparison of magnitudes. Chicken and egg, indeed.
And if you need to identify the formula "? ? ?" as an instance of the schema "P ? P", then you also have to have in place the apparatus of schemata and instances (those objects of Peter Smith's unforgiving gaze), which you presumably need both quantifiers and sets ? or at least classes of some kind ? to define rigorously. More chicken and egg.
And since we're wallowing in the muddy foundations [hide="*"] (like those of Wright's Imperial Hotel)[/hide], a quick question: somewhere I picked up the idea that all you need to add to, say, classical logic is one more primitive, namely ?, in order to start building mathematics. I suppose you need the concepts (but no definitions!) of member and collection as what goes on the LHS and RHS respectively, but that's it. And there just is no way around ?, no way to cobble it together from the other logical constants. Is that your understanding as well? Or is there a better way to pick out what logic lacks that keeps it from functioning as itself the foundations of mathematics?
Quoting TonesInDeepFreeze
Just a tendentious turn of phrase, not important.
Quoting fdrake
Kinda what I think. Also, at some point you'll have to say to the kiddies something like "group" or "collection" and just hope to God they know what you mean, because there is nothing anyone can say to explain it.
Quoting TonesInDeepFreeze
Certainly. I almost posted the same observations about the dual existence of logic courses and research in academic departments (logic 101 in the philosophy department, advanced stuff in the math department, and so on).
? ? I suppose another way of putting the question about formal logic is whether we could get away with thinking of its use elsewhere, not only in the sciences, but in philosophy and the humanities, as, in essence, applied mathematics.
Quoting TonesInDeepFreeze
Sure sure, my point was to suggest that logic could live here too, and I'm really not sure why it doesn't. Set theory is needed for the rest of math and so is logic. There's your foundations, all in a box, instead of logic coming from outside mathematics ? that's what I was questioning, am questioning. (I suppose, as an alternative to reducing it to something acknowledged as being part of mathematics, which I admit doesn't seem doable.)
You can give a few token examples of... collections... using blocks, hands of cards, sweeties, square of chocolate, desks in their class, and just hope that the kid can educe the idea of a collection through analogy. Eventually.
Though it is incredibly hard if someone struggles with analogies and abstractions.
Quoting Srap Tasmaner
Eventually the metalanguages terminate in predicates applied to undefined primitives and natural language statements. At some point that always happens. Even though such statements have clear conceptual content - though one might need to flesh that content out by derivation.
At the outset of talking about unions (U) and intersections (/\), we get an interesting consideration.
For any S, US = {x | there is a y in S such that x in y}
For any non-empty S, /\S = {x | for all y in S, x in y}.
Why non-empty?
U0 no problem. U0 = 0.
But why no /\0? Because:
Roughly put, the subset axiom is: For any set x and describable property P, there is the subset of x whose members are all and only those members of x that have property P.
Now, there is no set of which every set is a member. Why? Because:
Suppose there is a set V such that every set is a member of V. Then there would be the subset of V of all the sets that have the property of not being a member of itself. Then we have Russell's paradox.
So there is no such V.
Now, suppose there is /\0. But every set would be a member of /\0, as seen:
For all y, it is not the case that y in 0. So for all y, if y in 0 then x in y. So every x would be in /\0, so /\0 would be the universal set V, but there is no such set.
Note: I used the word 'because' in the sense of 'since' not causality.
It's curious when you notice that mathematics textbooks have no alternative to saying things like "Let x = the number of oranges in the bag", and if you don't say things like that, you might as well not bother with the rest. (For similar reasons, doing it all in some APL-like symbolism would work, but no one would have any idea what the symbolism meant, if you didn't have "? means is a member of" somewhere.)
And if you have natural language, you have how humans live, human culture, evolution, and all the rest. There's your foundations.
Don't know what you're driving at. People use variables outside of math books too.
I imagined Srap and I were talking about how the formalism in mathematics doesn't start at its "grounds", in the axioms. I imagine Srap and I are reacting to an imagined enemy of a formalist who thinks that mathematics is somehow "just" symbol manipulation. Or alternatively just awed at how the root of the formalism is in as something as messy as natural language, despite how set in stone - settable in stone - the concepts of mathematics seem to be.
Just that there's at least here a dependence of mathematics on natural language, which gives the appearance of being purely pedagogical, or unimportant "set up" steps (still closely related to the thing about logical schemata, from above).
Algebra books set up problems this way, with a little bit of natural language, and then line after line of symbolism, of "actual" math.
If you get nervous about there being such a dependency, you might shunt it off to something you call "application".
I'm just wondering if the dependency is ever really overcome, especially considering the indefinability of "set" for example.
I keep throwing in more issues related to foundations, sorry about that.
Yep, nice. But is "There is life" then the negation of "There is death"? If we pars "There is life" as "there is something that is alive" then it's negation is "it is not the case that there is something that is alive", which is not the same as "there is something that is dead". There are things that are neither alive nor dead, so being dead is not the negation of being alive.
Or do we pars the whole first assumption as "everything that is alive will die" in which case we have an implicit temporality in "will" and need to include time. But then nothing is both alive and dead at the same time.
That is, in setting the passage out as a series of proposition, the negation dissipates.
Yeah I think we're thinking about the same things.
Right. I think this is the nub. 'Not-A' should strictly be the negation of 'A'. We cannot say 'if something is alive, then it is dead' even if we can say 'if something is alive then it will be dead'.
Also, in ordinary language 'not-A' can alternatively be anything which is not A. As you point out death is not-A in the second sense but it not strictly the negation of life. The strict negation of life would be no life. Language is messy.
And Mathematics, also?
We understand each by what we do with it. Or rather, to understand a language is to be able to make use of it. Sets are not only defined by rules, but by our actually putting things into groups.
PI §201, yet again. There is a way of understanding a rule that is not found in setting it out but in following it.
Not transparently so, to me? Consistent systems capable of first order arithmetic can't contain their own truth predicate, so we don't put the predicate in. But natural language does contain its own truth predicate and behaves... well it doesn't disintegrate. That's at the very least a type distinction between consistent formal systems and natural language - one can contain its own truth predicate without being crap, one cannot.
It's more like the chicken and the egg (as you mention later in your post).
You can take the logic as given to base math on it.
And you can take a certain amount of math as given to base logic on it.
Choose your chicken or your egg.
Even formally: You need predicate logic as a basis for Z set theory. And, at least in usual formulations, you need at least some finitistic math to formalize the predicate calculus.
One way of thinking a way out of the bind is to take successive meta-theories, but that would be ad infinitum.
Another way is to point to the coherency: There is credibility as both logic-to-math and math-to-logic are both intuitive and work in reverse nicely.
Another way would be just to display the code for the formal theory without explanation or verification of any aspect of it, and then put in sequences of formulas and see which are ratified as proofs. That is, suppose you uploaded it to a highly intelligent life form, without explanation, and let those creatures discern that it works. Personally, though not necessarily philosophically, that tack doesn't appeal to me.
But, I don't see how to disagree that yes, ultimately, as humans, since we have finite time and can't in a lifetime escalate meta-theories infinitely, ultimately it will boil down to ostensive understanding, just as so much of thinking and use of language seems to do.
Quoting Srap Tasmaner
Most writers seem to view 'member' and 'set' (or 'class' depending on the treatment) as the base notions. But formally we can do it with just 'member'.
Quoting Srap Tasmaner
(1) 'e' is a non-logical constant.
(2) It is not precluded that we may define 'e' from primitives. von Neumann for example.
Quoting Srap Tasmaner
'tendentious' is definitely the Word of the Week. The runner-up is 'flows'.
Quoting Srap Tasmaner
I must misunderstand you? Famously, formal logic is not just studied in philsophy but applied in philosophy.
Just to be clear, one form of formalism is the extreme view that mathematics is mere symbol games. But formalism is not at all confined to such an extreme view.
/
Yes, even an extreme game formalist would have to admit that eventually we have to communicate in natural language. (Though he might try the "write the code and send it up in a spaceship to the advanced intelligence creatures" argument.)
And variables are used in both mathematics and everyday discourse. I can see the shape of an argument against the extreme game formalist based on the fact that variables (for example) probably originated naturally. So the argument is this?: Mathematics needs variables, and variables are ultimately understood naturally not formally.
I'm not so sure of this, since Kripke's theory of truth contains it's own truth predicate, and there is considerable work around its relation to arithmetic, I don't think we can yet rule out a Kripke-style first order arithmetic. I might be mistaken.
That is, the creativity of logicians is such that it might be better not to specify such a demarcation between formal ind informal languages, lest they invent a counter instance.
I don't know of anyone who thinks natural language conveyance of mathematics is unimportant.
Quoting Srap Tasmaner
'set' can be defined from 'element of'.
I like to be corrected -- it helps me learn. Alot of this is fuzzy in my head so I'm all ears -- formal training was an eternity ago, light, and now I just read logic books on my own in my free time for fun like a nerd.
Learning is one of the nice things about TPF.
Quoting Banno
I agree with on truth, or at least that's basically been an intuition that my other argument in the other logic thread relies upon, and I'm suspicious of substitution with respect to natural language -- it has more boundaries to it than we'd formally expect. That's why I conceded the point to @TonesInDeepFreeze about ironic statements, in natural language, don't fit the form of the OP.
@Srap Tasmaner -- My introduction to propositional logic and set theory came from a math class, so I do think there's some overlap between math and logic. What makes me hesitate to reduce logic to math has more to do with thinking about informal logic as still a part of logic, even though it doesn't behave in the same manner as formal logic -- at least by my consideration. I can understand a reductio without a formalization of it, and it always seems to me that that underlying, vague intuition of reasoning is basically what we check our formalisms against, in particular circumstances.
In one view, we have a formal object-language, and an informal or formal meta-language that includes the formal object-language.
I don't have the background to think through this unfortunately.
I'm not sure that I recall Tarski correctly (perhaps he does mention the notion of a 'consistent or inconsistent language'?) But usually languages are not consistent of inconsistent; sets of formulas are consistent or inconsistent. In this part, he's talking about an interpreted formal language. We have that there would be a contradiction (in the meta-theory, whether formal or informal) if such an interpreted formal language had its own truth predicate.
Yes, aside from paraconsistency, we would not comfortably bear contradiction, while we can bear paradox in natural language. But, we need to keep in mind that wide-open natural language doesn't provide the desiderata of formal languages.
Yes, and I rather like that. But as I understand it, Kripke's theory of truth involves one language, avoiding separating a meta language from an object language. It does this by only assigning a truth value to certain formulae, not to all.
The point is not that Kripke's theory of truth can be used as a basis for arithmetic (That seems to be a topic of some discussion amongst the academicians). So while fDrake is quite right,
Quoting fdrake
Kripke's system plays with consistency, creating a formal language that contains it's own truth predicate.
Hence, it might be premature to use "not containing it's own truth predicate" as a way to demarcate between formal and natural languages.
And a further point, whatever such demarcation might be offered, some clever logician might find a way to undermine it.
All speculative.
Interesting. Very much bears looking into.
By the way, I greatly enjoyed the video linked in the 'Logical Nihilism' thread. I have a lot of thoughts about it, and a lot of reading to do about it, but just not the time to put it together as a good post now.
So rounding back to your chat with , I'm reticent to place any firm boundary between formal and natural languages. Of course we could specify such a boundary, arbitrarily. That's cheating.
Very clever.
Fair. I was trying to convey the sense that there is this slightly annoying informal thing we have to do before we get on to doing math, properly, formally. And if you try to formalize that part ("We define a language L0, which contains the word 'Let', lower case letters, and the symbol '=', ..."), you'll find that you need in place some other formal system to legitimate that, and ? at some point we do have to just stop and figure out how to conceive of bootstrapping a formal system. And that bootstrapping will not be ex nihilo, but from the informal system ? if that's what it is ? that we are already immersed in, human culture, reasoning, language, blah blah blah.
I probably shouldn't have brought it up. It's another variation on the chicken-and-egg issue you pointed out.
Quoting TonesInDeepFreeze
This is a nice point.
Circularity need not be vicious. [hide="*"](I'm not thinking of the hermeneutic circle, though it has some pretty obvious applicability here.)[/hide]
In particular, it's interesting to think of this whole complex of ideas as being "safe" because coherent ? you can jump on the merry-go-around anywhere at all, pick any starting point, and you will find that it works, and whatever you develop from the point where you began will serve, oddly, to secure the place where you started. And this will turn out to be true for multiple approaches to foundations for mathematics and logic.
Well that's just a somewhat flowery way of saying "bootstrapping" I guess.
Now I can't help but wonder if there's a way to theorize bootstrapping itself, but I am going to stop myself from immediately beginning to do that.
Thanks for very much for the conversation @TonesInDeepFreeze!
Nice.
If you wade through everything I've vomited here in the last day or so, I think you'll find me half backtracking on that ? although I still tend to think there's something like a "formal impulse" that you can scent underlying mathematics and logic, so perhaps even our informal reasoning. It's a very fog-enshrouded area.
It's already been mentioned a couple times in this thread that "follows from" is often taken as the core idea of logic, formal and informal. Logical consequence.
Another option is consistency, and it's the story that Peter Strawson tells (or told once, anyway) for the origins of logic: his idea was that if you can convince John that what he said is inconsistent, then he'll have to take it back, and no one wants to do that. So the core idea would be not whether one idea (or claim or whatever) follows from another, but whether two ideas (claims, etc) are consistent with each other. (I should dig out a quote. He tells it better than I do.)
Do you know about the ultimatum game? It's a standard experiment design in psychology, been done lots of times in all sorts of variations. You take pairs of subjects, and you offer one of them, say, $100, on this condition: they have to offer their partner a share; if the partner accepts the offer, they get the agreed upon amounts of money; if the partner refuses, they get nothing. ? Okay, I'm telling you that story (which you probably already know) because it's famous for completely undermining a standard assumption of rationality. Since the participants start with 0, the partner should be happy to get anything, to accept $1 out of $100, instead of walking away with nothing. But that's not what happens. The offers have to be fair, something close to 50-50. Not quite 50-50 is usually accepted, but lowball offers almost never are.
And the point is this: evidently, whether it's evolution or a cultural norm, we have a sense of fairness. And it can override what theory might say is rational. (The target here is Homo oeconomicus, the rational agent.)
Similarly, we might hunt for "logical consequence" or "consistency" as some sort of ur-concept upon which logic is built.
Said in natural language? Includes using parentheses to mark arbitrarily deep nested sub-sentences?
In natural language, how would you say?:
?x ?y ?z ((P(x) ? ?u (Q(y) ? (R(u) ? ?v (S(v) ? T(z, v))))) ? ¬(?w (U(w) ? ?t (V(x, t) ? W(t, w))) ? ?p(X(p) ? ?q (Y(q) ? Z(p, q)))) ? (A(x, y, z) ? ?b ?c (D(b, c) ? (E(x, b, c) ? ?d (F(d) ? G(d, x, y)))))
And throw in some math and modal operators too.
?x ? ? ?y ? ? ?z ? ? ((x² + y² = z² ? ?u ? ? (sin(u) + cos(y) ? 1 ? (??? e? dt = e? - 1 ? ?v ? ? (v > 0 ? d/dv (v²) = 2v)))) ? ¬(?w ? ? (|w| ? 1 ? ?t ? ?? (log(t) ? 0 ? t ? 1)) ? ?p ? ? (p! = ????? k ? ?q ? ? (q ? 0 ? 1/q ? 0))) ? (A(x, y, z) ? ?b ? ? ?c ? ? (D(b, c) ? (E(x, b, c) ? ?d ? ? (F(d) ? G(d, x, y)))))) ? ?r ? ? (?H(r) ? ?I(r))
(I hope that's well formed and displays correctly - it was made by a bot.)
For all x there exists a y for all z such that if P is a property of x and there exists a u such that -- Q is a property of y or u is a property of R and allv's such that if v is S then the orderd pair z,v is T then it is not the case for all w such that w is U and there exists a t such that if t,x is V then t, w is W AND there exists p such that p is X and All q such that if q is Y then the ordered pair p q is Z OR x, y, z is A and for all b there exists a c such that (if the ordered pair c, b is D then x, b,c is E and there exists a d such that d is F and d, x, y is G.
Obviously.
I guess that' similar to the prisoner's dilemma.
Quoting Srap Tasmaner
Okay, but consistency is defined in terms of derivability (which, in first order, is equivalent with entailment).
Of course, you can use special formatting to do it, with a convention as to what it signifies. But is that natural? That is, try to do it spoken.
I did write it out while following the symbols though :D -- but I take your point that it's not something I'd ever say outside of logic.
Impressive that you did it without a bot. I just let the bot give me the English translation, but it too used specially formatting - bullet points and indentations.
But don't let the gratuitous complexity distract from the point. We could find examples in actual mathematics that might not be so complex but still tough. And then in the primitive language.
For writing, I would accept ordinary punctuation, but don't know about formatting or special characters given an ad hoc role.
Also, I recognize that the burden is not just to show that it would be difficult to use only English but that it would not be possible.
Cheers. I've been thinking and reading for three years. Still reading and thinking.
Quoting TonesInDeepFreeze
With difficulty... "For anything, there is something..." and beyond that it gets cumbersome and potentially ambiguous, but can we be sure it could not be put into a page of explanation in not-so-plain English? Is a software licence in a natural language or in a formal language? Should we ask @Hanover? He does lawyering, apparently. Or feed it into ChatGPT...
:wink:
The reason we have formal languages is that they make such things easier and clearer; a mediaeval logician would be in great difficulty.
Quoting TonesInDeepFreeze
Yep. We might stipulate a definition of natural language... is French a different natural language to English, or are both just dialects of one natural language? What about Lao?
I'm not offering an answer here, just pointing out that the difference between formal and informal languages is more intractable than it might appear.
Hey, that's my job! :D
Whether we can specify a form for "logical consequence" that will apply universally is the bone of contention in Logical Nihilism
We do have a definition of 'formal language': the set of well formed formulas is a recursive set*; and perhaps add unique readability. [EDIT: That might be only a terse synopsis of a definition that might need refinement and other clauses. And I have in mind mainly the kind of languages used in mathematical logic.]
* More exactly, the set of Godel numbers of well formed formulas is a recursive set.
/
Might be interesting to adduce a formal sentence and demonstrate somehow that it can't be said in English alone (not just that all known attempts failed).
It's related, yes.
Quoting TonesInDeepFreeze
Suppose I hold beliefs A and B. And suppose also that A ? C, and B ? ~C. That's grounds for claiming that A and B are inconsistent, but only because C and ~C are inconsistent. How would we define the bare inconsistency of C and ~C in terms of consequence?
Or did you have something else in mind?
Now it could be that the LNC, so beloved on this forum, functions as a minimal inconsistency guard, and from that you get the rest. ? This is a fairly common strategy with programming languages these days, to define a small subset of the language that's enough to compile the full language's interpreter or VM or whatever.
It could also be that the "starter versions" of consequence or consistency look a little different. I've been reading about some interesting work with gorillas, which suggests they grasp some "proto-logical" concepts. Negation, for example, is pretty abstract, but they seem to recognize and reason about rough opposites ? here/there, easy/hard, that sort of thing. Researchers have worked up a pretty impressive repertoire of "nearly logical" thinking among gorillas, though obviously their results are open to interpretation.
Anyway, suggests another type of bootstrapping.
( Might be worth mentioning that it looks like we're in the presence of one of Austin's trouser words, since the goal in Strawson's story is avoiding inconsistency, and that's what naturally came to mind above. )
I speak from the Anglo legal perspective, particularly American. Ambiguity in contracts feeds an industry, and even should there be clarity, ambiguity will be argued because the value of one's claim or obligation will be greatly affected by what the word means.
But in the law, we have a whole system to decide what things mean. And they mean what the person or people authorized to say it means.
But Americans like risk, so we keep things vague and subject to argument. Trials and hearings have the element of surprise, so compromises become of great value. We over pay sometimes just for certainty.
What this has to do with logic is that any argument goes so long as it's colorable. And this sparks creativity if you enjoy such chaos.
So how do you know what's what? You rely upon past decisions, and the art of the analogy and the ability to distinguish comes into play. Such is the significance of precedent. That we've been wrong for 100 years might hold more sway than a rigorous reevaluation. If you can't have clarity from the past, you'd have it nowhere.
Persuasion is the skill of the lawyer. Sometimes that has to with other than being strictly right. But what is "right" anyway?
Quoting Srap Tasmaner
Okay.
Quoting Srap Tasmaner
What sense of "consequence"? Entailment?
Do you mean how to define 'inconsistent'?
(First order in this post and generally in posts unless said otherwise.)
Or how to show that {C, ~C} is inconsistent? It's trivial. {C ~C} |- C & ~C. But yes, that uses conjunction intro, which is deduction. And since we have {C ~C} |- C & ~C, we have {C ~C} |= C & ~C. (soundness)
Or semantically, it's trivial to show that {C, ~C} is unsatisfiable. And we have that any unsatisfiable set is inconsistent. (completeness)
Or, we could define 'inconsistent' as "proves a formula C and proves ~C". Then, even more trivially, {C ~C} |- C and {C ~C} |- ~C. But even that uses a deduction rule (whatever you call it - inferring a sentence by virtue of it being in the set of premises.) And since we have {C ~C} |- C and {C ~C} |- ~C, we have {C ~C} |= C and {C ~C} |= ~C.
