The Propositional Calculus
A forum on philosophy ought have threads on the basics of logic.
I am no logician, although I did do a few undergrad courses half a century ago and have read a bit since. So what I write might be wrong. This is at least partly about seeing what I can remember. There are real logicians hereabouts, and many more folk who think they understand logic but have no idea. Your task might be to learn how to recognise them. This thread might help
So here is a thread on propositional logic, to play with on rainy days. Starting with rules of syntax, it might grow to explain modus ponens, axiomatic systems and perhaps consistency and completeness. If that works we might start another thread on first-order logic.
https://openlogicproject.org
I am no logician, although I did do a few undergrad courses half a century ago and have read a bit since. So what I write might be wrong. This is at least partly about seeing what I can remember. There are real logicians hereabouts, and many more folk who think they understand logic but have no idea. Your task might be to learn how to recognise them. This thread might help
So here is a thread on propositional logic, to play with on rainy days. Starting with rules of syntax, it might grow to explain modus ponens, axiomatic systems and perhaps consistency and completeness. If that works we might start another thread on first-order logic.
https://openlogicproject.org
Comments (127)
Why not just send everyone to Peter Smith's blog?
Sometimes folk say that it is about statements, because a proposition is the intangible thing that "it is raining" has in common with "il pleut". For the most part the terms "proposition" and "statement" will be interchangeable.
What's pertinent is that propositional calculus treats the propositions as whole things, not getting in to the details of the things named or there attributes. That's the job of predicate calculus, with all the x's, a's, f's and funny quantifiers.
What we want to do is to examine the relations between these propositions, rather than their contents. And we don't want to deal with any particular propositions, like "snow is green" or "Fred is English". So here we will just write a letter for any proposition. And following convention we will start at "p", I suppose because "proposition" starts with a p. The proceeding alphabetically, we can write p, q, r, and so on, with the presumption that this sequence does not finish at z but goes on as far as you like, including to infinity and beyond.
Quoting Banno
I disagree. I've done my share of teaching some basic logic here, helping folks with homework, answering questions. We should do that. But we should point people learning from scratch, especially on their own, to better resources.
In fact, for many sorts of questions, stackexchange is a better resource than us.
I understand why you might want to brush up, but why should we watch?
Maybe I just don't understand what sort of conversation you expect to have by presenting textbook material. Are we going to do philosophy of logic? If so, why the textbook review?
Next we need some rules about what one can write.
There's a few other symbols. These are ~, &, v, ? and ?. They have names in English, but for now we might leave that aside and treat them just as arbitrary signs. There are rules for how we can use these symbols. We can't just write anything, like
Some string of symbols are well-formed, others are not.
As already mentioned, we are allowed to write any lower-case letter, like
or
and so on.
And lets invent symbols for any well-formed formulae( wff): ?, ? and so on
Now we can write the rule:