/
Df. a set of sentences is inconsistent if and only if it proves a contradiction.
Th. a set of sentences is inconsistent if and only if it entails a contradiction.
Df. a set of sentences is satisfiable if and only if there is an interpretation in which all the sentences are true.
Th. a set of sentences is inconsistent if and only if it is not satisfiable.
I don't know what you mean by "minimal inconsistency guard".
Roughly that the LNC could enforce a narrow, specialized sense of consistency ? that P and ~P are inconsistent, for any P ? and this would be enough to bootstrap a more general version of inconsistency that relies on consequence, so that with a fuller system you can say A and B are inconsistent if A ? C and B ? ~C. It's a bootstrapping technique; start with special cases and leverage those to get the general. Special cases are easier, cheaper, in this case don't require additional resources like consequence.
It's probably all too speculative to do much with. Most of the ideas I've had in the last few minutes just recreate the fact that you can build the usual collection of logical constants with negation and one of the others (unless you want to start with the Sheffer stroke). If I were to say, maybe we need both consistency and consequence as core ideas ? that's almost all that would amount to.
I was thinking, though, that there might be a way to get negation out of a primitive sense of consequence ? not the material conditional, just an intuition of what follows from what ? something like this: any given idea (claim, thought, etc.) has a twin that is the one thing guaranteed under no circumstances to follow from it, and that would be its negation. You could define ~P roughly by partitioning the possible consequents into what can and can't follow from P, but the two buckets are different: what can follow from P might initially be empty, who knows; but what can't never starts empty.
If, like the gorillas, you didn't already have the abstract concept of negation, the bucket we're going to use to define negation would probably be full of stuff ? given any P, that bucket will have stuff that ~P follows from, in addition to ~P itself, maybe, sometimes. Example: if P is "It's sunny", our bucket of things that don't follow includes "It's cloudy", "It's nighttime", "It's raining" ? all different things that "It's not sunny" follows from.
Don't spend any time trying to make sense of all this. It's just me thinking on the forum again.
We have that. You want to use that to define inconsistency in general without using the notions of semantic or syntactical consequence?
Quoting Srap Tasmaner
We already have:
If G |- A -> C and G |- B -> ~C, then Gu{A B} is inconsistent.
I don't see what you're bringing.
Quoting Srap Tasmaner
Or Nicod dagger.
Quoting Srap Tasmaner
Not getting it.
We define consistency from provability. (We could also define it from satisfiability.) Why is that lacking?
Quoting Srap Tasmaner
How do you know there is only one thing?
Quoting TonesInDeepFreeze
Quoting TonesInDeepFreeze
Sorry. Obviously I haven't managed to make clear what I'm trying to do here, probably because I've been writing a bunch of stuff I ended up scrapping, so I probably think I've said things I haven't.
I'm trying to figure out how we could bootstrap logic or reasoning, informal at first, of course, what we would need to do that, what the minimum is we could start with that could grow into informal reasoning. I'm not proposing an alternative to the logic we have now. So
Quoting TonesInDeepFreeze
is not the kind of question I was addressing at all.
For example, my last post suggested a way you might leverage a primitive understanding of consequence or "follows from" to piece together negation. I don't know if that's plausible, but it hadn't occurred to me before, so that's at least a new idea.
Quoting TonesInDeepFreeze
At first probably not! But you can see how a bunch of ideas that all point to "not sunny" might eventually get you there.
And as I noted, there's some reason to think other great apes already have the ability to reason about pairs of near opposites, even without an abstract concept of negation. I was imagining a way some sense of consequence might get you from such pairs to genuine negation.
Like I said, all very speculative, and probably not worth your time.
I tried to put something together along the lines you have in mind.
The best I came up with is this:
(1) For any sentence P, the set of all sentences is partitioned into two sets: (1) the set of sentences that follow from P, call it C(P) and (2) the set of sentences that do not follow from P, call it N(P). Then instead of sentences, consider sets of sentences, let the negation of C(P) be N(P).
But that's not what you want. So, maybe we would consider the set of sentences not compatible (my word) with P such as "raining" is not compatible with "sunny" (putting aside sun showers). But that uses "not".
So I thought of this:
(2) For any sentence P, the set of the set of all sentences is partitioned into two sets: (1) the set of sentences Q such that {P Q} is satisfiable, call it C*(P) and (2) the set of sentences Q such that {P Q} is not satisfiable, call it N*(P). Then instead of sentences, consider sets of sentences, let the negation of C*(P) be N*(P). But that uses "not".
But then I thought that we should just leave it up to gorillas; and that does seem to work.
Much of classical math existed before the introduction of set theory. So, no. Modern math is another thing.
Yeah that's an interesting idea!
I guess we could assume that nothing in N(P) would follow from anything in C(P), because follow-from would already have that sort of "transitive" property that we're used to.
I've tried to work out some consequences of this, but it's still not clear to me. (I had a whole lot of ideas that just didn't work.) It's interesting though.
Quoting jgill
Yeah, I get that. Looking at the reconstruction of math using set theory is one way to hunt for the difference between math and logic, that's all. Maybe not the most interesting way.
Depends on what 'needs' means.
Mathematics pretty much needs sets to work with. But if one denies that mathematics needs to be axiomatized, then mathematics does not need the set theory axioms.
If one affirms that mathematics needs to be axiomatized, then the usual axiomatization is set theory.
I would think 'follows from' is reflexive and transitive, but not symmetric.
I would need to doublecheck these (and depends on knowing more about 'follows from'):
C(P) is consistent if and only if P is not logically false.
If P is contingent, then N(P) is inconsistent.
P is logically false if and only if N(P) = 0. (explosion)
If P is logically true, then N(P) is inconsistent.
You should think of a word for 'follows from' so that it is not conflated with other common senses.
I suggest 'P raps Q' (equivalently, 'Q raps from P') instead of 'Q follows from P'.
('raps' from 'Srap')
Yep.
Am I right in understanding that the definition you gave of formal languages is strictly syntactic? It is formal iff it follows some rule for being well-formed?
If not, how does it differ?
(When I write, 'well formed formula', take that as short for 'well formed formula of the language'.)
Of course, people may have different ideas about what 'formal' means. But at least I think we would find that, for the most part at least, such things that are considered formal languages - such as languages for formal theories, computer languages, etc. - have in common that well formedness and other certain other features are algorithmically checkable.*
If a set is recursive, then there is an algorithm to determine whether something is or is not in that set. So if the set of well formed formulas is recursive, then there is an algorithm to determine whether a given sequence of symbols is or is not a well formed formula.
The desideratum is that it is algorithmically checkable whether a given string is or is not a well formed formula.
And, yes, that is all syntactical.
And the formation rules are chosen so that indeed they provide that the set of well formed formulas of is recursive. So, the rules are given as recursive definitions.
And the inference rules are recursive relations. So the set of proofs is a recursive set. So it is machine checkable whether a sequence of formulas is or is not a proof.
The point is that, with a formal language and formal proof, it is utterly objective whether a sequence of formulas is indeed a proof. A computer or a human following the instructions of an algorithm may (at least in principle) objectively check whether a given purported proof is indeed a proof.
And, yes, all of that is syntactical.
/
* We can also look at notions of 'formal' prior to the advent of recursion theory. And we can look at the general study of modern 'formal languages' that is mostly aimed at formal linguistics and computer science.
So I find myself back at some foundational questions. Is there always one and only one answer to the question of an argument's being valid? And closely related, what is the logical structure of an argument, in contrast to its syntax, grammar, and semantics.
In effect, those who claim that the argument in the OP is invalid are inadvertently suggesting that there is more than one way for an argument to be valid - that it is formally valid, in propositional calculus, but that in some other logic it is invalid. In some cases, maintain this goes against some folk's own view as expressed elsewhere.
Or they may be saying that the logical form of the argument is other than that shown by parsing it in propositional calculus. In that case, there would be more than one logical form for even such a simple argument.
Good post. I may have fallen too far behind in this thread, but I don't think we have to choose between logic and physics to explain such an argument. Physics provides us with a particular kind of logic which makes the argument sound.
I want to say that "flows from" or validity in logic is a specific kind of inferential relation and justification. Your story about Billy fulfills that inferential relation, albeit with some tacit premises.
Quoting Srap Tasmaner
But why? Given the explanation, can we deduce that Billy is not at work?
I agree that the consequence relation ("follows from") is hard to formalize. Or rather, I think it is impossible to formalize.
(Feel free to ignore this post if the thread has moved too far away from it.)
That is true, but shouldn't there be a distinction not just between "valid but not sound" but also between "valid but incoherent"?
For example:
If P then not Q
P
not Q
This is valid. It is sound if P and ~ Q are true. Unsound if not.
If P and Q are the same thing such that:
If P then not P
P
Not P
This is valid and not sound, but also not coherent.
As in, "If I went to the store, I did not go to the store, and I went to the store, so I did not go to the store." That is valid, but meaningless. I have no idea what you did, whether you went to the store, didn't go to the store, and I can't understand how your going to the store made you not go to the store."
And that was the debate for 20 pages I suppose. The pluralism might not be over "validity" if you wish to protect that term to only reference formal structure, but perhaps over soundness if you want to speak of what synthetically is false versus what is analytically false.
This conversation is pedantic and legalistic if I'm understanding it correctly. We all can agree with what truth tables show and what logic dictates, but the battle might be over terms, but I might misunderstand because that was the extent of my disagreement.
The incoherently true statement is also distinct from the vacuously true statement. As in, "if Tokyo is in Spain, then the Eiffel Tower is in Bolivia." There the antecedent cannot ever be satisfied, so it can never be true, but it's impediment to truth is due to a synthetic falsehood, but that's unlike the OP where the antecedent is premised to be false.
I'll let you guys better explain it to me if I've misunderstood this, but the contradiction and the incoherence that follows is what trips this issue up to me at least.
It would help if you provided a definition of 'coherent' such that its a matter of form alone.
We do have the definition per form alone of 'inconsistent' (in sentential logic, both equivalent with unsatisfiable, and reducible to per form alone).
The set of premises of the above argument is inconsistent.
/
(0) An argument is valid if and only if there is no interpretation in which all the premises are true and the conclusion is false.
Ways (0) holds:
(1) The set of premises is not satisfiable and the conclusion is logically true.
(2) The set of premises is not satisfiable and the conclusion is contingent.
(3) The set of premises is not satisfiable and the conclusion is logically false.
(4) The set of premises is satisfiable and the conclusion is logically true.
(5) The set of premises is satisfiable and the conclusion is contingent, but there is no assignment in which all the premises are true and the conclusion is false.
(6) Every member of the set of premises is logically true and the conclusion is logically true.
Ways (0) does not hold:
(7) The set of premises is satisfiable and there is an interpretation in which all the premises are true and the conclusion is false.
(8) Every member of the set of premises is logically true and the conclusion is not logically true.
/
We could coin the word 'revonah' (suggesting the opposite of what Hanover likes), and say:
An argument is revonah if and only if its set of premises is not satisfiable.
(1), (2) and (3) are revonah.
Quoting Hanover
Again, valid/invalid in ordinary formal logic pertain to the entailment relation. Indeed, it would be foolish to look for information about the truth of the premises and conclusion merely from consider of validity, except to see that there are no interpretations in which all the premises are true and the conclusion is false.
But, of course, one may hold that the world 'valid' should not be used if it doesn't comport with certain everyday and philosophical senses, though, personally, I understand the notion in ordinary formal logic and allow that words have different special senses in various fields of study.
Quoting Hanover
The conditional is vacuously true in all interpretations in which 'Tokyo is in Spain' is false. But it is not the case that 'Tokyo is in Spain' is false in all interpretations.
Quoting Hanover
You might understand if you read an introductory textbook in formal logic. You wouldn't have to accept the material, but at least you would see how it operates.
Tones made the response I would have - what is "coherent"? The argument is coherent, in so far as it is consistent with propositional logic.
So again, you seem to want two types of validity, one formal and the other informal.
Quoting Hanover
It isn't meaningless. We have an idea of what it would be to go to the store, and not to go to the store. Yep, you can't do both.
What would it be for an argument not to be "coherent", in your terms? Do you just mean "valid yet unsound"? I'm not seeing what the introduction of "coherent" adds.
I'm not happy with my response to the question of distinguishing between formal and informal languages.
Your challenge could be taken as: Provide a definition such that any language is exactly one of: formal and informal.
(1) I chose the attribute of having a recursive set of formulas ('formula in the sense of 'well formed formula' in logic). But that I think it should be more general: the set of well formed expressions is a recursive set.
(2) What about formal/informal blends?
(3) Even with ordinary formal languages for logic, there may be other considerations that are required to hold for formality other than that the set of expressions is recursive. (Especially the notion of 'an effectivized language'.)
(4) Other complications.
That argument doesn't seem for me to work.
A logical monist could say that certain supposed laws of entailment are not correct and thus not laws of logic. It doesn't follow that the monist would be in contradiction if she also said that there are certain laws of entailment that are the only correct laws of logic.
That is, it doesn't seem to me that in denying that certain supposed laws are correct one has to agree that that there are different competing sets of laws that are all correct.
I don't know what you mean.
The most basic "structure" is that an argument is an ordered pair, with the first coordinate being a set of sentences and the second coordinate being a sentence.
Another way: An argument is a non-empty set of sentences with exactly one of the members designated as the conclusion.
Well expressed; and my hunch is that we cannot provide any such clear cut distinction. So we might stipulate that formal languages are those with recursive formation rules. I can imagine a contrarian logician developing a system that undermined some aspect of that - perhaps, by some novelty, having an uncountable number of formation rules, or some such oddity.
This speculation came about after struggling with two SEP articles. The first was Logical Consequence, which I read with a view to trying to get a handle on what the recent thinking is on what it is for a conclusion to follow from a premise. This led me to the article on Logical Form, were I ran afoul of differentiating Syntax, grammar and semantics. I might have been clearer if I had, after that article, asked what logical structure is.
Quoting TonesInDeepFreeze
Perhaps one might ask, is that designation arbitrary? Why this sentence rather that that one? Is there more, such that the designated sentence is in addition a Logical Consequence (whatever that is) of the others?
An example: supose we have the sentences {p, q, r} and designate r as the conclusion. Is that an argument, or is there something more, such that in addition, r is the "logical consequence" of {p.q}? This seems to be the sort of thing that relevant logic is chasing. If we have {"Sydney is in Australia", "Some swans are black", "The cat is on the mat"} and designate "The cat is on the mat" as the conclusion, do we then have an argument, albeit a very bad one? Or is there more to an argument than just a grouping of sentences with one designated as the conclusion?
My hunch is that we can; but my amended attempt might not be satisfactory.
Quoting Banno
Even just an uncountable set of symbols knocks it out of being formal. For example, in logic, we can have languages with uncountably many symbols (and useful to have for certain purposes, such as in model theory to derive a model upon which to base non-standard analysis), but such a language is not considered formal, since there's no such thing as an uncountable recursive set.
Quoting Banno
Yes. It should be.
Quoting Banno
No, because that would be defining 'valid argument', not 'argument' in general.
Quoting Banno
Don't know how it goes specifically with relevance logic. But my guess is that even in relevance logic, 'argument' would not mean just valid argument.
I take the idea as being as general as possible: The one thing all arguments have in common is having a set of premises and a conclusion. (Sometimes a set of premises and a non-empty set of conclusions*.) Then we find definitions of various notions of validity: whether classical, intuitionistic, relevance, multi-value, etc.
* But I've heard of a notion in which the set of conclusions could be empty.
P ? ¬P
? ¬P ? ¬P
? ¬P
P
? ¬P
Or more simply:
¬P
P
? ¬P
It's not raining and it's raining therefore it's not raining.. So yeah, it's "incoherent" in that its premises are inconsistent.
Deduction should allow you to pass, by valid inference, from what you know to what you did not know. Yes?
In mathematics, these elements are well-defined. What do we know? What has been proven. How do we generate new knowledge? By formal proof.
Neither of these elements are so well-defined outside mathematics (and formal logic, of course). There is no criterion for what counts as knowledge, and probably cannot be. And that defect cannot be made up by cleverness in how we make inferences.
I see no reason to question the traditional view. "Our reasonings concerning matters of fact are merely probable," as the man said. There is deduction in math and logic; everyone else has to make do with induction, abduction, probability.
That is -- making shit up and then seeing if it works(and finding that it usually does not). Though in school I call it "Guess and check"
Accepting that definition of "incoherent," we can then say we have (1) valid and coherent arguments and (2) valid and incoherent arguments.
We can also have (3) valid and sound arguments and (4) valid and unsound arguments.
Would you agree that:
A. All 3s are 1s, but not all 1s are 3s?
B. All 2s are 4s, but not all 4s are 2s.
C. No 1s or 3s are 4s or 2s.
D. No 4s or 2s are 1 or 3s.
(Venn diagram is: 3 is a circle within the 1 circle and 2 is a circle within a 4 circle).
The OP is a 2, but not all 2s are a 4, so just calling it valid but unsound doesn't capture its special class.
Maybe we should could call 2s a "NotAristotle" after the creator of this thread. Or, is there already another name for 2s.
Disagreement with what I've said here?
Arguments can be:
1. Valid, consistent, and sound
2. Valid, consistent, and unsound
3. Valid, inconsistent, and unsound
4. Invalid
OP's argument is (3), and is an example of the principle of explosion.
No 3 is a 4 because no argument can be both valid and invalid.
I get that, but a 3 permits explosion, which can force anything anywhere.
Well, you could have the valid but unsound argument:
1. It is raining
2. It is not raining
3. Therefore, arguments can be both valid and invalid
But regardless of how you get there, the conclusion "arguments can be both valid and invalid" is false.
Are you claiming that knowledge does not exist outside mathematics? I don't see why "the elements being less well-defined" results in any serious problem here. This comes back to the Meno question I have posed to you elsewhere. One could answer that question by denying that knowledge exists.
Quoting Srap Tasmaner
Sure, and haven't we achieved that with Billy?
Can we say the conclusion is valid or do we reserve the term "valid" only to argument forms and not to conclusions?
Premises and conclusions are either true or false.
Arguments are valid if the conclusion follows from the premises.
Arguments are sound if they are valid and the premises are true.
The statement "that's a valid conclusion" does make sense, so I would think a listener who hears that would realize immediately that the person speaking isn't using the term "valid" as a term of art, but must mean something else.
While I think it's defensible to say that "knowledge does not exist outside mathematics," I don't think I have to, to show the difficulty.
Mathematical knowledge, to borrow Williamson's term, is "luminous": that is, when you know that P, you know that you know that P. That may put it too strongly: there are cases where you think you have a proof, but you don't; there are cases where someone has provided a proof, but it's complex enough that it takes a while for people to confirm that it is a proof. Nevertheless, there is an alignment of the process of knowledge production and knowledge justification, and a single standard governs both.
Outside of mathematics, there are no standards of either that garner universal approval, much less guarantee that production and justification are measured by the same standard. We may have knowledge, but in general we cannot know when we do and when we don't, and thus we cannot know when our valid arguments are sound and when they are not.
I'll throw in a side issue that emphasizes the difference. It is a wise saying that experiments which are not performed have no results. And yet, in mathematics your hypotheses can be so sharply defined that they do: a difficult theorem like Fermat's last theorem might be solved piecemeal ? you prove that if lemma X were the case, then you could prove theorem T, and then you look for ways to prove X. That is, in mathematics, it's not that unusual to prove a conditional, without knowing whether the antecedent is in fact true. I think the independence results in set theory are also different from the sort of thing we can ever hope to achieve in empirical investigations.
I'm not in love with this story. It would be nice to retreat instead to some sort of common sense that of course we know things and deduce more things in everyday life. Sure. But part of that common sense is also that there are exceptions, we turn out not to know what we think we do, we turn out not to be justified in making the inferences we do. So I end up back in the same place, because we already have a name for this sort of rule that generally works but has exceptions: that's probability. ? Philosophical attempts to close the gap and specify, in some vaguely scientific way, exactly the criteria for knowledge and inference, so that we can be on ground just as solid outside of mathematics, have not only universally failed, but there are reasons to think they must fail.
I do not see a way around making some kind of distinction here. Either only mathematics (and logic) gets knowledge and deduction ? and everything else gets rational belief and probability ? or there are two kinds of knowledge, and two kinds of deduction. Pick your poison.
Mathematical knowledge and empirical knowledge differ so greatly they barely deserve the same name. Obviously the history of philosophy includes almost every conceivable way of either affirming or denying that claim.
Whatever @Michael meant, I don't take it as a definition. It only states:
If a set of statements is inconsistent, then it is incoherent.
It doesn't say:
A set of statements is inconsistent if and only if it is incoherent.
More generally, an expression may be incoherent but not inconsistent. Expressions that are not syntactical are incoherent but they're not even statements, so they are not even in the category of things that are consistent or inconsistent.
By using 'incoherent' rather than 'inconsistent', we lose the information that the premises are not merely incoherent, but they are, more to the point, inconsistent.
Also, @Michael, as I understand him, meant scare quotes. Indeed, I don't see the analysis of this particular matter in ordinary formal logic as being in regard to a wider rubric of 'incoherent' (that includes both not-syntactical gibberish and syntactical inconsistency) but rather in regard to inconsistency.
Also, personally, in this context, I like to mention satisfiability rather than consistency, since they are equivalent only in first order logic, and, even more basically, mentioning satisfiability rather than consistency underscores that we don't need to have a particular, or even any, deductive calculus in view.
/
I suggested the neologism 'revonah' for an argument that has an unsatisfiable set of premises.
but maybe a neologism that is more technical sounding would be better:
Df. An argument is sat-premised if and only if the set of premises is unsatisfiable.
Df. An argument is unsat-premised if and only if the set of premises is unsatisfiable.
Quoting Hanover
Soundness is per each interpretation. But let's say we're confining to just one interpretation, so we don't have to say 'per the interpretation':
(1t) sat-premised and valid
Not every sat-premised argument is valid.
Not every valid argument is sat-premised.
(2t) unsat-premised and valid
Every unsat-premised argument is valid.
Not every valid argument is unsat-premised.
(3t) sound ['sound' is defined as 'valid and all premises are true', so 'sound argument' is redundant]
Every sound argument is valid.
Not every valid argument is sound.
(4t) unsound and valid
Quoting Hanover
(C) and (D) are WRONG (see below).
These are all CORRECT except those marked WRONG:
(A1) For any argument, if it is (3t) then it is (1t).
(A2) It is not the case that, for any argument, if it is (1t) then it is (3t).
(B1) For any argument, if it is (2t) then it is (4t).
(B2) It is not the case that, for any argument, if it is (4t) then it is (2t).
(C1) For any argument, if it is (1t) then it is not (4t). WRONG.
There are arguments that have a satisfiable set of premises but there is at least one false premise. This is a key point in ordinary formal logic. Consider:
{"Macron is German"} is satisfiable but "Macron is German" is false. This is a key point in ordinary formal logic: A set of premises may satisfiable but still have falsehoods. Consider:
"Macron is German" is false per ordinary facts, but there are interpretations in which "Macron is German" is true.
(C2) For any argument, if it is (1t) then it is not (2t).
(C3) For any argument, if it is (3t) then it is not (4t).
(C3) For any argument, if it is (3t) then it is not (2t).
(D1) For any argument, if it is (4t) then it is not (1t). WRONG.
There are arguments that are unsound but have a satisfiable set of premises. This is a key point in ordinary formal logic: For example:
"Macron is German" is false per ordinary facts, but there are interpretations in which "Macron is German" is true.
(D2) For any argument, if it is (4t) then it is not (3t).
(D3) For any argument, if it is (2t) then it is not (1t).
(D3) For any argument, if it is (2t) then it is not (3t).
Yes, so this has inconsistent premises:
1. It is raining
2. It is not raining
3. Therefore, is is raining
And this has incoherent premises
1. Red fast what
2. Glooblefooble
3. Therefore it is raining
Indeed.
EDIT: But "Red fast what" and "Glooblefooble" are not even premises since they are not statements. So it's not even an argument, since {"Red fast what", "Glooblefooble"} is not a set of statements.
Usually, we don't say that arguments are consistent/inconsistent. Sets of sentences are consistent/inconsistent.
All combinations:
(1) sound (thus satisfiable set of premises) and valid
(2) satisfiable set of premises, unsound, and valid
(3) satisfiable set of premises, sound, and invalid
(4) satisfiable set of premises, unsound, and invalid
(5) unsatisfiable set of premises (thus valid and unsound)
Explosion is the property of a set of statements entailing all statements. But it's still the case that no argument can be both valid and invalid.
There are two definitions:
Df. An argument is valid if and only if there are no interpretations in which all the premises are true and the conclusion is false.
Df. A statement is valid if and only if it is true in all interpretations.
We sometimes say 'the statement is a validity' synonymously with 'the statement is valid'.
I appreciate the deep analysis, but can you just draw me some Venn diagram circles with 1, 2, 3, and 4s on them and then I can see what can be what? It's easier on my visual brain.
We're actually debating what terms each of us can make up and the best terms that would describe whatever we're trying to say. I'll defer to yours with my backwards name and provide myself a translator so we can speak the same language. In truth, I think we largely follow what each other are saying at this point.
What I mean by "incoherent" is that which is "expressed in an incomprehensible or confusing way; unclear." @Michael's rendition of what "incoherent" might look like includes gibberish, which is a new additon to this conversation, so it might require an entirely different term. We could then start inserting such non-linguistic items such as the smell of lilac and that weird feeling of deja vu in as premises. Everyone loves a good emoji as well, so that could go in there too.
In any event, "Gloobelfooble" could indeed be a statement, inasmuch as A can be statement and Q can be a statement.
If Gloobelfooble, then Q
Gloobelfooble
Q
That's ambiguous. It could mean two things:
(1) A certain argument that ends with that conclusion is valid.
(2) The conclusion is valid (i.e., it is a validity).
You make a valid point.
I get that joke. Thank you.
Of course. Nice.
Quoting TonesInDeepFreeze
Cool. So we have {p, q, r} with r designated as the conclusion, and that's an argument, and then in addition if it is a valid argument, r is also the logical consequence of {p, q}. Thanks for clearing this up.
You have a preference for the model-theoretic account of logical consequence, if I've understood aright. it has an intuitive appeal for me. On that, account, an argument is valid iff there are not counter-examples. The SEP article notes "One of the main challenges set by the model-theoretic definition of logical consequence is to distinguish between the logical and the nonlogical vocabulary" and suggests that this might be overlooked if we "limiting the admissible models for a language". This was my puzzle. What follows refers to that article.
Another issue is the difficulty that "the actual world is not accounted for by any model", but this seems to me to be misleading; sure "each model domain is a set, but the actual world presumably contains all sets, and as a collection which includes all sets is too large to be a set", but it doesn't follow that there are any particular things int he world that we cannot include in our model. That is, while we may not be able to model everything, we can model anything.
I take it that the "Tonk" argument undermines proof-theoretical accounts by showing them to be arbitrary. My realist tendencies play a role here.
Frankly I haven't yet been able to follow the argument for bringing proof-theoretic and model-theoretic perspectives together, but there is some appeal in that, and I gather that the result would be a win for logical pluralism.
None of this is 'tight" enough for a firm conclusion, but do you have any thoughts?
I'm not going to draw diagrams.
Quoting Hanover
I'm not debating that. We can make any defiinitions we want. And I am not claiming that the definition of 'valid' in ordinary formal logic is suited for many everyday senses of 'valid'.
Quoting Hanover
Under that definition, I don't take contradictions to be incoherent.
Jack Shaklemoff is in Kansas and Jack Shaklemoff is not in Kansas.
That is clear, comprehensible and not confusing.
And I understand that there are no interpretations in which it is true.
Moreover, consider some set of premises that are very complicated and so that it is not at first apparent whether the set is inconsistent. I don't have to wait until it is proven that the set is consistent to understand it as a set of premises. Consider:
The set of axioms of PA along with "Every even number greater than two is the sum of two primes."
I don't know whether or not that is a consistent set of sentences. But even if later we find a proof that it is inconsistent, then it still was and still will be a clear, comprehensible and not confusing set of sentences.
Quoting Hanover
That's not what @Michael meant. He didn't mean 'Gloobelfooble' as a name of a sentence or as a variable ranging over sentences, but rather as just a meaningless expression.
I reference it because it is rigorous, captures a common and basic intuition I share with logicians and mathematicians, and it seems the most prevalent account so that my remarks are understood in a context people know about.
But I don't claim it is the only credible account or even the best one. And of course, the intuitionist notion of model differs from the classical account, and the intuitionist notion fascinates me as do all the alternative logics though I wish I had more time to study them.
Quoting Banno
I haven't read that article in full, so I'm only off the cuff here:
Of course, models are relative to languages. "for all models" has as tacit that there is a particular language L that is addressed, so really it is, "for all models for language L".
"the admissible models for a language".
There is the notion of admissible models of set theory, but I am not familiar with a general notion of admissibility.
SEP says: "each model domain is a set, but the actual world presumably contains all sets, and as a collection which includes all sets is too large to be a set (it constitutes a proper class), the actual world is not accounted for by any model (see Shapiro 1987)."
Of course, every domain is a set, and there is no set of all sets, so there is no domain that has all sets as members. But I don't know what it means to say "all sets are in the real world". The matter raised is interesting, but I don't know enough about it. Anyway, I haven't premised anything I've said on the claim that there is a mathematical model of all of the "real world".
Quoting Banno
I don't know enough about it.
I do know (let '*' stand for 'tonk'):
From P infer P*Q, and from P*Q infer Q
So, if 'infer' is transitive then from P infer Q.
So, from any statement, we may infer any statement.
What argument is being made about that?
Yeah, I baulked at that too.
From what I understand, if we allow TONK as a rule then any statement is provable. If we define logical consequence in terms of proof, and since any statement is provable by TONK, any statement is a logical consequence of any other. TONK shows that not just any rule will do for a proof-theoretical approach.
There are subsequent developments in Proof-Theoretic Semantics...
Quoting Proof-Theoretic Semantics
But these seem ad hoc to me... I may be just misunderstanding them.
I'm not (or mustn't...) drawing any conclusions here, just trying to make some sense of what turns out to be a surprisingly varied and lively debate.
Regarding 'formal language' from, 'Notes on Metamathematics' by William Goldfarb:
"A formal language is specified by giving an alphabet and formation rules. The
alphabet is the stock of primitive signs; it may be finite or infinite. The formation
rules serve to specify those strings of primitive signs that are the formulas of the
formal language. (A string is a finite sequence of signs, written as a concatenation
of the signs without separation.) In some books, formulas are called well-formed
formulas, or wffs, but this is redundant: to call a string a formula is to say it is
well-formed. A formal language must be effectively decidable; that is, there must
be a purely mechanical procedure, an algorithm, for determining whether or not
any given sign is in the alphabet, and whether or not any given string is a formula."
What I said is right along those lines. His account is in context of mathematical logic, but perhaps it generalizes with any needed tweaks.
If an epistemological theory leads us to think we don't know anything, isn't that just evidence that the theory has gone astray?
You know you are reading this.
How do you state that contest?
Roughly, Davidson inverts Tarski, keeping truth constant in order to derive meaning. He hoped to translate natural languages into extension formal language - basically first-order calculus. So truth-theoretical.
Wittgenstein dropped meaning in favour of examining what we are doing. This was loosely linked to introduction rules by Gentzen, roughly that they give logical connectives meaning. Proof-theoretical semantics followed.
But if I had the answer, I'd have that PhD.
I do understand the difference between taking 'true' as defined and taking 'true' as primitive.
Good post. This is a clear representation of the variety of univocity that I would oppose. I don't think we have to pick a poison. They are different but not altogether different. I only would have been happier if you had said, "...[they] differ so greatly they don't deserve the same name."
But this should be bookmarked as a jumping-off point for a substantive thread. [Do they deserve the same name?]
I already did previously. There's no contradiction in that case because it is vacuously true. But that's a bit of a bait and switch.
I guess my gripe is that I would expect any statement to be logically consistent under all values of the antecedent. The fact that logical inference ignores it because under one of the values of the antecedent it does make sense is all very counter-intuitive to the point I feel the need to reject it.
If I uttered: "If it is raining then it is not raining." ... If formal logic is "mappable" onto ordinary language, then you should be able to infer "oh okay, it's not raining." But no one speaks like that and no one would make such an inference. At least, no one would consider such an "argument" "valid." That being so, while I would prefer there not to be equivocal definitions of validity, it appears that there are, one formal, the other informal.
You're not talking about validity there, you're talking about the truth of an "if ... then ..." premise.
In propositional logic "if Michael is American then he is the President" is true, but in "ordinary" language it isn't.
Your real concern is with material implication.
If "P" is false then "If P then Q" is true.
I am not American, therefore, "I am American" is false, therefore "If I am American then I am the President" is true.
And yet here we are. Turns out folk do speak like that.
If this thread is not long enough, then it is long enough.
The word 'valid' is equivocal.
There are different definitions and understandings of the word 'valid'.
One of the formal definitions of 'valid' is a common one. That definition is not equivocal.
Meanwhile, among the everyday senses of 'valid', which has a definition that is not equivocal?
What is the edited conditional?
The conditional is "If Michael is American then Michael is president".
That conditional is true since "Michael is American" is false. There's no need to mention any aspect of greatness of citizenship, nor "maybe", nor "end up".
@Banno maybe you have thoughts about that too?
I disagree with regards to ordinary language, because we ordinarily reject contradictory premises for sake of avoiding contradiction; we naturally reject A whenever A implies (B And Not B) for any proposition B.
One isn't inferring Not A in such cases, rather one is establishing a consistent set of premises for subsequent inferencing. This is reflected by the fact that the case you find to be problematic, is actually an alternative axiom used in the definition of negation in intuitionistic logic.
So you would say that a reductio ad absurdum is not an inference in the proper sense?
Originally I wrote "if I am American then I am the President", but I changed "I" to "Michael" to avoid any debate about indexicals.
It is an inference in the syntactical sense of implication, but not in the semantical sense of implication as ordinarily used by scientists and legal practitioners who are in the business of inferring facts as opposed to uninterpretable sentences.
In a consistent deductive system , If the sign "Not A" is either taken to be an axiom, or is inferred as a theorem, then it means that the sign "A" is non-referring and hence meaningless in that it fails to denote any element of any possible world among any set of possible worlds that constitutes a model of the axioms. By symmetry, the same could be said of the sign "Not A" being meaningless if A is taken as an axiom, but by model-theoretic traditional the sign A is said to not denote anything in a model if ~A is provable.
For instance, let the sign "A" denote the proposition that the weather is wet in some possible world. If "A" is deductively assumed or proved, then A is a tautology, meaning that the logical interpretation of "A" is stronger than being a mere possibility and denotes the weather being wet in all possible worlds. On the other hand, if "~A" is provable, then no possible world is wet, in which case the sign "A" fails to refer.
In conclusion, A and ~A can only both be meaningful if they both stand for possible but unnecessary states of affairs, in which case neither are provable. So the OP's problem isn't a problem, because the signs of the implication A --> ~A aren't simultaneously meaningful.
A --> NOT A says: all worlds that satisfy A also satisfy NOT A.
But in Kripke semantics, a world satisfies NOT A if and only if it doesn't satisfy A. So the set of worlds S that satisfy this condition is empty. A forteriori, there aren't any worlds in S satisfying A. Therefore
NOT A is true, and A refers to nothing.
This goes back to my pedantry comments. I can't see how it could matter if we designated a name for that special class of modus ponens described in the OP, where it is structurally consistent with modus ponens but is logically inconsistent. This thread strikes me as more of a primer in formal logic nomenclature than in logic qua logic.
Still, it also appears that the conclusion is an unwarranted logical leap from the premises, so that is why I think there might be room to argue that the argument is not valid according to some informal definition of logical validity. That is to say, the conclusion doesn't follow or doesn't lead to the conclusion. I understand that this is not the definition of validity formally speaking.
The conclusion logically follows, as has been explained many times.
P ? ¬P
? ¬P ? ¬P
? ¬P
P
? ¬P
Or more simply:
¬P
P
? ¬P
The only issue is that people misunderstand what "P ? ¬P" means. It doesn't mean what "if ... then ..." means in ordinary English.
If it is raining then it is not raining
Therefore, either it is not raining or it is not raining
Therefore, it is not raining
It is raining
Therefore, it is not raining
Or more simply:
It is not raining
It is raining
Therefore, it is not raining
If it is raining then it is not raining.
Therefore, it is not raining.
Who in there right mind would conclude the conclusion from the premises in a conversational setting?
(It is a different argument from the original argument in the first post).
They probably wouldn't, because the grammar of ordinary language does not follow the rules of propositional logic.
In propositional logic, the following is a valid argument (specifically, it's a tautology):
P ? ¬P
? ¬P
In propositional logic, the following is a valid argument:
P ? ¬P
? ¬P"
Exactly. And if someone wouldn't make such an inference, I am suggesting that that is a logical mistake of some sort, which is a way of saying the argument is not valid.
There's no logical mistake? It's just the case that "if ... then ..." in ordinary English doesn't mean what "?" means in propositional logic.
A ? B means B or not A
If I punch you then you will cry does not mean you will cry or I won't punch you.
This uses the inclusive OR which is also not so standard in English.
If that's been said once, it's been said a thousand times... which is not once.
I consider the logical conditional a performative, as exists in an algorithmic way.
Consider, "If X = 4, then Y = 7." That is , if we set X at 4 then Y is set at 7. We could not program if we could not make such statements. If P then Q results in the occurence of Q when P is the case necessarily. I consider this an analytic operation and consistent with computer logic in programming (as far as I know about programming).
I consider the linguistic conditional not an indication of what is or what will be, but a hypothetical counterfactual that does not indicate, but hypothosizes. Because it does not indicate, we don't speak in the indicative mood, but in the subjunctive, as in what we wish, hope, or hypothesize about.
As in: "If I were President, I would lower taxes." This is not represented as P -> T. That would overstate the meaning of my speculative statement. Note the "were," not "was." This is a counterfactual (it hypothethesizes an antecedent that did not occur), not a logical conditional.
"If I was President, I lowered taxes" makes more sense as a formal conditional.
If I was President, I lowered taxes
I was President
ergo I lowered taxes
But not:
If I were President, I would lower taxes
I were President
ergo I would lower taxes
What does it mean that I were President versus I was President? I think the meaning is critical in changing from the formal indicative conditional to the non-formal linguistic subjunctive conditional.
My thoughts at least.
I know Banno; I am not disagreeing with the formal validity of that argument.
Quoting Banno
I don't disagree with that either. But the argument A ? ~A ? ~A clearly does not translate into natural language very well (I don't think there is any way to translate it in a way that renders the translation sensible and "logical"). And yet, the argument is valid formally speaking.
Michael suggested that the argument is not sound in ordinary language. I think he may be right. However, even arguments that are not sound can still be valid such that we can understand how the speaker reached their conclusion (though we may point out to them that such-and-such premise is not true). For example, if someone argued:
1. P
2. P?Q
Therefore, Q.
We might correct them, "well, actually ~Q." "Your reasoning is spot on and logical, it just happens to be that ~P, so while your reasoning is valid, the argument you presented is unsound."
On the other hand, "If it is raining, then it is not raining, therefore it is not raining" sounds like an unwarranted leap that is not logical when we consider it in an informal way. The problem isn't just that the initial premise is unsound (within an informal context); the problem is that the argument just doesn't make sense and is not logical, so soundness aside, that is why I call it "not valid" informally.
Yes, there is a difference between an unsound argument that arises from an incorrect fact as opposed to one that arises from a contradiction.
- If I go to the store, I will buy milk, I went to the store, so I bought milk. That's true, unless I forgot to buy milk.
- If I go to the store, I will not have gone to the store, I went to the store, so I didn't go to the store. That statement is never true regardless of what I do. The reason it's never true is because "If I go to the store, I will not have gone to the store" is logically equivalent to "I did not go to the store."
SO you want to introduce a new form of validity, that depends not on the explicit structure of the argument but on your intuition. Ok.
The formal meaning of negation in intuitionistic logic refers to the syntactical inconsistency of the negated sign, rather than to a purported semantic counterexample denoted by the negated sign. Classical logic inherits the same meaning of negation from intuitionistic logic, except for infinitary propositions that appeal to the Law of Excluded Middle, which have no scientific or commonsensical application. So we should stick to discussing negation in intuitionistic logic, before proceeding to other formal logics such as affine linear logic, whose concept of negation is closer to ordinary use. In such cases (A --> Not A) --> Not A is not derivable, corresponding to the fact that Not A obtains the same semantic status of A.
But can we elucidate the meaning of (A --> Not A) --> Not A in the systems for which it is valid, by appealing to the mutually exclusive states of the weather? Suppose that a weather forecaster said "It is raining in Hampshire therefore it is not raining in Hampshire". Jokes about the english weather aside, wouldn't you assume that they were talking about anything apart from the weather in Hampshire? in which case your abstaining from assigning a meaning to their words would resonate with the formal meaning of negation in intuitionistic and classical logic.
As for formalities,
(A --> ~A) --> ~A is little more than the obvious identity relation ~A --> ~A, due to the fact that ~A is definitionally equal to A --> f , where f denotes absurdity. So we at least have
(A --> f) --> ~A
But the only means of obtaining f from A is via the principle of explosion (A And ~A) --> f. And so it is sufficient that A implies ~A.
(A --> ~A) --> ~A
And since the converse direction is immediately true, we could in fact define the negation of A to be the fixed point of the expression X => (A --> X) that Haskell programmers call a Reader Monad.
~A = A --> ~A
~A = (A --> (A --> (A --> ..... ) ))
which serves to highlight the meaning of Negation As Failure (NAF); A proof of ~A amounts to a finite proof that the right hand side doesn't converge, which represents an infinite failure to prove A by random search. But if we haven't managed to prove either A or ~A using our available time and resources, then we are at liberty to declare ~A by decree and reason accordingly, in which case ~A serves to nullify any hypothesized A by turning it into ~A, so as to ensure consistency with our failure to decide the issue, at least for the time being...
Our logical intuitions are basic, or foundational for doing logic, much in the way that having a functional ear is foundational for making a musical symphony.
One could argue that P->Q and P together implies not-Q, but translating that into natural language with the conditional spoken as an "if...then..." (or A and not-A therefore A and not-A) will be very difficult and I would say impossible, and that's because logic relies on meaning maybe just as much as meaning relies on logic.
All that to say that, at least informally A->notA therefore not-A may not be valid after all if our starting point is a set of meaningful natural language propositions.
That doesn't imply that formal logic is merely "academic" because it clearly has application to fields like computer science and mathematics.
But it may imply that some definitions we use in formal logic may be reviseable or at least more fungible then we previously thought.
That's really sad.
P
P->Q
Therefore Q.
Or that this argument is logical:
All men are mortal.
Socrates is a man.
Therefore, Socrates is mortal.
Or that this argument is logical:
If it rained yesterday then the lake is swollen today.
It rained yesterday.
Therefore, the lake is swollen today.
You have placed yourself outside of rational discourse.
Cheers.
Can an underivable argument be valid? (I suppose you would say "yes" because the "underived" (unconditioned) constituents of the argument are mere possibilities).
I would think many people would apply a truth table to the argument (A->notA therefore not-A), as I did, and see based on that, that the premise is only true when "not-A." Maybe "infer" is too strong a word for the conclusion of not-A.
The conclusion does not seem to "follow" or be a "logical" conclusion when we apply the argument to ordinary natural language.
So I guess what I'm wondering is whether an underivable (or meaningless) argument may be regarded as logical? Or are soundness and validity insufficient for a logical argument? Or is meaning related to soundness?
It seems, to me, as though what is meant is critical to determining whether an argument is logical.
Your post is hopelessly confused because you don't know the basic concepts.
If you would just read a little bit in an introductory textbook, in print or online, you would know.
All my remarks pertain to ordinary usage such as found in basic textbooks:
There is no such thing as a "derivable argument" or "underivable argument". The expressions "derivable argument" and "underivable argument' make no sense.
An argument is a pair, with the first component called 'the set of premises', and the second component called 'the conclusion'. An argument is valid or invalid, and sound or not sound. There is no such thing as an argument being "derivable" or "underivable".
Derivability pertains to proof. An argument is not itself a proof. An argument is a pair, with the first component called 'the set of premises' and the second component called 'the conclusion'.
What is underivable or not, is a conclusion from a set of premises. And that pertains to proof.
(1) Example:
set of premises:
{P, Q, R -> S, ~(P & R)}
conclusion:
S
That is an argument (it happens to be invalid). Just a set of premises and a conclusion. It's not a proof.
(2) Example:
set of premises:
{Q -> R, S, P, P -> Q}
conclusion:
R
That is an argument (it happens to be valid). Just a set of premises and a conclusion. It's not a proof.
A proof (in Hilbert form) is a sequence of sentences such that each sentence is a premise or follows by an inference rule from previous sentences in the sequence:
(3) For example (with the applications of the rules annotated):
1. P -> Q (premise)
2. Q -> R (premise)
3. P (premise)
4. Q (from 3, 1 by modus ponens)
5. R (from 2, 4 by modus ponens
That is a proof. It's a proof of R from the set of premises {P, P -> Q, Q -> R}. From that proof, we establish that the argument (2) is valid, since the proof has premises only from those of (2) (we didn't need to use the premise S, by the way) and the last line of the proof is the conclusion of (2).
I don't know what that is supposed to mean.
An antecedent is the part of a conditional that comes before '->'.
The argument under discussion:
A -> ~A
A
therefore, ~A
The only conditional there is:
A -> ~A
Its antecedent is:
A
There are two values we can assign to A: true, false
If A is false, then A -> ~A is true.
If A is true, then A -> ~A is false.
But A -> ~A is consistent in either case. It does not prove a contradiction, and it is satisfiable, since there is an assignment in which it is true, viz. the assignment that assigns false to A.
"The fact that logical inference ignores it because under one of the values of the antecedent it does make sense"
Ignores what?
A -> ~A doesn't make sense to you. But we didn't say A -> ~A makes sense only when A is false. A -> ~A makes sense whether A is true or A is false.
You're mixed up as you don't know the basic concepts. Reading just a little in a textbook in the subject would help you.
No sentence that proves a contradiction is valid. And no set of sentences that proves a contradiction is satisfiable.
And a sentence or set of sentences proves a contradiction or does not prove a contradiction irrespective of any assignment of truth values. That is, if a sentence proves a contradiction, then there is simply no assignment in which that sentence is true. And if a set of sentences proves a contradiction, then there is no assignment in which all the members of the set are true.
Quoting Benkei
Again, if a set of premises proves a contradiction, then there is no assignment in which all the premises are true.
But, I guess what you mean is this:
Consider all and only those arguments in which the conclusion is not contradictory.
Okay, say an argument is an N-argument if and only if its conclusion is not contradictory. And say an argument is an S-argument if and only if its set of premises is satisfiable.
So "what happens"?
Any argument is an N-argument if and only if it's an S-argument.
Yes:
Df. ~P stands for P -> f
where 'f' is primitive.
But, just to note, that can be a definition in classical logic too.
Also, TonesInDeepFreeze, an argument where all the premises are false and the conclusion is false would necessarily be valid; is that correct?
I was thinking of:
P->not-Q
not-P
Therefore,
not-Q.
Assuming that all the premises are false and the conclusion is false, the argument must be valid. Is that correct?
(1) You say "in a consistent deductive system" but your remarks wouldn't apply to ordinary sentential or predicate systems, but rather, more specifically to modal systems. So, your remarks don't obtain as to deductive systems in general.
(2) With ordinary models for modal propositional logic, sentence letters themselves are members/not-members in worlds. But that ~A is an axiom or theorem doesn't entail that A is meaningless in any given model. Rather, as in classical semantics (but by more complicated considerations) A is false if and only if ~A is true.
(3) I don't know your definition of 'tautology' in modal logic. In propositional logic, a sentence is a tautology if and only if it is true in all models. I am not familiar with a notion in modal logic that being assumed makes the sentence a tautology.
(4) You said [paraphrase:] A stands for "wet in some world", then assuming A yields "wet in all worlds". That would be (where 'p' for possibly and 'n' for necessarily):
pA -> nA
And that is not generally (if at all) considered a validity.
[EDIT:] By the way, (A -> ~A) -> ~A is intuitionistically valid, perforce so is ((A -> ~A) & A) -> ~A.
As the argument forms are intuitionistically valid:
{A -> ~A}, therefore ~A, perforce {A -> ~A, A}, therefore ~A.
Classically by classical models; intuitionistically by intuitionistic models.
It makes sense in the sense of having a truth value.
Quoting NotAristotle
No, quite incorrect. Egregiously incorrect. That you say that shows that you haven't paid attention to the numerous explanations given in this thread, let alone that you haven't paid attention to the most basic articles available on this subject.
Quoting NotAristotle
Not correct at all. It goes exactly against the definition of 'valid'.
Where can one read an account of ordinary modal logic, ordinary intuitionistic logic or basic Kripke semantics in which that is the case?
Of course.
But the ordinary formal definition is itself not equivocal. It is definite. It gives an 'if and only if' with a definiens in which all the terms are themselves defined back to primitive rubric.
Meanwhile, there are other formal definitions that differ from the ordinary formal definition. And they may also be not equivocal, though probably more complicated than the ordinary definition.
Meanwhile, there are different informal senses. If there is one in particular that you propose as being definite enough to avoid the kind of subjectivity equivocation in everyday discourse, then you're welcome to state it.
Having a false premise and a false conclusion does not in and of itself make an argument invalid. You have forgotten or did not understand the definition.
Whatever "structurally consistent" means there, a clear and simple way to say it is: The argument is and instance of modus ponens.
And no instance of modus ponens is inconsistent. What are consistent or inconsistent are sets of sentences. What is inconsistent in the argument is the set of premises.
Quoting Hanover
This thread strikes me as having posters in it that are commenting on formal logic while knowing virtually nothing about it.
The original thread question would naturally be taken by many people to pertain to ordinary formal logic. But it can also be taken to pertain to other informal contexts, including everyday speaking and reasoning, and also can be discussed in context of alternative formal logics.
But when answers were given in terms of ordinary formal logic, certain posters commented as to the formal logic, while knowing virtually nothingabout it. Thus, correct explanations of it are exemplary meaningful, informative and generous posting.
I've said it maybe fifty times in this forum: Ordinary formal logic with its material conditional does not pertain to all contexts. But that is not a basis that one should not say how ordinary formal logic handles a question and not a basis that one should not explain ordinary formal logic to people who are talking about it without knowing about it.
Your question is answered by looking at the method of truth tables.
That is in chapter 1 of any book in 'Logic 1'.
I don't mean examples of valid arguments, I am referring to the definition itself.
I was re-composing my post while you were posting your reply.
Meanwhile, do you have any thoughts about offering your own unequivocal definition of 'valid argument'?
They don't imply a definition of validity.
If you read chapter one, you'll understand that we have:
(1) a definition of 'valid argument'
(2) a definition of 'interpretation' such that an interpretation assigns truth values to sentence letters
(3) a stipulation for how the truth value of a compound sentence is reckoned per an interpretation, so that for any interpretation and any compound sentence, we can reckon the truth value of that sentence.
(4) To determine whether an argument is valid or not, apply (1), (2), (3).
And, if I recall, I even showed in this thread how to set that up with truth tables.
/
In a nutshell: I defined 'valid argument'. And earlier in this thread, I defined 'true in an interpretation' for compound sentences. So apply the definition of 'valid argument' by considering the truth values of the compound sentences per each of the interpretations.
What relevant rules? What makes a rule relevant? Whose rules? What if people use different sets of rules from one another? What if the rules are unclear or ambiguous?
What is the meaning of a sentence? How do we unequivocally, let alone objectively, determine the meaning of sentences? What theory of meaning? What if people take different meanings of sentences from one another? What if someone takes 'valid' to mean causal connection but they don't take causality and meaning to be the same? What about people who consider 'valid' to require that all the premises are true?
Of senses of 'valid' different from yours, are they wrong? Or can there be different reasonable senses of 'valid'? The ordinary formal sense cannot be among those different reasonable senses?
The meaning of the premise and conclusion depends on the expressions used (I guess this definition isn't unequivocal as it would only apply to ordinary natural language, not to formal logic). I don't know any theories of meaning so I can't answer that. If the meanings differ, then I'm not really sure what the result would be, seems like communication is out the door let alone logic if we can't agree on the same meaning of words and sentences.
What is an example of rule that if it weren't followed then the conclusion would be different? Different from what?
The rules of formal systems in mathematical logic and computing are not just clearly communicated, but they can be checked algorithmically for correctness of application.
What are some examples of your rules that are communicated differently?
What if two people both like the same rules, but have different intuitions as to whether they're being correctly applied?
How are your rules for your propositional logic different from those in ordinary formal logic?
Of course, the meanings depend on the expression. But I'm asking, given an expression, what determines its meaning, or its meanings?
People disagree about meanings often. We can't do logic with expressions because people disagree about their meanings?
You say a relevant rule is on such that if all the premises are true then the conclusion is true.
That is the ordinary definition of 'valid' in formal logic.
And it doesn't say anything about meaning other than truth and falsehood.
So what are you adding to the ordinary formal definition or how are you disagreeing with it?
You're circular. You say that an argument is valid only if the rules used are relevant, but also rules are relevant only if they are truth-preserving. But truth-preserving is the same as valid.
I don't understand the second question.
Third question answered as correctly used rules is defined.
I don't know the difference between propositional logic and ordinary formal logic so I do not know how to answer this one.
The meaning of an expression depends on what the speaker intended by it - natural language I would think would go along way in dissolving confusion over what is meant.
Right, where there is disagreement over a meaning, that meaning is not well-formed and not suitable for logical operations. I would expect something like that to be true for your definition of validity as well.
I guess I agree with the ordinary definition of valid in formal logic. That is not the definition you cited earlier in the thread - the definition that I am suggesting an alternative to.
I do not see truth preservation as synonymous with validity; I defined validity as rule following; a rule is followed correctly if it preserves truth; I didn't define validity as truth-preserving. Truth preservation is a consequence of validity, namely, following the relevant rules correctly.
(2) A set of premises can prove more than one conclusion. So what is "the" conclusion that "should be"?
(3) I wrote: "What are some examples of your rules that are communicated differently?"
I meant: What are some examples of your rules that are communicated clearly?
(4) How do you know what the speaker intended? What if there is not a particular speaker? People disagree about what speakers intend often. And people misunderstand and disagree as to what was said often. You say your answer goes a long way to making clear what the meanings are. Well, yes, often it's pretty clear what a speaker intends, but not so often that it would determine which arguments are valid, since too often it is quite unclear what was intended.
(5) Yes, if the expression is equivocal, then either we reject it of choose one of the candidates for its meaning. But rather how would we determine in an objective way whether there is or is not equivocation and, if so, which candidate to choose?
Formal logic does not presume to know how sort out many of the difficulties in everyday speech; only that if we are given sentences that have a formal relation, we can determine validity of arguments. That is the very point. Suppose someone writes words from a language I don't know:
tarabalu bock meras dan pelrere bosoundo tam.
erofereht, pelrere bosoundo garom
As long as I know that 'dan' means 'and' and 'erofereht' means 'therefore', I can say, "On the assumption that the foreign language expressions are declarative statements, then the argument is modus ponens, and, as an instance of modus ponens, it's valid".
That is, if I know what are the connectives and how we reckon the truth of compound statents, I can tell you about the validity of the argument.
But you have not given such forms, but rather, we have to know the meanings first before determining validity.
So, your offer requires sorting out all the problems about 'meaning' much more than the formal method does.
(6) What definition of 'formal logic' have I given that contradicts any other definition I've given?
(7) An argument form is truth preserving if and only if, for every instance of the form, there is no assignment of truth values such that the premises are all true and the conclusion is false. That is, mutatis mutandis, the same definiens as for 'valid argument'.
(8) Yes, you defined validity as rule following, but then you defined proper rule following as being truth preserving. That's your circularity. You say, a rule is proper only if it is truth preserving, and an argument is valid only if it uses only proper rules. But truth-preservation is validity.
Notice that you didn't say anything about the meanings of P and Q, even if they were translated to a natural language.
Rather, you mentioned only the meaning of the conditional (and I would mention also the meaning of 'not'.)
That's an example of formal logic. We stipulate how the connectives determine the truth or falsehood of compound statements depending on all the assignments of the sentence letters or atomic statements, without having to consult otherwise as to the meanings of those atomic statements.
Nope. You say that your notion of validity is based on proper use of rules, but your notion of proper use of rules goes through the notion of rules being truth-preserving. But truth-preserving is validity, even as you defined yourself "If the premises are true then the conclusion is true".
That is equivalent to saying: A rule is correctly used only if application of it never leads from true premises to a false conclusion.
And "never leads from true premises to a false conclusion" is validity.
You've managed to define validity in terms of correct rules and correct rules in terms of validity.
2. Provided that a set of all conclusions follows the rules correctly and is exhaustive of all such conclusions, that set encompasses all legitimate conclusions.
3. logical operators.
4. We would have to ask the speaker to clarify.
5. Noted, let's set aside questions concerning meaning; the second definition may have more problems then I can resolve.
6. Okay.
7. So then "the truth of the premises guarantees the truth of the conclusion" is the same as "there is no interpretation (assignment of truth values) such that the premises are all true and the conclusion is false" ?
8. So I think what I am trying to say is that the definition of validity is following the rules correctly. And that following the rules correctly is defined by rule-following that results in truth preservation. Such that, truth preservation is a consequence of rule following, and it is the rule following itself that is responsible for the validity. In other words, the premises themselves don't guarantee the truth of the conclusion, rather the following of the rule(s), given that the premises are all true, is what guarantees the truth of the conclusion. Put another way, truth preservation does not make the argument rule-following, but rule-following is what makes the argument truth preserving. (Truth preservation does make the rule-following "correct.") Not sure if that totally makes sense.
You use "must"; is that in addition to "is"? Example:
If Bob is smart then Bob knows English history.
In ordinary sentential logic, that is false in an interpretation if and only if Bob is smart and it is not the case that Bob knows English history.
What is your sense of "must"?
Is it that there are no possible circumstances in which Bob is smart and Bob does not know English history? That is, there are no interpretations in which Bob is smart and Bob does not know English history? Or does "must" mean something else for you?
As I shared already, note the difference in ordinary formal logic:
Here 'P' and 'Q' are variables ranging over sentences:
P -> Q
is true in a given interpretation if and only if either P is false in that interpretation or (inclusive 'or') Q is true in that interpretation.
and a different notion:
P entails Q
if and only if there is no interpretation in which P is true and Q is false.
Symbolized :
{P} |= Q
(2) Do you mean, "as long as the rules are followed (applied) correctly" or do you mean "as long as the conclusion follows correctly from the rules"?
And I asked you already, what is "the correct conclusion" when there may be many correct conclusions? And you skipped answering
And "exhausts"? A lot of different conclusions may follow from a given set of premises. How would you know that you exhausted them? I asked that already, and you skipped answering.
(3) Connectives are not rules. Rather, we have rules for connectives. What are your rules for the connectives?
(4) What if the speaker is not around to clarify? What if the speaker is too confused himself? What if there's not a particular speaker but rather the statement is a general public statement? I asked you about that already, and you skipped answering.
(5). Okay, so take out meaning. Now, how is your offer different from ordinary formal logic?
(7) Yes, they are equivalent. The former is a less rigorous way of saying the latter.
(8) You say, "the definition of validity is following the rules correctly. And that following the rules correctly is defined by rule-following that results in truth preservation."
And, as I've shown, that is circular. You have not given an answer to that other than eliding that validity and truth-preservation are the same.
(9) Whether a rule is truth-preserving or not is not based on whether it happens to be a rule, but rather on the fact that it is truth-preserving. One can make any rule one wants to make, and it will or will not be truth-preserving not on the basis that one says, "It's a rule" but rather on the basis that any application of the rule is truth-preserving.
So, still in your situation:
To define 'valid' (truth-preserving) you appeal to correct rules. But what is a correct rule? Well, it's one that is valid (truth-preserving). That's circular.
Let's compare with a non-circular approach in ordinary formal logic:
First, define, 'is an interpretation'.
Second, define 'is true in an interpretation' and 'is false in an interpretation'.
Third, define 'valid argument' without mentioning inference rules, but only mentioning interpretations and true and false.
Fourth, we state rules that are correct in the sense that they provide for only valid arguments.
Meanwhile, I hope you're looking up the method of truth tables.
My point is that we know that If P then Q, where P = A and Q = not-A, implies a contradiction where P is true because Q will be true and both A and not-A will be the case. It is counterintuitive to assert that "if it rains then it doesn't rain" and "it rains" therefore "it doesn't rain" is a valid argument. So formal logical inference appears to ignore the obvious rejection any normal person would have with the natural language sentence without going through a logical proof (the resulting logical contradiction is rather obvious).
So we know the premisse is unsound but it seems to be of another order when it's unsound due to a logical contradiction then say because it fails to take into account the fact there are additional causes for a consequent to happen (any time really where correlation isn't causation). I guess "formal" in formal logic really is the main point of that system.
2. Rather than a correct conclusion, all we need are conclusions that follow the relevant rules, any and all such conclusions are legitimate.
3. I refer to connectives as rules.
4. Then we are out of luck.
5. I drop the truth preservation condition for validity.
8. If we drop the truth preservation part of the definition, it is not circular. An argument is valid where it follows the relevant rules. Period. I don't think it is necessary for me to stipulate that a rule be followed "correctly," just that it be followed.
"If P then Q" means "not P or Q".
"If A then not A" means "not A or not A".
"not A or not A" is not a contradiction.
Quoting Benkei
"If it rains then it doesn't rain" means "it doesn't rain or it doesn't rain".
So the argument is:
P1. it doesn't rain or it doesn't rain
P2. it rains
C1. therefore, it doesn't rain
Notice that P1 and P2 cannot both be true. If P1 is true then P2 is false; if P2 is true then P1 is false.
C1 is irrelevant. It could be anything, as per the principle of explosion.
* For argumentation, I suggest studying both informal and formal logic. Informal for practical guidance; formal for appreciation of rigor. I don't have particular texts in informal logic to recommend. For formal logic, I recommend 'Introduction To Logic' by Suppes, though I don't know whether it is available online.
* A -> ~A does not imply a contradiction.
You are not heeding the many explanations already in this thread.
(Frist though, if a sentence implies a contradiction, it does so no matter what interpretation is adduced.)
Next, having to repeat this again:
A contradiction is a sentence of the form P & ~P (or P and ~P both as lines in a proof).
A sentence or set of sentences is inconsistent (i.e. contradictory) if and only if it implies a contradiction.
A -> ~A is not contradictory. But {A, A -> ~A)} is contradictory.
* My guess is that for the most part, people do take the material conditional and "If it rains, then it does not rain" as counterintuitive. The point you make about this has been recognized over and over and over in this thread.
But the material conditional has use in mathematics and computing ('P->Q defined by '~(P & ~Q)'). It doesn't have to be intuitive to everyday speakers to be make sense to many (most, I surmise) mathematicians, logicians, and modern philosophers in the field of logic, and in other formal contexts. That people around town don't countenance the material conditional doesn't entail that it is not the case that it is useful and makes sense in other contexts.
Meanwhile, there are other alternative formal definitions for "if then".
And there are various competing notions of "if then" in everyday use.
* It's been explicated, in detail, over and over in this thread that, yes, there is a difference between a set of premises that has one or more false members but is not inconsistent and a set of premises that is inconsistent.
Quoting Benkei
I don't know what that is supposed to mean.
Again, the distinction I adduced:
P -> Q
is true in a given interpretation if and only if either P is false in that interpretation or (inclusive 'or') Q is true in that interpretation.
and a different notion:
P entails Q
if and only if there is no interpretation in which P is true and Q is false.
It seems you take P -> Q to mean "P entails Q".
Quoting NotAristotle
What are the "relevant rules"? What are your rules?
Quoting NotAristotle
That's not the usual notion, but let's see what we get from it:
As rules, what are these connectives?:
~
&
v
->
<->
Quoting NotAristotle
So, with your offer, we can't examine validity for a vast amount of everyday argumentation.
Quoting NotAristotle
You do now, after I showed you that your definition of validity in terms of 'relevant rules' in terms of truth-preservation is circular.
So, what now are your definitions of 'relevant rules' and 'valid argument'?
8. If we drop the truth preservation part of the definition, it is not circular. An argument is valid where it follows the relevant rules. Period.
Not period. Again, what is your defintion of 'relevant rule'?
Quoting NotAristotle
So, for you, an argument is valid if it follows some rule or another, irrespective of whether the rule is truth-preserving or correct in any aspect?
Anyone can state any rules they want, including ones that are not truth-preserving, and even ones that permit contradictions from a set of satisfiable premises. Pretty much logical anarchy.
This is not responsive to anything I've said. I know you want to keep saying over and over what formal logic dictates, but my post was referring to how ordinary language handled conditionals and how that was distinct from formal language. I didn't suggest we should jettison formal logic or that it lacked value because it was distinct from the ordinary way we speak.
I really like this book for informal logic:
https://www.cambridge.org/core/books/arguments-about-arguments/27835C37D9CEDFA6BE9EFD11CA2DA5A3
It is exactly responsive to your post. You said:
Quoting Hanover
So, I addressed that.
And I didn't say that you said that formal logic lacks value.
As to the difference between the material conditional and informal notions of the conditional, that point has been gone over and over and over. If there is something more you want to say about, no one is stopping you.
And I gave you information about modus ponens, consistency and arguments too, to clear things up for you after your confused comment about them.
That table of contents looks pretty good. But I wonder whether it is best as a very first book to read.
Very inviting.
Why are you telling me that no one is stopping me?Quoting TonesInDeepFreeze
Thank you for reminiscing, but that's not what my last post was about.
To emphasize that, for example, nothing I've said is a barrier to you adding whatever else to the subject you might have to add other than what has already been gone over.
Quoting Hanover
Yes, I recall that you ignore good information you need to remedy your confusions.
Quoting Hanover
Whatever your post was "about", it included a confusion regarding modus ponens and consistency.
Quoting Hanover
All three of the above are incorrect.
A -> ~A is not contradictory.
A is true or false depending on a given interpretation.
~A in that argument is the conclusion.
2. I take your question to be what would a rule be, how is it defined? I would define a rule as a member belonging to a set that exhausts all "truth possibilities." I would add that the following of a rule may not result in a contradiction.
A rule relating two different variables would have (I think) 15 possible truth configurations. The rules must at least enable all those possibilities to be instantiated (though perhaps it may exclude possibilities that are necessarily contradictory).
3. "Some proposition is not the case"
Both propositions must be true
Either proposition must be true
If the one proposition is true, so must the consequent proposition
Both propositions are either both true or both false.
5. Valid argument = following the rules, where rules are defined as those operations that enable each truth possibility to be instantiated but that do not result in a contradiction by following that rule.
8. Not logical anarchy; the rules must enable all truth possibilities to be instantiated except that the rule may not result in a contradiction if it is followed.
This way of defining validity may be preferable because it deals with cases such as A->not-A therefore Not-A that are intuitively illogical; such an argument does not involve the following of a rule, and so it is not valid.
Similarly, A, A->not-A therefore not-A another intuitively illogical seeming argument would not be valid because the following of the rule results in a contradiction.
What on earth would the meaning of P -> Q be such that it differs from not-P or Q? It's not like P -> Q is in any way a natural language sentence. We're already relying upon substitution (with the variables P and Q) so what prevents us for substituting "not-P or Q" for "P-> Q" when they are logically equivalent?
It has the same meaning in truth-functional terms, but it does not have the same meaning qua modus ponens (or modus tollens). See, for example:
Quoting Leontiskos
Quoting Michael
Only if "or" is used inclusively. The ordinary language use is often exclusive, or ambiguous.
So this doesn't help.
...except perhaps to provide an example of how formal logic works out the ambiguities of natural language. Which is part of what it is for.
A ? ¬A ? A ? ¬A
So I'm asking -- what else is there? What does the full meaning say?
I'm sympathetic to informal logic taking priority over formalization, though it gets difficult because of the inherent ambiguity in informal logic. When I say "If I touch the stove then my hand will burn" I'm not talking in terms of material implication or disjunction at all, but a causal relationship between action and event.
But cashing out that causal relationship into the formalization is not in any way easy. Which is why we usually rely upon the informal.
But once we start introducing symbols and substitution I tend to believe we're not really talking about this imprecise, though more frequently utilized, informal reasoning.
You are talking in terms of the first premise of a modus ponens, and that is what the material conditional is in many logics. If there is a difference between modus ponens and disjunctive syllogism, then there is a difference between A?B and ¬A?B. You can of course say that there is no difference and that "informal logic" and formal logic are infinitely separated, but I think that is to put the cart before the horse. Something which has nothing to do with human reasoning is not logic, and so I would say that if someone is talking about something which is wholly separate from human reasoning then they are not talking about logic (and besides that, they are not paying any attention to the historical development and motivations behind formal logics).
The difference is the shape of the little squiggly marks.
I think there's a difference and I've committed to indications for the difference -- in the recent posts substitution has been the criteria I've been using.
I don't think that makes them entirely separable.
There is no difference in terms of substitution.
(The thread already linked goes into these questions in detail. <This post> would be one example of that.)
So when we start using things like "P" to represent any proposition whatsoever this relies upon substitution. Without a notion of substitution we'd not be able to make sense of variables.
Or, what's probably a better way of stating this, informal substitution is subject to more criteria than formal substitution is. Informally substitution is usually reserved for mathematics and statements of that form -- which is what formal logic is very much like.
But since A -> ~A uses symbols it's more appropriate to call this a formal construction of material implication, which we can write the truth-tables out for and easily conclude it's valid, but unsound, as said.
I opined on some of this earlier in the thread*, but I think you are talking about the argument of the OP, not merely the sentence (A?~A).
* It depends on what we mean by "valid" and it depends on the argument for why it is or is not valid.
You could say that, but you would end up having to admit that "P does not imply Q" cannot be formalized in any way whatsoever, at least in propositional logic. See the thread that I have been referencing for this topic.
Whether or not the two expressions are semantically equivalent in a meta-logical sense depends on how one is using them. But I think your intuition is correct and defensible, namely that they are not semantically equivalent in a meta-logical sense. Trouble is, you can never prove that at the level of the object language.
"If P then Q" means "not P or Q" presumes an inclusive OR.
Fair, so I suppose I should "if P then Q" means "not-P or Q or both".
Where G is a set of sentences and Q is a sentence, "G entails Q" is symbolized:
G |= Q
I.e, there is no interpretation in which all the members of G are true and P is false.
If G is a singleton {P}, then we sometimes write:
P |= Q
I.e., there is no interpretation in which P is true and Q is false.
Then note:
P -> Q |= ~P v Q
and
~P v Q |= P -> Q [corrected in edit]
Whatever you mean by "meaning", the sentences are equivalent in the sense above.
Quoting NotAristotle
See later in this post for my reply to your supposed rules given lately.
Member of what sets? What sets are you talking about?
And what does "exhausts all truth possibilities" mean?
And an application of rule may not result in a contradiction? You said previously that you don't define 'correct' for rules. So consider the rule of conjunction-intro:
From P and Q infer P&Q.
If P is A and Q is ~A, then apply the rule to get A & ~A.
Conjunction-intro is not a rule for you now?
Quoting NotAristotle
There are 16 2-place Boolean functions.
Quoting NotAristotle
What does it mean to "instantiate" in that regard?
What we do have is this:
All 16 2-place Boolean functions are realized by certain sets of connectives. And if all 2-place Boolean functions are realized, then all n-place Boolean functions are realized for all n.
That is not regarding inference rules, but merely the definitions of the connectives and the truth evaluation of compound sentences.
Quoting NotAristotle
Those are not rules for arguments. Those are just the standard clauses in the ordinary definition for the truth value of compound sentences. (Except your use of "must", and note that you left out "must" for the first and fifth.)
And it's not clear what you mean. Do you mean "must be true" as "is necessarily true"? With modal operator n for 'necessarily', taken at face value, your formulations seem to be:
~P is true if and only if P is false. (So far, so good!)
P & Q is true if and only if both nP and nQ.
P v Q is true if and only if either nP or nQ. (And, unless you tell me otherwise, I take 'or' as inclusive).
P -> Q is true if and only if P necessarily implies Q. (?)
P <-> Q if and only if either both P and Q are true or both P and Q are false. (Good!)
Quoting NotAristotle
What does "enable truth possibilities to be instantiated" mean?
And what if a rule allows a contradictory sentence (but that itself is not a contradiction) to be derived? Note that in predicate logic there is no mechanical procedure to determine whether any given sentence is or is not contradictory. So, with your offer, there would not be a mechanical procedure to determine whether an argument did use only your rules. But, of course, we may consider such a logic.
Quoting NotAristotle
That might be helpful if you define "enable all truth possibilities to be instantiated" vis-a-vis rules.
Quoting NotAristotle
It doesn't derive a contradiction. So in what way does it fail to "enable all truth possibilities to be instantiated"?
Quoting NotAristotle
No, it doesn't result in a contradiction. The conclusion is ~A, which is not a contradiction. Yes, the premises are inconsistent, but your definition of "rule" doesn't disallow inconsistent sets of premises, only required is that application of the rule doesn't allow a conclusion that is a contradiction. The particular application you mentioned doesn't derive a contradiction. But other applications do derive contradictions. I know of no rule that disallows deriving contradictions, since rules don't disallow inconsistent sets of premises. However, as mentioned previously in this thread, one might require rules to not have inconsistent sets of premises. But the catch ... for predicate logic, it would not be algorithmically checkable to see whether a rule was applied, since it is not algorithmically checkable to see whether a set of sentences is inconsistent. Though, I suppose you mean for all your rules to be informal anyway.
What does that mean?
https://www.umsu.de/trees/#(((A~5B)~1A)~5B)~4(((~3A~2B)~1A)~5B)
A
therefore, B
is not the same argument as
~A v B
A
therefore, B
That doesn't vitiate that A -> B and ~A v B are equivalent.
Depends on what 'implies' means.
It is not the case that if P then Q
is formalized
~(P -> Q)
that's in the object language
It is not the case that P entails Q
is formalized
P |/= Q
that's in the meta-language
According to you, what is the full meaning of P -> Q?
But is there anything more to it than the difference in the shapes of the letters?
The ordinary clause is:
P v Q is true if and only if either P is true or Q is true. ('or' inclusive)
P -> Q is true if and only if either P is false or Q is true ('or' inclusive)
What specificity is lacking?
(1) An argument is valid if and only if there are no interpretations in which all of the premises are true and the conclusion is false
(2) An argument is valid if and only if every interpretation in which all of the premises are true is an interpretation in which the conclusion is true
The dispute comes down to a claim that, without justification I had applied the material conditional in the meta-language.
I have explained that in ordinary formal logic, the material conditional is used for both the object language and meta-language, especially since the meta-language itself is formalizable and logicians don't eschew the material conditional merely on account of moving to the meta-language.
I cited numerous definitions of 'valid', some using (1) and some using (2). They are equivalent, though a writer may choose one or the other and not mention the other or the equivalence, since the material conditional is indeed the sense of "if then" in ordinary formal logic, whether object-language or meta-language.
But I happened to come across a text that does mention the equivalence:
"A set of sentences G implies or has a consequence the sentence D if and only if there is no interpretation that makes every sentence in G true, but makes D false. This is the same as saying that every interpretation that makes every sentence in G true makes D true." - Computability And Logic - Boolos, Burgess and Jeffrey.
(Very minor and not material technical disclaimer: The quote above is followed by a technical exception, but also an explanation of how that technical exception dissolves upon understanding that 'every interpretation' may be taken to mean 'every interpretation that assigns denotations to all the nonlogical symbols in whatever sentences we are considering'.)
Hmm interesting, I think my position is that the formal conditional is meaningless then, insofar as it is just symbol manipulation.
Quoting Leontiskos
I have tried to formalize it and can't seem to do so; this is an approximation:
(A v ~A) ? (~B v ~A)
When (B and A) are both true, the expression seems to be false. On the other hand, the negation of that expression seems to imply that (A and B) must both be true. If the conditional is construed as only being true when A and B are true, then the negation of the initial expression maps onto A?B. Perhaps that could be written as, it is not the case that A does not imply B therefore A implies B. (Though if that were the case then A?B would be logically equivalent to A^B, although not meta-logically equivalent).
But then I don't mind saying "P does not Imply Q" can't be formalized.
I may have mispoken, but to me the full meaning of "If P then Q" captures the fact that "P does not imply Q" can still be true even though not-P v Q can still be true. But then I now think P->Q is a meaningless expression so saying it "means" the same think as not-P or Q is unsubstantiated.
What are the "full meanings" of "If P then Q" and "P does not imply Q", according to you?
And what is the difference, according to you, between "the meaning" and "the full meaning"?
You still have not stated any rules.
The argument:
A -> ~A
therefore A
makes use of the interpretative clauses for '->' and '~'.
But I have not mentioned a rule, since the above is merely an argument and not a proof.
Quoting NotAristotle
You didn't read a word I wrote about that.
That's very incorrect.
Why don't you just read one chapter in an intro textbook? Is there some reason you won't read even a few pages of a book or article to inform yourself on the subject? Are you allergic to books and articles or something? Have a phobia of them or something?
As for the instantiation of truth possibilities by the rules, what I mean is that the possibilities for what is true and what is false are arrayed across a truth table. The rules must account for all the ways that those truth possibilities can be instantiated. So for the expression A v B, the truth table is T, T, T, F. On the other hand, T, F, F, F, is A ^ B. Every possibility wherein T is present must be uniquely accounted for by the rules. So T, F, F, F, and F, T, F, F, and F, F, T, F, and F, F, F, T, must all be "achievable instantiations" based on the rules we bring to the variables. If A v B were the only rule we applied, then not all of the truth possibilities could be instantiated, does that answer?
Is that your offering? So, the conditional is false when A is true and B is false (that is ordinary), when A is false and B is true, and when A is false and B is false?
Then we would have A -> B is true if and only if A & B is true, So "if then" to you is just conjunction?
I think you meant:
P -> Q |= ~P v Q
and
~P v Q |= P -> Q
?
What is the difference between "followed" and "present"? And, according to you, how does it vitiate what I said about contradiction.
Quoting NotAristotle
The ordinary rules do not eschew any rows in any truth table.
Quoting NotAristotle
You didn't read a word I wrote about realizability.
A v B
is a sentence, not a rule.
Yes, my typo. Thank you.
While you are slipshod.
You've not shown any illogic I've committed. Meanwhile, you are slippery mess of informal illogic.
What, according to you, is the difference? Yes, they are different formulas, but equivalent according to your own definition. Just waving "meta-logic" like some kind of magic wand is nothing.
The ratio of my substantive typos to good information and good arguments is 1:10000.
The ratio of your confusions, circularity, and ignorance to good information and good arguments is 1:1.
What I responded with --a rule must have been "followed" not merely be "present" and the use of a rule may not result in a contradiction means that the use of a rule, or I guess you would call it an operator or connective, whatever you call it, must not result in a contradiction. A->not-A, when this rule is applied and followed, that is, when it is true that "A" and the rule "A->not-A" is actually applied, a contradiction results, specifically "A and not-A."
By "actually applied" I mean that the rule, or connective, does work in leading to the conclusion.
The "following" of a rule versus it's being merely "present" can be illustrated by the following example:
A->B
B^C
Therefore, C.
In this example, the rule A-> B does not do any work, so even if it did result in a contradiction, the fact that it doesn't do any work in the argument and isn't followed or actually applied, means that the argument could still be valid.
To wit,
B
Therefore A?B
Formally valid.
Water was added to the lake.
Therefore,
If it is cloudy out, then water was added to the lake.
Informally not valid.
as well as -
A ^ B
Therefore, (A?B).
Formally valid.
Kangaroos are marsupials and Paris is the capital of France.
Therefore,
If kangaroos are marsupials, then Paris is the capital of France.
Informally not valid.
This issue is directly parallel to the earlier discussion about the nature of validity:
A: "If the premises are inconsistent then the argument is valid."
B: "Validity has to do with the conclusion following from the premises, and inconsistency is not evidence that the conclusion follows from the premises."
If one cannot recognize that something can follow from something else in different ways, then they will not be able to recognize the difference between a material conditional and a disjunction; and this is similar to the way that if one cannot recognize the essence of validity, then they will not be able to exclude or even recognize degenerative cases.
That ((P?Q)?Q), therefore P is not valid, whereas ((A?¬A)?(P?Q)?Q), therefore P is valid, does seem strange to me. Inconsistent premises don't seem to have anything to do with whether the argument "follows." Although I have a feeling that Tones will have something to say about that.
I am a man and I am not a man. Therefore I am rich.
The argument is valid; the conclusion follows from the premise. We can show this in four parts:
1. If "I am a man and I am not a man" is true then "I am a man" is true.
2. If "I am a man" is true then "I am a man or I am rich" is true.
3. If "I am a man and I am not a man" is true then "I am not a man" is true.
4. If "I am a man or I am rich" is true and if "I am not a man" is true then "I am rich" is true.
("I can give a valid argument moving from A to B," is not the same as, "The argument that was given is valid.")
If it turned out that validity required more than what that definition suggests (I think it does), then the argument you stated may well turn out to not be valid, as I think is the case.
Maybe another way of coming at this is as follows - the conclusion is true. Period. Under that understanding, "there is no interpretation where the conclusion is false" ergo there is no interpretation s.t. all the premises are true and the conclusion is false. But the conclusion being true does not seem to guarantee that the argument is valid. But with Tones' definition, it would. Similarly, inconsistent premises also guarantee the validity of the argument according to Tones' definition, but that also seems problematic.
You seem to be putting the cart before the horse.
It's not the case that the word "valid" means something and then we try to give a proper description of this meaning, and that we disagree on the proposed definition.
Rather, a bunch of logicians got in a room together and decided that if an argument's conclusion follows from its premises using the rules of inference then they will name this type of argument "valid". And that if the premises are also in fact true then they will name this type of argument "sound".
But that argument isn't valid.
Quoting NotAristotle
That is not what "sound" means.
It's not. You might as well claim that mathematicians are wrong to define the "=" sign as meaning "is equal to".
No.
No, it doesn't.
P1. If I am a man then I am mortal
P2. I am a man
C1. Therefore, I am mortal
This is an invalid argument:
P1. If I am a man then I am mortal
P2. I am a man
C1. Therefore, I am English
Both premises and both conclusions are true, but the second conclusion doesn't follow from the premises (whereas the first conclusion does).
These are two different arguments, and the validity of the first does not ensure the validity of the second.
P1. "I am a man and I am not a man" is true
P2. If "I am a man and I am not a man" is true then "I am a man" is true.
P3. If "I am a man" is true then "I am a man or I am rich" is true.
P4. If "I am a man and I am not a man" is true then "I am not a man" is true.
P5. If "I am a man or I am rich" is true and if "I am not a man" is true then "I am rich" is true.
C1. Therefore, "I am rich" is true
P2 through P5 can be kept implicit as they simply express the commonly understood rules of inference.
No, they are two different arguments. One involves inferential reasoning and the other does not.
In simple terms, given these two premises:
P1. If I am a man then I am mortal
P2. I am a man
You can use the rules of inference to derive the conclusion "I am mortal" using a priori reasoning, but you cannot use the rules of inference to derive the conclusion "I am English" using a priori reasoning.
It's one argument that uses deductive reasoning to derive the conclusion from the premise.
P2 - P5 simply make explicit the rules of inference and can normally be left unsaid.
Argument 1:
Quoting Michael
Argument 2:
Two different arguments.
You want to claim that argument 2 is an enthymeme of argument 1. But it need not be. And the question at hand is whether argument 2 is valid independent of argument 1.
That is well said.
Perhaps we we disagree about what may be considered a rule of inference. Unless you think an argument that is invalid only coincidentally doesn't follow? Or is it invalid because it does not follow?
Logicians coined the term "valid argument" as a shorthand for "an argument with a conclusion that can be derived from the premises using a priori reasoning".
Logicians coined the term "sound argument" as a shorthand for "a valid argument with true premises".
The "argument 1" sits in between the premise and the conclusion of "argument 2" to make a single argument:
P1. "I am a man and I am not a man" is true
P2. If "I am a man and I am not a man" is true then "I am a man" is true.
P3. If "I am a man" is true then "I am a man or I am rich" is true.
P4. If "I am a man and I am not a man" is true then "I am not a man" is true.
P5. If "I am a man or I am rich" is true and if "I am not a man" is true then "I am rich" is true.
C1. Therefore, "I am rich" is true
P2 through P5 are normally left implicit as they are simply stating the commonly accepted rules of inference.
My question is, if I use a priori reasoning, how can I conclude that "I live in Antartica" (assuming that is true) based on the premise "Pluto is a planet and Pluto is not a planet". How does the conclusion "follow?" I saw your reasoning from the earlier argument, I'm just wondering what rule of inference leads to this conclusion.
To be more specific, it seems to me that in the argument you stated, P1, P5, and C1 cannot all be true. That is, if C1 is true then P1 cannot be true. And if P1 is true then C1 cannot be.
This is dumb beyond belief.
The short-circuiting is still the same. Once you have ~P in hand, you know that ~P v anything is true; you also know that P & anything is false. When you have ~P, no P without Q is true for every Q, because there are no P; you'll never find a P unaccompanied by Q, because you'll never find a P.
We have a whole thread on this idea:
Quoting bongo fury
Quoting Leontiskos
Yes. I'll rephrase the argument in propositional logic:
P1. P ? ¬P
P2. P ? ¬P ? P (conjunction elimination)
P3. P ? P ? Q (disjunction introduction)
P4. P ? ¬P ? ¬P (conjunction elimination)
P5. (P ? Q) ? ¬P ? Q (disjunctive syllogism)
C1. Q
Notice that P2 - P5 are all rules of inference; they are implicit in every argument (along with every rule I didn't write out) and so don't need to be said.
I am going to limit myself to serious interlocutors.
I am being serious. Read up on the principle of explosion.
Quoting TonesInDeepFreeze
(Your contention that argument 2 cannot ever exist without argument 1 is magical, ad hoc thinking. There is nothing serious about it.)
He's talking about something else. I'm taking about this:
P1. P ? ¬P
C1. Q
This is literally the principle of explosion:
---
Quoting Leontiskos
See the section titled "Proof" which includes all my additional steps P2 - P5. They don't need to be made explicit because they are inherent rules of inference.
It says "the existence of a contradiction (inconsistency) in a formal axiomatic system is disastrous", which it is.
Given that P ? ¬P entails Q, we must be sure that we never allow for P ? ¬P to be true.
The principle of explosion is simply the acknowledgement that if we apply the rules of inference to a contradiction then we can derive any conclusion we like. That is simply an a priori fact about propositional logic.
Obviously almost nobody will accept that a contradiction can be true.
Although there are dialetheists like Graham Priest who argue that they can.
My comments concern ordinary formal logic unless stated otherwise:
The question was:
Is the following argument valid?
(1)
A -> ~A
A
therefore ~A
The answer is:
Yes, (1) is valid.
Df 1. An argument is valid if and only if there are no interpretations in which all the premises are true and the conclusion is false.
That is equivalent with:
Df 2. An argument is valid if and only if every interpretation in which all the premises are true is an interpretation in which the conclusion is true.
(By the way, there is no mention of inference rules in those definitions. The definition of 'valid argument' is couched only with regard to truth, falsehood and interpretations.)
Then note that there are no interpretations in which both the premises A -> ~A and A are true (see the truth table), perforce there are no interpretations in which both the premises A -> ~A and A are true and the conclusion ~A is false.
However, we show: For any inference rule, and for any interpretation, there is no application of the rule that allows deducing a falsehood from true premises. One such inference rule is modus ponens, so the validity of (1) can also by shown by this proof:
1. A -> ~A
2. A
3. ~A (1, 2 modus ponens)
But the definition of validity provides that if a set of premises is inconsistent, then any argument with that set of premises is valid (call that 'explosion concerning arguments'). So, since the set of premises (A -> ~A, A} is inconsistent (see the truth table), the argument with ~A as conclusion is valid.
I already explained that the only time a rule yields a contradiction is when it is applied to an inconsistent set of formulas. So, if you want to define 'valid argument' so that no valid argument has a contradictory conclusion, then stipulate that no valid argument has an inconsistent set of premises.
You ignore information given you.
Quoting NotAristotle
A -> ~A is a sentence. It's not a rule.
First, you conflated connectives with rules. Now you conflate sentences with fules
You are hopelessly ignorant and confused about even the basic concepts:
connective
sentence
rule
Quoting NotAristotle
Again, A -> B is a sentence, in this case it's a premise. Again, A -> B is not a rule.
I'm guessing what you mean is that a valid argument has no premises that could be excluded and still have the set of premises entail the conclusion.
But what you said above is actually the opposite of that, as you wrote "the argument could still be valid".
And note that your stipulation of not having unneeded premises would leave us without the monotonicity principle.
Quoting NotAristotle
You still have not defined 'informally valid'. You abandoned your first attempt after I finally got you to see the circularity in your attempt. Then your subsequent attempts have been nonsense even to the extent that you conflate the notion of 'sentence' with that of 'rule'.
Quoting Michael
Quoting Leontiskos
Michael's reasoning is correct there and doesn't contradict anything I've said.
The definition of validity entails that the principle of explosion is valid.
Quoting NotAristotle
It's not "my definition" in the sense that I am proposing it as the only acceptable definition or asserting that there can't be a better definition. Only that it is the standard definition, is clear, is understood by mathematicians, logicians, and philosophers, and has applications in those fields of study, and makes sense to me in certain formal contexts.
Quoting Michael
No mention of rules of inference is in the definition.
This is what I said:
Quoting TonesInDeepFreeze
And we can add:
(3) Th. If a set of sentences G is inconsistent, then for any P,
So, since {A -> ~A, A} is inconsistent,
A -> ~A
A
therefore ~A
is valid.
Quoting NotAristotle
WRONG. You egregiously misrepresent the article.
That Wikipedia article does not say that the principle of explosion is disastrous. What it does say is that explosion makes any inconsistent axiomatization disastrous. And the point is that if you have an inconsistent theory and explosion then you have a trivial theory in the sense that every sentence is a theorem. One approach is to not have explosion but to allow inconsistent theorems. But in ordinary logic, we have the law of non-contradiction, so one would eschew inconsistent theories even if not for explosion.
Explosion is not incompatible with the law of non-contradiction. Rather, retaining explosion but eschewing inconsistency upholds non-contradiction. On the other hand, eschewing explosion but retaining inconsistency does not uphold non-contradiction.
All of your posted confusions and now a blatant misrepresentation of a cite. You are egregious.
Quoting NotAristotle
Arguments are not contradictory or not. Sentences or sets of sentences are contradictory or not.
Quoting NotAristotle
That is directly false. You don't know what you're talking about. You're an ignoramus spouting misinformation and confusion while you won't even read a single page in a book or introductory article on the subject.
That's fine if you want to say that the strange way you want to apply your definition is based on explosion, but this is a new claim. Earlier in the thread you said that the two are "similar," not that one presupposed the other:
Quoting TonesInDeepFreeze
What you are apparently saying now is that someone who does not understand the principle of explosion cannot apply the definition in the way you prefer.
My point here has been that validity has to do with "follows from." If you think that your idiosyncratic application of your definition of validity is permissible because "anything follows from a contradiction" (i.e. explosion), then you have not disagreed with me that validity has to do with "follows from."
(I won't belabor the point of how strange it is to count on explosion enthymeme to understand a foundational definition.)
Tones is the one being idiosyncratic... :grin:
I have not applied the definition in any strange way.
And the definition is not based on the principle of explosion. Rather, the definition implies the principle of explosion. You have what I said backward.
Quoting Leontiskos
I am saying no such thing.
And it's not just "the way I prefer". The definition of 'valid argument' entails the principle of explosion, no matter what I prefer.
Quoting Leontiskos
I addressed the characterization that is is "my definition'.
And there is nothing idiosyncratic. I stated the standard definition and showed that it immediately entails explosion.
In any case, what I said (in whatever words) is that the (1) the definition of validity entails that (2) an inconsistent set of premises entails any conclusion. (1) and (2) are not equivalent.
And a while ago you claimed that I illegitimately claimed the equivalence of the two wordings of the definition based on using the material conditional in the meta-language. So I pointed out that of course ordinary logicians use the material conditional in both the object-languages and in the meta-languages. And I even quoted a text saying explicitly that the two wordings are equivalent.
Your interpretation of your definition presupposes explosion.
Quoting TonesInDeepFreeze
So you would say that someone who does not understand the principle of explosion can apply the definition in the way you prefer?
I haven't said anything about 'presuppose'. Rather, I have shown that the definition of validity (semantic) entails the (semantic) principle of explosion. As for rules (syntactic) one can embody the principle of explosion as a rule without reference to semantics or a notion of 'validity'.
But, as I've told you at least a dozen times, we also go on to prove the soundness and completeness theorems, that is, an equivalence between entailment (semantic) and deducibility (syntactic).
It only "entails" it because it has presupposed it. Else you do disagree with Michael, who thinks that your construal of your definition is nothing other than a tacit appeal to the principle of explosion.
But you are stuck in your quibbles again. When you figure out how exactly the principle of explosion relates to your definition, feel free to get back to me.
We have a definition of validity. Then we show that that definition entails the principle of explosion.
It's not my concern to sort out what is in your mind about presuppostion.
Quoting Leontiskos
Whether that is or is not a correct characterization of anything he said, all I said is that a certain argument he gave is correct.
Did you see the colon at the end of that sentence?
I was editing my post, dropping the comment about a link, while you posted yours above.
Well, if your strange interpretation of your definition is not based on explosion, then we are back to square one, and it is simply wrong. "If the premises are inconsistent then the argument is valid by definition (and this does not presuppose the principle of explosion)," is just a terrible interpretation of the definition of validity.
Arguments are not valid in virtue of being inconsistent (lol). You obviously won't admit this, but most TPFers are able to recognize its truth. Indeed, there is only one person who has agreed with you in this.
Quoting TonesInDeepFreeze
Quoting Leontiskos
Yes, explosion is similar, in the context of what I posted preceding, with ""any argument with an inconsistent set of premises is valid". Call that (*).
(1) Michael mentioned a particular argument. It is a correct argument. (*) is consistent with that.
(2) The definition of validity and the principle of explosion are not equivalent. The latter follows from the former, but not vice versa.
(3) I've made no claim about "presupposes".
(4) You would do very well to reread that post.
It's not a definition of validty! It's not supposed to be definition of validity!
You are terribly confused and not paying attention to what I've said over and over and to what I said in the last few posts.
Quoting Leontiskos
Arguments are not consistent or inconsistent. Sentences or sets of sentences are consistent of inconsistent.
You are confused, as usual.
Here is what Michael said:
Quoting NotAristotle
Quoting Michael
You came in and said, "Michael's argument is valid," but you haven't at all reckoned with what was really said. I invite you to do that. NotAristotle made an observation about your construal of validity, and Michael defended your construal of validity with recourse to the principle of explosion. Reckon with that.
"If the premises are inconsistent then the argument is valid by definition," does not mean that the definition of validity is equivalent to the premises being inconsistent. It only means that any argument whose premises are inconsistent is valid in virtue of the definition of validity. Your boat is being swamped by your quibbles. :roll:
(1) You don't know that. (2) Even if it were true, it doesn't prove much. (3) I wrote again in my first post today that I am not arguing what the definition should be. But I have said what the ordinary definition is and what follows from it.
READ exactly what I wrote:
Quoting TonesInDeepFreeze
I referred exactly to this:
"The argument is valid; the conclusion follows from the premise. We can show this in four parts:
1. If "I am a man and I am not a man" is true then "I am a man" is true.
2. If "I am a man" is true then "I am a man or I am rich" is true.
3. If "I am a man and I am not a man" is true then "I am not a man" is true.
4. If "I am a man or I am rich" is true and if "I am not a man" is true then "I am rich" is true."
Michael
That is what I said is correct.
Quoting Leontiskos
(Michael thinks your construal of validity is true in virtue of the principle of explosion. You explicitly say that it is not. You are obviously disagreeing with Michael. Stop being disingenuous.)
Edit: And Michael's interpretation is not unreasonable (except for the fact that it is a misrepresentation that has been addressed earlier in the thread). Your interpretation is irrational without recourse to the principle of explosion. Michael's general approach is much more rational than your own:
Quoting Michael
It's not an interpretation of the definition (as if I intepret it to be the definition of as if I interpret the defintion in some strange way. Rather, it is simple inference from the definition. That is a critical point not a "quibble".
There are a lot of things to untangle in a discussion. I didn't purport to vindicate everything the poster has said. And I am not, at least at the present, interested in untangling whether the way you represented what he said is correct or not. In that post by me, I mentioned a particular thing he posted and I said it is correct.
You focused on something which no one contested. I am inviting you to focus on the point at issue.
(Further, the reason NotAristotle is so confused is because Michael is failing to recognize that he is justifying validity in a different way than you are; and you are aiding and abetting his failure. NotAristotle made an argument against your view and Michael defended your view, falsely believing himself to be holding the same view. NotAristotle therefore ends up with constant ignoratio elenchus from Michael.)
No, YOU quoted a certain argument by him and followed with a comment about me, as if that argument is not compatible with what I said. So I made clear that that argument is not incompatible with what I said.
You cited me in your disputes (and without linking my name). So I exercised the prerogative to make clear that that particular argument is not incompatible with anything I've said.
You failed to understand the point I was making to Michael. I invite you again, for the third time, to go back and reckon with that point. If you can't see it by now then you are surely closing your eyes to what is plainly obvious.
(It is fascinating that in the last few hours I disagreed with Michael twice, and then both you and Banno attempted to agree with Michael in order to disagree with me, despite the fact that you ultimately disagree with Michael. Michael is aware that Banno disagrees with him, even if he is not yet aware that you do. But the whole thing is a bit comical. You and Banno are not doing philosophy, you are doing a gossipy "Contradict Leontiskos" game. :grin:)
Oh, for Pete's sake. I just wrote that I am not, at least at this time, interested in sorting out the disagreement between you and him. Rather, at that juncture, I posted to make clear that a particular quote of him (quoted by you followed by mention of me and something I wrote) is correct and not inconsistent with anything I've said. That's it. That's all I posted. Get it now? [Edit: Also, I said, "The definition of validity entails that the principle of explosion is valid." And that is correct.]
Yes, you cherry-picked a highly tailored and dialogically misrepresentative quote in order to go out of your way in affirming an idea that no one was even contesting.
I laid it all out here: go look:
If you don't want to be honest and reckon with the actual object of the conversation, I'm sure no one will be surprised.
I didn't "cherry pick" anything. And I didn't "misrepresent" anything. YOU picked out a quote by him, and cited ME in connection with that quote by him. So I quite rightly exercised my prerogative to make clear that what he wrote is correct and is not in contradiction with anything I've written.
I laid it all out here; go look: https://thephilosophyforum.com/discussion/comment/948275
I don't know what passages you're referring to. You've not shown that anything I've said is incorrect. Your picture of this thread as some kind of tag team match doesn't interest me.
I've addressed the subject of this thread in detail.
My intellectual credibility does not require that I sort through your own disagreements with a poster, nor even that I sort through your notions in and of themselves.
Again, I'm not interested, at this juncture, in untangling Michael's role. I am especially not interested in commenting on his posts vis-a-vis you in between with your characterizations of his posts and your characterizations of my posts and your characterizations of how they compare.
NotAristotle is confused because he knows virtually nothing about the subject, not even chapter one of any book or first material in an article, whether standard or alternative - doesn't even know the differences among connectives, sentences and arguments. And talk about cherry picking, that one by NotAristotle from Wikipedia is a doozy! It was a terrible misrepresentation and failure to even read the context cited by the article, resulting in him flirting with the very point of view he claims to oppose! Talk about what is NOT philosophy!
I didn't give an "interpretation" of the definition. I stated the standard definition. The definition doesn't make "recourse" to the principle of explosion. Rather, from the definition, it is simple to show that the principle of explosion is valid.
That can't be more clear: (1) State the definition. (2) From the definition, infer the principle of explosion. In that order: (1) then (2). How cannot someone not understand that?
Including the prior steps:
(1) we define an 'interpretation'
(2) we define 'true in an interpretation'
(3) we define 'argument'
(4) we define 'valid argument' (mentioning only truth, falsehood and interpretations)
(5) we define 'inconsistent'
(6) we define 'unsatisfiable'
(7) we easily show that an inconsistent set is unsatisfiable
(8) we easily show that the definition of 'valid argument' implies the principle of explosion (that is, that the principle of explosion is valid)
The principle of explosion was not mentioned, assumed, invoked or recoursed to. Only that it was proven to be valid from the definition of 'valid argument'.
I gave details in an earlier post.
From the definition of validity, we show that the principle of explosion is valid.
Then, if an argument has an unsatisfiable set of premises, we thereby show that it is valid, no matter what the conclusion is. (i.e. an instance of the principle of explosion).
Also, from the definition of validity, we show that modus ponens is a valid argument form.
I pointed out that the particular argument in the original post happens to be an instance of modus ponens, so the argument is valid on that basis too. That doesn't contradict showing that the argument is valid on account of explosion. Both are correct: it is valid on account of modus ponens and it is valid on account of explosion. And it valid on account of both of those because both of those are valid on account of the definition of validity.
That modus ponens is a valid argument form is shown from the definition of 'valid argument'; and that explosion is a valid argument form is also shown from the definition of 'valid argument'. The validity of modus ponens and the validity of explosion are both consequences of the definition of 'valid argument'.
And that is the case no matter what I say, no matter what Michael says, and no matter how you characterize any relationship between what Michael says and what I say.
/
Again, since people say things like "Tones's view":
When I report how it goes in ordinary formal logic with the standard definition of 'valid argument', I have not thereby claimed that ordinary formal logic (with the usual definition of 'valid') is the only reasonable logic, or the best one in all situations, or the best one for a philosophical understanding of logic and validity, or that it accords with all of everyday reasoning, or even that is more than hardly found in everyday reasoning, or that is the only one worthy of study or adopting. Alternative formal logics abound and are, in my opinion, worthy of study and adoption in the contexts they are suited for. But ordinary formal logic is appreciated by, studied, applied and discussed greatly among mathematicians, logicians, computer scientists, and philosophers. It is at the heart of mathematical logic that axiomatizes the branches of mathematics, including computability, including the very invention of, and improvements to, the digital computer. But, meanwhile, ordinary formal logic is fair game to critique. But critiques of any logic are mindless, and not even close to philosophy, when they are premised in ignorance and confusion about how it actually goes in the logic. And corrections given to such ignorance and confusion are not themselves a form of endorsement of the logic nor do they constitute claims that it is "right", unless one does go on to endorse the logic. For one to explain what actually happens in the logic is not in and of itself to say that people should adopt the logic - but rather, at least that they should not attack it on false bases. By analogy, one doesn't have to agree with the constitution of a country just to study it and report what it actually says. Moreover, one may rebut arguments against a logic without endorsing the logic, as one's rebuttals may be merely that the arguments are not good, without opining on whether the conclusions are correct or incorrect.
There is repetition in the above, but it is there to drive these points that keep getting overlooked no matter how many times they are stated and especially to, hopefully, foreclose against someone yet again putting words in my mouth to make it appear that I've adopted a position that in fact I have not adopted.
No I don't. I'm saying that P ? ¬P ? Q is valid and that P ? ¬P ? Q is called "the principle of explosion".
Much like P ? Q, P ? Q is valid and that P ? Q, P ? Q is called "modus ponens".
At this point it's obvious that both of you are more interested in being contrarians than actually meting this out, so I'll leave you to it.
Tones is just super precise in his expressions and Michael is a little more intuitive. Once you understand the issue, you'll see that they're saying the same thing.
Do you understand why the argument is valid?
No I'm not.
Quoting Leontiskos
Quoting Michael
As I said, there aren't two arguments; there is one argument:
P1. P ? ¬P
P2. P ? ¬P ? P (conjunction elimination)
P3. P ? P ? Q (disjunction introduction)
P4. P ? ¬P ? ¬P (conjunction elimination)
P5. (P ? Q) ? ¬P ? Q (disjunctive syllogism)
C1. Q
But we don't have to write out P2 - P5 because they are all necessarily true; they are some of the rules of inference. We can leave it as:
P1. P ? ¬P
C1. Q
I haven't said that this has something to do with every kind of valid argument. It has nothing (necessarily) to do with modus tollens or modus ponens, for example.
Susie gives an argument. Her premises are inconsistent. Is her argument valid? Do not presuppose the principle of explosion.
A. Yes, Susie's argument is valid.
B. No, Susie's argument is not valid.
C. We do not yet know whether Susie's argument is valid.
It's valid. I don't even know what you mean by "not presupposing the principle of explosion".
(2) As to validity, I said that the standard definition of 'valid argument' implies that any argument with an inconsistent set of premises is valid. That it is correct: The standard definition implies that any argument with an inconsistent of premises is valid.
(3) I said that the standard definition of 'valid argument' does not mention the principle of explosion, and I listed the definitions and points that the definition of 'valid argument' does rely on. Then from the standard definition of 'valid argument' we easily show that the principle of explosion is valid. That is correct.
(4) There is no "contrarianism" in any of that.
Semantic: Explosion as an argument form is valid. It is easy to show that explosion as an argument form is valid by noting that the definition of 'valid argument' implies that explosion is a valid argument form.
Syntactic: Explosion as an inference rule, depending on the system, can be either a primitive rule or a derived rule.
" ¬?x(P?Q) "
where x is an interpretation, P is "all premises are true" and Q is "the conclusion is false."
Is there something problematic about writing the definition of validity that way?
Why?
First, why do you ask?
The statement you seem to have in mind is:
(1) It is not the case that there is an interpretation in which all the premises are true and the conclusion is false.
One way to write that:
Let Px stand for "x is an interpretation in which all the premises are true".
Let Qx stand for "x is an interpretation in which the conclusion is false".
Then the statement is:
(2) ~Ex(Px & Qx)
But neither (1) nor (2) capture the definition of 'is a valid argument', which, if we spell out the quantifiers is:
For all T(T is a valid argument
if and only if
(T is an argument
and
for all N(if N is an interpretation, then it is not the case that
((for all p(if p is a premise of T, then p is true per N))
and
for all c(if c is the conclusion of T then c is false per N)))))
Symbolized:
Let Vx stand for x is a valid argument (the definiendum)
Let Bx stand for x is an argument
Let Dx stand for x is an interpretation
Let Rxy stand for x is a premise of y
Let Txy stand for x is true per y
Let Uxy stand for x is the conclusion of y
Let Fxy stand for x is false per y
AT(VT
<->
(BT
&
AN(DN ->
~((Ap(RpT -> TpN))
&
Ac(UcT -> FcN)))))
(I hope I got that all correctly, including the parentheses.)
Because of the reasoning explained here.
And so you are appealing to or presupposing the principle of explosion when you claim that Susie's argument is valid, are you not?
See the ? Q at the end? That means that Q follows from the bit before.
Weve already established that (P ? Q) ? ¬P is true, so therefore Q is true.
Do you know if you are appealing to the principle of explosion? Because I asked if you are "appealing to or presupposing the principle of explosion."
I dont know what you mean by appealing to the principle of explosion.
Its like saying that we appeal to modus ponens.
We use modus ponens to derive some conclusion and we use the principle of explosion to derive some conclusion.
This is all a priori reasoning based on logical axioms, not some a posteriori proposition that is possibly false.
I asked why you think Susie's argument is valid. You gave an argument which you admitted is a presentation of the principle of explosion. Clearly you think Susie's argument is valid because of the principle of explosion, just as we might think that a conclusion follows because of modus ponens. Someone who makes an inference based on a rule of inference is appealing to that rule of inference. This should not be so hard.
An inference is presupposed when it is interpreted as the tacit reasoning of an enthymeme. You are supplying Susie's argument with a rule of inference that she does not explicitly present. You are interpreting it as an enthymeme and supplying what you see as the implicit inferential steps.
Here's how I would write it:
1. P & ~P (premise)
2. P (from 1, conjunction elimination)
3. P v Q (from 2, disjunction introduction)
4. ~P (from 1, conjunction elimination)
5. ((P v Q) & ~P) -> Q (theorem)
6. ((P v Q) & ~P (from 3, 4, conjunction introduction)
7. Q (from 5, 6, modus ponens)
Or, have explosion as either a primitive rule or derived rule in a natural deduction system:
1. P & ~P (premise) {1}
2. Q (explosion) {1}
A or B
Not-A
Therefore, B (disjunctive syllogism)
E?A?(B?¬(C?D)) is the definition. I know that if we are being precise it is not, but thematically would this work for the definition of validity.
But that doesn't work if A and not-A are both true. That's my point. The proof doesn't work. The proof only works if you ignore that A is also true.
I can only guess, but I think that is what Tones meant by referring to that step as a "theorem."
That is not a rendering of my formulation.
I adduced it as a previously proven theorem to help you see how the final step would by an application of modus ponens, so you'd have another way to look at it to see that the steps are correct. We could also do it this way:
Rule (disjunctive syllogism): If P v Q occurs on a line, and ~P occurs on a line, then infer Q.
1. P & ~P (premise)
2. P (from 1, conjunction elimination)
3. P v Q (from 2, disjunction introduction)
4. ~P (from 1, conjunction elimination)
Q (from 3, 4, disjunctive syllogism)
The definition involves quantification, so I wouldn't reduce it to a merely sentential formula.
(1) In no interpretation are both A and ~A true.
(2) Having A & ~A as a premise, thus being able to have A as a line and ~A as a line, does not vitiate use of disjunctive syllogism:
The rule is: If P v Q is on a line, and ~P is on a line, then infer Q.
The rule is NOT: If P v Q is on a line, and ~P is on a line, and P is not on a line, then infer Q.
Sure, and that's the same puzzle of the OP. I see your point.
See:
Quoting Leontiskos
In this case:
We have the premise P & ~P.
We want to get P v Q and we want to get ~P, so we can apply disjunctive syllogism to get Q.
We get P v Q by first getting P from P & ~P by conjunction elimination, then P v Q from P by disjunction introduction.
We get ~P from P & ~P by conjunction elimination.
/
The rule is: If P v Q is on a line, and ~P is on a line, then infer Q.
The rule is NOT: If P v Q is on a line, and ~P is on a line, and P is not on a line, then infer Q.
I think @NotAristotle is right insofar as the rule is ambiguous. There is no magical rule-book of logic that settles this issue, and in practice someone who contradicts themselves is responded to with a reductio.
Youre right, it shouldnt. Which is why I dont understand why you are taking issue with what I am saying.
It is simply an a priori fact that from p and not p one can derive any conclusion, and so any argument with p and not p as premises is valid.
If P v Q is on a line, and ~P is on a line, then we may put Q on a new line.
Or better, without "we", "may" and "put", the rule may be stated:
A deduction from a set of formulas G is a sequence of formulas such that:
If a formula P appears on a line, then it is either a member of G or there are formulas on previous lines such that there is a rule such P comes from those formulas.
And among the rules is: P v Q, ~P |- Q.
What disjunctive syllogism is is settled by any ordinary textbook that has it as a rule. Whatever particular wording is used in an ordinary textbook, it amounts to the rule that from P v Q and ~P we may infer Q.
You think it's valid because of explosion. It's that simple. Again:
Quoting Leontiskos
Sure, but a premise that is not doing any work in an argument is not a premise of the argument. It is an unrelated proposition. You want to say that if one half a contradictory pair is doing work, then the other half is implicated.
I'm sure you could find that in a textbook, but one must recognize that such textbooks presuppose that the premises are not inconsistent.
Are there any introductory textbooks that talk about the principle of explosion?
It's not a matter of what I "regard" to be the case.
Oh come on! Get a textbook that uses disjunctive syllogism. You won't even look at a textbook yet you are challenging me to cite one! Not playing your idiotic game.
Sure it is, unless you are the Source of Truth Itself.
What I've said is correct, not merely because I said it.
Disjunctive syllogism is in lots of textbooks. You're ridiculous.
This is metalogic, but note that validity is meant to show how conclusions rightfully follow from premises. It is meant to provide us with a way to think correctly, and increase our knowledge.
Anything follows from an explosive system, and yet not anything follows with respect to correct thinking. This means that explosion is an aberration (along with the contradiction that it flows from). In propositional logic contradictions are supposed to be eliminated (via reductio), not utilized.
So is an explosive argument valid? In one sense it is, and in one sense it is not. It does not provide us with the thing that the notion of validity is meant to provide, but it is nevertheless valid in a certain (arguably degenerative) sense.
There are some logicians in these parts who view logic as mere symbol manipulation, without any relation to correct reasoning. For these logicians an explosive argument is uncontroversially valid.
I said "principle of explosion" not "disjunctive syllogism"
"Not playing your idiotic game"
Then I accept your unconditional surrender.
You linked to my post about disjunctive syllogism.
Yes. But why do you ask, unless you'll read one?
So we have something like tiers of sophistry:
What's interesting about the "medium sophistry" is that Michael has detached logic from humans in a remarkably thoroughgoing way. He is basically saying, "If a conclusion is inferentially reachable from the premises, then the argument is valid, even if the argument does not present the necessary inferences." He collapses an argument and an enthymeme into one thing, which doesn't make any sense in the end. Pace Michael, inferences (or lack thereof) are part of the argument itself.
(@NotAristotle)
(1) define 'is an interpretation'
(2) define 'is true in an interpretation'
(3) define 'is an argument'
(4) define 'is a valid argument' (mentioning only truth, falsehood and interpretations)
(5) define 'is inconsistent'
(6) define 'is unsatisfiable'
(7) show that an inconsistent set is unsatisfiable
(8) define the principle of explosion
(9) show that the definition of 'valid argument' implies that the principle of explosion is valid
And note that, in this sequence, (4) precedes (8) and (9).
Although, I would say there's a "logical floor" where no further arguments or definitions can settle whether an inferential rule is "necessary;" that is why I refer to "logical intuition" - so I would say that while modus poenens fits into a set of rules, I am skeptical that the move itself can be justified using argument. That it really is logical is basically a matter of faith that the way we're thinking is "correct."
Maybe there's an evolutionary argument to support "correct thinking" although that would assume that passing on genes correlates with correct thinking or something like that, which would still leave an open question of whether the thinking is "normatively" right. Maybe we might be able to conclude that it's "logical enough." or something along those lines. But it would be interesting if logical thinking could somehow be proven scientifically, and yet that would seem to also be a very circular argument.
You asked me a question. I gave you a rigorous detailed answer. What was the purpose of your question?
And you would do well to re-read that Wikipedia article you cited about explosion, to see that you misrepresented what it says and to see that the passage you cited is itself based on a passage in a site about paraconsistency, which is a context that may, depending on the formulation of such a system, allow that contradictions can be other than false, which is the opposite of what you claim to hold. Indeed, explosion, which you reject, is the antithesis of paraconsistency.
Hmmm.
Do you make any distinction between premises and inference rules?
Quoting Leontiskos
I'm trying to understand this. Are you arguing against the cut rule?
In practice, we show only the inferential steps we don't assume the audience can fill in for themselves. The overwhelming majority of mathematical proofs are not "complete", don't show every single step. For good reason.
As a practical matter, if your audience can't fill in the missing steps, they may not find your argument persuasive. But if you can show them the missing steps on demand, you should be on the same page.
You could, but I am not.
Quoting Srap Tasmaner
My point is that argument 1 and argument 2 are different arguments. Argument 2 could be an enthymeme form of argument 1, but it need not be. Michael somehow thinks it needs to be.
This is important because if we cannot speak about argument 2 apart from argument 1, then we cannot even understand the difference between Michael and Tones (and at this point in the thread Michael looks to be actively suppressing the emergence of that difference in a very strange way).
Edit: On my view no argument is demonstrably an enthymeme just in virtue of its material constituents. On Michael's view this is apparently not right, and therefore validity has to do with possible inferences, not documented inferences. Frege's judgment-stroke and the difference between inference and consequence seems relevant here, although Michael is pushing consequence even further than Frege's opponents would.
I was trying to understand how the definition implies that in terms of symbolic logic. I think I understand how the definition could imply that an argument with inconsistent premises must be valid according to the definition you stated, and I think you will agree with me that if the conclusion is necessarily true, then the argument must be valid, according to the definition you stated. And, if the premises are inconsistent and the conclusion is necessarily true, then such an argument must again be valid according to the definition you stated.
I'll use the notion of 'satisfiable' (there is an interpretation in which all the members the set are true, and 'unsatisfiable' denoting the negation of that) rather than 'consistent' (there is no deduction of a contradiction from the members of the set, and 'inconsistent' denoting the negation of that), to keep the matter all semantical, and as it is an obvious and easy to show theorem that if a set of sentences is inconsistent then it is not satisfiable.
Here I changed some variables from previously, to avoid using T as both a variable and relation symbol, and to make the role of others more clear. Hope I don't make any typos:
We already have (1) below:
(1) Definition of 'is a valid argument':
For all g(g is a valid argument
if and only if
(g is an argument
and
for all i(if i is an interpretation, then it is not the case that
((for all p(if p is a premise of g, then p is true per i))
and
for all c(if c is the conclusion of g then c is false per i)))))
Symbolized:
Let Vx stand for x is a valid argument
Let Bx stand for x is an argument
Let Dx stand for x is an interpretation
Let Rxy stand for x is a premise of y
Let Txy stand for x is true per y
Let Uxy stand for x is the conclusion of y
Let Fxy stand for x is false per y
Ag(Vg
<->
(Bg
&
Ai(Di ->
~((Ap(Rpg -> Tpi))
&
Ac(Ucg -> Fci)))))
(2) Then we want to show that, for any argument g, if there is no interpretation in which all the premises of g are true, then g is valid:
Ag((g is an argument
&
~Ei(Di & Ap(Rpg -> Tpi))) -> Vg)
It's merely a tedious, routine exercise to do the proof in a system of the first order predicate calculus.
(3) Also, we want to show that, for any argument g, if there is no interpretation in which the conclusion is false, then g is valid:
Ag((g is an argument
&
~Ei(Di & Ac(Ucg -> Fpi))) -> Vg)
It's merely a tedious, routine exercise to do the proof in a system of the first order predicate calculus.
(4) And you want to also show that, for any argument g, if there is no interpretation in which all the premises of g are true, and there is no interpretation in which the conclusion of g is false, then g is valid.
But that is implied, a fortiori, from (2) and (3), given this theorem of sentential logic:
((P -> Q) & (H -> Q)) -> ((P & H) -> Q)
Leontiskos does not name who he means, so it behooves me to speak for myself.
(1) I'm not a logician and (2) I do not regard logic as mere symbol manipulation.
As for professional logicians, I'd be interested to know of one who regards logic as mere symbol manipulation.
By following the links to the posts, the poster referred to disjunctive syllogism.
In that context, what set of premises does Leontiskos think textbooks "presuppose" to be consistent?
If the poster meant explosion, then still, in that context, what set of premises does Leontiskos think textbooks "presuppose" to be consistent?
The principle of explosion is that from an inconsistent set of premises any conclusion follows. There's no "presupposition" that the set of premises in such an argument is consistent. It wouldn't even make sense otherwise.
If you say that logic is not merely symbol manipulation, then what do you say it is?
"(2) Then we want to show that, for any argument g, if there is no interpretation in which all the premises of g are true, then g is valid:
Ag((g is an argument
&
~Ei(Di & Ap(Rpg -> Tpi))) -> Vg)"
This might not make too much of a difference, but it seems to me that (if we use the definition of validity you stated)... that there being no interpretation s.t. all premises are true does imply an argument is "valid."
But that the definition of validity implies that there being no interpretation with all true premises implies that the argument is valid - that I am not so sure about, because I do not see how a definition can imply anything.
It seems like two definitions are often included in for validity, and often both are presented side by side:
1. An argument is valid when it is impossible for the premises to be true and the conclusion false; and
2a. The conclusion follows from the premises.
2b. The conclusion is contained in the premises.*
Normally, these overlap. If we accept our inconsistent premises as true we can move to the conclusion (but of course they aren't true). We can use disjunctive syllogism and disjunction introduction to move from our inconsistent premises to whatever we'd like. But this relies on a certain notion of implication and explosion, both of which have been controversial in the history of logic (if nonetheless mainstream), precisely because they seem counterintuitive and don't seem to capture natural language reasoning or notions of "good reasoning."
My thoughts were that a combination of relevance conditions for implication and changes to avoid explosion could perhaps get us to a case where an argument is valid under definition 1 but not 2? That is, we'd have inconsistent premises but no inferences connecting them to our conclusion even if we did affirm all the premises.
Can there be such a counter example where the two diverge?
I mentioned quia demonstrations vs. propter quid demonstrations earlier. Supposing that the two definitions do rightly overlap, it would seem like 1 would be a quia demonstration (going from effects backwards), while 2 actually gives us the "why." But in natural language, with our penchant for equivocal and analagous predication, fuzzy terms, and lack of clarity, I can see why people would like to hold to 2 over 1 even if they thought they properly overlapped. We might say, "1 is simply a consequence of 2."
*I am not sure how 2b works with explosion. I have seen Floridi and D'Agastino argue in the context of the "Scandal of Deduction," from an information theoretic lens that there is a certain sense in which some conclusions aren't contained in their premises, with some forms of inference injecting new information.
When you claim, "There are some logicians in these parts who view logic as mere symbol manipulation", do you include me, thereby claiming that I view logic as mere symbol manipulation?
You said you asked for a symbolization of the definition for this reason:
Quoting NotAristotle
I gave you symbolizations of (1) the definition of 'valid argument' and (2) "if there is no interpretation in which all the premises are true, then such an argument is valid".
With those symbolizations it is merely an exercise to prove (2) from (1).
But your reply is to say that you don't see how a definition can imply anything! If you don't understand how definitions are used in proofs, then that is what should be discussed first, not belaboring symbolizations. But once I did give you the symbolizations, if you knew anything about basic symbolic logic, then you would be able to finish the exercise by showing the proof of (2) from (1) and thus not have to wonder how definitions imply things. Meanwhile you won't even look at a book to see how definitions and proofs work in first order logic.
You are oblivious to how very irrational you are.
Here's you at a restaurant:
Waiter: Welcome to TPF Bar & Grill, may I take your order?
NotAristotle: Yes, I'll have the ribeye steak, rare, cooked with an extra amount of salt.
cut to:
The waiter delivers the steak, cooked to order, rare and with extra salt, very nicely set on a beautiful china plate, with gleaming utensils.
Waiter: Here's your steak, sir, as you ordered it - a beautiful ribeye, rare, with plenty of salt. I hope you enjoy it.
NotAristotle: Take this back. I can't eat this. I'm a vegetarian. And I'm on a low-sodium diet.
Waiter: I don't understand, sir. It's what you ordered.
NotAristotle: Just take it back.
Waiter: Very well, sir. I'll bring you a menu to order something else.
NotAristotle: Yes, bring me a menu so that I can not read it. No, never mind, just bring me a ribeye steak, rare, extra salt.
NotAristotle: Was that so hard? ... thank yo-- what the hell is this?
Waiter: it's the ribeye sir, rare, with extra salt.
TonesinDeepFreeze: it's what you ordered NotAristotle, just eat it.
NotAristotle: I don't even like steak, why would I order it?
Tones: don't ask me.
Michael: stop making a scene NotAristotle, you do this every time!
NotAristotle: Leontiskos, do you want the ribeye?
Leontiskos: Not really, no.
Banno: check please.
When you wrote it, you were referring to unnamed posters. Was I one of them or not?
Your intent when you wrote it is not affected by anything I say retroactively.
But you did order it.
You asked for a symbolization to see how the definition of 'valid argument' implies that explosion is a valid argument. I gave you exactly what you asked for.
Is photography mere photon manipulation?
I don't know if you fall into that group. It's hard to spot photographers. They disguise themselves, blending into their environment. That's why I am asking you a question. You have disavowed such a view in the past but I don't understand what alternative route you purport to take.
When you wrote, "There are some logicians in these parts who view logic as mere symbol manipulation", who were you referring to?
(I meant 'posters' not 'photographers'.)
When I wrote it I wasn't sure whether you fit the bill or not. Time to answer my question:
Quoting Leontiskos
Who did you think fits the bill?
It's a simple question.
Why do you get to keep ignoring my question? Why do you think you are entitled to an answer to your question?
It's mathematics without the math. :roll:
Incidentally, this is my mother's favorite kind of mathematics. :smile:
Here's some informal logic that is not "mere symbol manipulation":
You said that there are some here who view logic as mere symbol manipulation. So, in such a small domain as this one, you could easily list them. And, since by your claim that there are "some", there is at least one that you have in mind. So, if you can't list any other than me, then we may infer that you meant me.
But you left it open, thus it is insinuation. But you don't have the integrity to say who you mean.
/
To maintain that I don't think logic is mere symbol manipulation, it is not required for me to say what logic is. To maintain that basketball is not mere players' statistics, I don't have to tell you what basketball is; whatever it is, I know that it is not mere players' statistics.
/
At some point, time and interest allowing, I may write a post with more about my own sense of the scope of logic. In any case, I use and recognize informal logic as well as formal logic, and I don't take formal logic to be mere symbol manipulation. That is apparent even by the fact that I have discussed, in your presence, certain English sentences vis-a-vis symbolization, as I even did a few posts ago.
Moreover, so many posts I have written about mathematics and logic, written mention a scope that is not at all confined to mere symbolization.
/
Who did you mean ? If you won't say, then I'll take it you don't have the guts to say, as you are sneaky insinuator. "Joe McCarthy" Leontiskos saying, "I have in my hand a list of posters who view logic as mere symbol manipulation".
Leontiskos: There are people here who think acetone is merely oxygen.
TonesInDeepFreeze: I don't think acetone is merely oxygen.
Leontiskos: What else do you think is in acetone? You must answer that for me to decide whether I meant you when I said that there are people here who think acetone is merely oxygen.
TonesInDeepFreeze: Whatever acetone is, I don't say it is merely oxygen, Who are you claiming thinks acetone is only oxygen.
Leontiskos: It is time for you to answer my question.
TonesInDeepFreeze: You first made the claim that there are people here who think acetone is merely oxygen. If you were undecided about me when you made that claim, then which of the people here, do you claim to think that acetone is just oxygen? It's a simple question, and would be honest to answer rather than being a sneaky insinuator.
Leontiskos: How dare you ask me to name the people I claim to think acetone is just oxygen, when you won't write a post about what your notion of logic is?
TonesInDeepFreeze: Well, this started with you making the claim. So, it is natural to first get clear who you meant. And, if you can't say a single person other than me, then that leaves only me.
Leontiskos: What car do you drive?!
Leontiskos insinuated rather than openly stating.
To insinuate instead of openly stating is dishonest.
Leontiskos has been dishonest.
That is logic that is not mere symbol manipulation.
Here's more logic that is not mere symbol manipulation:
Even just one counterexample refutes a universal generalization. So the argument above refutes that I regard logic to be mere symbol manipulation. And this argument does too!
I actually said, "When I wrote it I wasn't sure whether you fit the bill or not." So no, not primarily you.
Quoting TonesInDeepFreeze
Rather, you asked for information that you would misuse and I did not give it to you. I only gave you the information that pertained to your person.
Quoting TonesInDeepFreeze
Pretty amazing how many miles you will run to avoid a simple question.
Quoting TonesInDeepFreeze
Paranoid, much?
Quoting TonesInDeepFreeze
Not a great analogy, to say the least.
Quoting TonesInDeepFreeze
The formula for acetone is (CH[sub]3[/sub])[sub]2[/sub]CO. Notice how simple that was.
- Three posts on this? Have a drink, or take a nap or something.
This all comes from the conversation that emerged after you insisted that NotAristotle needs to follow a rule:
Quoting TonesInDeepFreeze
That characterization is a good example of logic-as-symbol-manipulation, and @NotAristotle's difficulty had to do with the nature of logic or inference. This question of "symbol manipulation" is therefore quite relevant to the question of how to understand validity vis-a-vis explosion. Burying your head in the sand and refusing to address the heart of the issue is not a great look.
I think that's right. I think you've got the horse pulling the cart.
Quoting Count Timothy von Icarus
I don't see this as relevant. I don't see that propositional logic has a causal or metaphysical direction in any obvious sense. For example, why think that (1) would be a quia demonstration? Because the "effect" of a valid argument is the necessary relation between premises and conclusion that (1) captures? It's hard to see this as an "effect" in any strict sense.
Quoting Count Timothy von Icarus
On the standard (non-Tones) interpretation of (1), I would say that this represents a kind of static, non-directional construal of validity. It's about possibility space, not primarily deducibility. On this standard reading of (1) I think (1) and (2a) are coterminous, are they not? I think (1) properly captures a necessary formality of valid arguments; I just think Tones interprets it badly. On the standard interpretation, there is nothing mistaken about (1).
The effect issue is sort of ancillary. The issue is that 1 only follows from 2 given elements of logic that seem to be more a bug than a featurethat do not comport with common standards of "good reasoning."
As Priest says:
Now, what is now orthodox comes out of people being uncomfortable with where logic had been previously, fixing perceived problems, so if those moves were properly motivated, others attempts for satisfactory resultions seem like they should be too.
Agree.
This is tangential (in that it is about logic but doesnt really relate to the original post), but what would you say about this argument? Is it viciously circular? --
if modus ponens is logical then any argument of the form [P, P->Q] implies Q.
modus ponens is logical.
therefore, "any argument..."
In Aristotelian terms, we would say that (1) is a proper accident of validity. It is not the essence of validity, and yet every valid argument will possess the character of (1).
Quoting Count Timothy von Icarus
:up: That's what I've been saying for months. :smile:
Quoting Count Timothy von Icarus
I don't know if I quite followed that.
I think the degeneration of logic has a lot to do with what said.
See my post <here> and the excerpt contained therein <here>.
My version of the argument is missing the inductive element that would cause the argument to be justified, if still circular. It's like a track record argument for perceptual abilities.
Perhaps, in addition to an inductive argument for modus ponens, an argument from coherence can be made. For instance it seems that if modus ponens failed, then MT or RAA would also fail.
1. If MP could be false, then RAA could be false.
2. But RAA is not false.
Therefore neither is MP.
(MT isnt a premise, however the argument is structurally MT). That is to say if MT is veridical, and so is RAA, then that would guarantee the truth of MP.
Yes, but I think that all arguments are, structurally, modus ponens. This goes back to the earlier point about whether all arguments are modus ponens, or whether all arguments utilize a material conditional. Tones is claiming that the metalogical inference uses a material conditional, and is not merely a modus ponens, and that this is why he thinks inconsistent premises automatically* make an argument valid.
Quoting NotAristotle
You're right that the conclusion utilizes modus tollens, but here is the way that modus ponens is operating metalogically:
When I deny that the '?' in the first premise is a material conditional, what I mean is that no legitimate metalogical move is available whereby the degenerative uses of the material conditional are utilized. It is only the logical connector needed for a modus ponens, not a material conditional in its full degenerative sense. So there is no permissible metalogical argument as follows:
* Note that "automatically" is my word, not Tones'. Let us preempt his quibble.
(1 ^ ~1) ? 2
? 2"
Agree, I think; correct me if I have this wrong: by metalogical I take you to mean a logical "move" (such as MP) that is not identical to its truth function.
Apparently, it is not just arguments with contradictions that are problematic.
If it is settled that any premise in an informal argument is demonstrably false, it is unclear whether such an argument's conclusion can be true and yet the argument still be valid, where a valid argument is signified only as an argument that operates with the material conditional. If all valid arguments use the material conditional, arguments with some false premises could seem to still have a true conclusion.
But this seems wrong, at least to me. If any premises are false, a valid argument will result in a conclusion that is necessarily false, according to my non-standard understanding of validity in an informal context.
You may agree. But if you do, then any argument that is valid will turn out to be, in the relation of premises to conclusion, either [true true], or [false, false]. But that is the truth function of equivalence. Indeed, were you to exclude [F, F] as a degenerate case, your resulting truth functionality for a valid argument [T, T] would be truth functionally equivalent to "conjunction." You may argue that either of those truth functionalities is the case, and yet that an argument is still structurally but metalogically MP, although what you meant by calling an argument structurally and metalogically MP would be unclear to me.
In any case, I am not sure I agree that an argument is MP in any formulation, as putting an argument in terms of MP would seem to lead to the result that every argument had an "infinite regress" of premises. What I mean is:
P
P?Q
Therefore Q
Is really..
(P^(P?Q))?Q
P?Q
P
Therefore Q
Is really...
((P^(P?Q)?Q)^(P?Q)^P)?Q
(P^(P?Q))?Q
P?Q
P
Therefore Q
Ad infinitum.
Yes, something like that.
Quoting NotAristotle
That seems intuitively correct. This may be close to what @Count Timothy von Icarus was fishing for. The idea is that valid arguments preserve falsity, and not just truth.
But in fact this does not turn out to be correct. For a counterexample,
Quoting NotAristotle
This paragraph is unclear to me, but the degenerate case of the material conditional that I am thinking of is [F X]. [F F] does not strike me as degenerate.
What is at stake here is a direction of evaluation. "The antecedent is false, therefore the conditional is true," is parallel to, "The premise is false, therefore the argument is valid." It is not the value of the conclusion that is at stake, but the validity of the argument (which has to do with guarantees regarding the value of the conclusion).
Quoting NotAristotle
I think every argument does have an "infinite regress" of premises in that way. This is just to say that logical inference (modus ponens) is not capturable in formal or truth-functional language. Trying to capture it in that schema results in an infinite regress.
More simply, modus ponens can be thought of as "follows from," and every inference relies on the notion of "follows from."
Yes, agree. :up:
Quoting Leontiskos
That inferences relies on -follows from- is surely true of deductive arguments; that inductive arguments rely on inference would seem to be true, but I do not know if I would characterize the inference in an inductive argument with -follows from-.
Quoting Leontiskos
Incorrect.
P1. If I am a human woman then I am a human
P2. I am a human woman
C1. Therefore, I am a human
The argument is valid. It's modus ponens. P1 and C1 are true. But P2 is false.
Also:
P1. If I am a woman then I am English
P2. I am a woman
C1. Therefore, I am English
The argument is valid. It's modus ponens. C1 is true. But P1 and P2 are false.
If I were to represent your first argument symbolically, the first one would be:
P?Q
~P
Therefore Q.
But that is clearly not a valid argument. So why is it that the way I've represented your argument does not align with the original non-symbolic argument?
These are two different arguments:
P1. If I am a human woman then I am a human: P ? Q
P2. I am a human woman: P
C1. Therefore, I am a human: Q
P1. If I am a human woman then I am a human: P ? Q
P2. I am not a human woman: ¬P
C1. Therefore, I am a human: Q
The first is valid, P1 is true, P2 is false, and C1 is true
The second is invalid, P1 is true, P2 is true, and C1 is true
You should have read beyond the first few sentences of that post.
I think it depends on what "logical" is supposed to mean.
I would maybe think of these issues as somewhat analagous to software bugs. Video games are a good example. Some classic games that are very well received are also very buggy. You can break them, either making them trivial or else just causing crashes or all sorts of bizarre behavior.
The game still serves its purpose. It does what we want. We just know, "don't do that or you will break it." And if "that" is not something we're likely to do by accident, it really isn't a huge problem. Yet we still might want to patch the bug, but this can also be done in ways that are straightforward and "make sense," or in ways that just seem like ad hoc papering over, just like you can do good body work on a car and restore it, or just pull out the Bondo and patch it.
But when it comes to "correct reasoning," we are talking about something essential to human flourishing, freedom, and even the rise and fall of civilization. So probably want to get to the bottom of any bugs.
Explosion [I]seems[/I] like a bug. Suppose we think common paradoxes of self-reference involve situations where statements are really both true and false. Yet even if this is so, we will likely think that this does not constitute a good reason to think that [I]everything[/I] is both true and false.
Yep, and this goes to @Srap Tasmaner's notion of "degenerate cases" (of, say, the material conditional). If one does not see logic as teleological, then there can be no degenerate or non-degenerate cases.
Note too that formal logic is supposed to involve no rules that require interpretation. But once we introduce "degenerate cases," we have introduced a rule or norm of logic that requires interpretation. This is why formalists dislike the notion of degenerate cases.
For the record, it's not my notion, it's what mathematicians call them. A triangle with interior angles of 180/0/0 would be a degenerate triangle. It allows you to say that any three points in a plane determine a triangle instead of saying that any three non-colinear points do. Mathematicians are generally pleased when they don't have to make special rules to cover edge cases.
Hmmm . . . so the common triangle is a non-degenerate triangle. I doubt I ever used the expression, "degenerate". But I see it's popular on Wikipedia.
However, I've always liked the expression "indifferent fixed point". Rather than "neutral fixed point".
Who would have thought the ridiculous expression A -> -A would go for one K posts?
I was reviewing this thread and it occurs to me that I disagree with the contention that the argument I stated:
Quoting NotAristotle
is metalogically an MP argument. In fact, I think the argument is metalogically neither MP nor simply making use of a material conditional. Instead, I think it is independently an MT argument structurally and does not collapse into MP. If it were a conditional metalogically, it would have to be a degenerative case. Instead of:
1 and 2 therefore 3.
1 and 2.
Therefore, 3.
The initial argument I forwarded would, I think, be more like:
1 and 2 then not 1.
1 and 2.
Therefore not 1.
But, whether such an argument is valid, an argument of this form surely is not convincing, and is therefore a bad argument. And so, if I am correct that the initial argument concerning RAA, MT, and MP is a good argument, then it must not be a degenerative instance of a conditional and must instead be a true to form example of MT structurally and irreducible to MP.
That's modus ponens.
I don't know what that is. I was just referring to the form. From ~1 you can next derive ~(1 & 2), so now you have a contradiction.
I don't remember what the point of all this was supposed to be.
It's usually taken as an inference rule, if that's what you mean.
Quoting NotAristotle
Given MP as an inference rule, can you derive MT? That is, is MT a theorem?
The bottom line shows modus ponens. It doesn't matter what A and B mean:
Now replace B with ¬A. Only one line has both A ? ¬A and A true, and on that line ¬A is also true. That's all it means for the argument to be valid.
However, two of the lines are removed by the laws of excluded middle (the top) and non-contradiction (the bottom)
That gives us:
A ? ¬A
? ¬A
Or
A
? ¬(A ? ¬A)
So the argument is unsound.
I continue to think that modus ponens is always operating metalogically:
Quoting Leontiskos
Quoting NotAristotle
Which was this:
Quoting NotAristotle
Quoting NotAristotle
The most obvious problem is that you seem to be misrepresenting your own argument. Your argument is a modus tollens that metalogically comes to this modus ponens:
You are mixing up the antecedent of 1 with 1 itself. But I think my phrasing is preferable since there really is a 3 in the object-level argument, namely the conclusion. The conclusion is not reducible to the premises, and neither is it "not 1." Once we recognize that the conclusion, namely 3, is the same as "not [antecedent of 1]," we see that the two construals are identical, and that both utilize modus ponens.
I still agree with what I said about the metalogical question .
* Note that we could generalize the form of this modus ponens representation of modus tollens:
(Edit: for the root issue, see my post <here>. If someone like Tones thinks RAA equally entails two completely different conclusions and makes no recourse to explosion, then I think it is obvious that what I have said about RAA is correct. Namely: The RAA inference is not as metalogically secure as the modus ponens inference, and the introduction of something which can be construed as semantically equivalent to Falsum helps show this.)
I would not say I misrepresented my own argument, I would say I miswrote your representation of my argument.
Would you agree that your representation of my argument:
Quoting Leontiskos
could also be written as follows...
A. not-3 then not-2. And 2. Then 3.
B. not-3 then not-2. And 2.
C. Therefore 3.
If you agree that this is accurate, it seems to me that we can see that the argument will be correct because of modus tollens. In particular, we can see that the "A" premise is already agreeable because of modus tollens. If you had instead forwarded premise "A" to be "not-3 then not-2. And 2. Then not-3...," then it's clear that the conditional would not be doing any of the work. The modus tollens makes all the difference does it not? It confirms the truth of the conditional in premise "A" that serves as the basis for the argument's modus ponens. In other words, premise "A" is true "a priori" (if I can use that term here) because of the modus tollens logic, and the truth of that premise gives the basis for the rest of the argument. Premise "B" becomes the only questionable premise. Given it's truth, the argument necessarily works because premise "A" thanks to modus tollens, cannot be questioned.
So maybe you are right that any argument can be written metalogically as a modus ponens, but I think it cannot be so written without the logical inferences that the argument require, in this case a modus tollens is necessary to the argument and cannot be written off as being a hidden modus ponens.
What do you think?
Here is your argument:
Quoting NotAristotle
Here is my construal:
Quoting Leontiskos
Here is your construal:
Quoting NotAristotle
Isn't it clear that your construal is mistaken? Try substituting 1, 2, and 3 into each of our construals and see what happens. 1, 2, and 3 are defined in your original argument.
-
Quoting NotAristotle
No, I don't follow this. And you'll need parenthesis and clearer operators if you are trying to transform my propositions. For example, "X then Y" is not a standard usage. You are either omitting 'ifs' or confusing 'then' with the implication sign (?).
Quoting NotAristotle
But I don't see how your construals are retaining the modus tollens. None of them seem to use modus tollens at all...?
The point about the ubiquity of modus ponens is related to the "therefore" of all arguments. "Therefore" means something like "Hence it follows from the preceding," and that "follows from" move is arguably identical with modus ponens.
A. ((¬3 ? ¬2) ^ 2) ? 3
B. ((¬3 ? ¬2) ^ 2)
C. ? 3
Premise B and conclusion C complete the modus tollens. Premise A seems to be something extra. And actually, I think it would make more sense to make premise A a second conclusion as herein:
B. ((¬3 ? ¬2) ^ 2)
C. ? 3
C2. ((¬3 ? ¬2) ^ 2) ? 3
It is a second conclusion because it is more of a conclusion derived from premise B, rather than an independent premise that is doing work in the argument.
I do not mind saying my argument metalogically can involve modus ponens, but only incidental to and dependent on it first requiring modus tollens.
A
¬A
therefore,
(B?¬B)
being "valid." I guess you would say "yeah, but principle of explosion..." but the principle of explosion is also nonsensical to me.
Again, I see no modus tollens there. The inference needed to achieve C is modus ponens, not modus tollens.
Second, let's do the substitutions that I suggested you do. We start with your construal:
Quoting NotAristotle
And we substitute the premises into their places:
That doesn't make any sense to me. It doesn't even include (1) at all, and it doesn't look anything like my own construal:
Quoting Leontiskos
...which would be:
That's valid and intelligible, at least if we don't fret too much about the "could be" modal idea that I have not been fretting about.
What do you think "valid" means? It just means that the conclusion can be deduced from the premises using the rules of inference. It doesn't mean that the argument is sound or cogent.
Quoting NotAristotle
1. Either I am a man or pigs can fly
(1) is true if either "I am a man" is true or "pigs can fly" is true. Therefore, we can deduce (1) from the premise "I am a man". This is called disjunction introduction:
P1. I am a man
C1. Therefore, either I am a man or pigs can fly
Separately, we can deduce that if (1) is true and "I am a man" is false then "pigs can fly" is true. This is called disjunctive syllogism:
P1. Either I am a man or pigs can fly
P2. I am not a man
C1. Therefore, pigs can fly
So what happens when we combine the two?
P1. I am man
C1. Therefore, either I am a man or pigs can fly (from P1)
P2. I am not a man
C2. Therefore, pigs can fly (from C1 and P2)
From the contradictory premises "I am a man" and "I am not a man" we have deduced the conclusion "pigs can fly". We can deduce anything from a contradiction. This is part of the reason why we agree that contradictions are impossible.
Quoting Leontiskos
All I am doing is giving more detail to "1."
1 = (¬3 ? ¬2)
Given that replacement, premise 2 of the argument as well as the conclusion in line 3 may be written as:
(¬3 ? ¬2) ^ 2
? 3
My understanding of modus tollens is that it is a logical operation of this form:
(¬A ? ¬B)
B
? A
But that is the same form as
(¬3 ? ¬2) ^ 2
? 3
or written otherwise
(¬3 ? ¬2)
2
? 3
that is why I refer to the second and third lines of the argument as a "modus tollens."
Using the terms you suggested, the modus tollens would be:
(1 ^ 2)
? 3
which of course is no modus tollens at all. However that is only because without the detailed replacement for "1" we cannot see that the second and third lines are a veritable modus tollens.
I explained it above.
P1. I am man
C1. Therefore, either I am a man or pigs can fly (from P1, using disjunction introduction)
P2. I am not a man
C2. Therefore, pigs can fly (from C1 and P2, using disjunctive syllogism)
If P1 is true then C1 is true. Therefore, the inference is valid.
You seem to be suggesting that if both P1 and P2 are true then it's possible that C1 is false?
Right.
Which doesn't make any sense. C1 is false if and only if both "I am a man" is false and "pigs can fly" is false. Yet by stipulation "I am a man" is true; that's P1.
So, as per the rules of logic, if both P1 and P2 are true then C1 is true. The inference is valid.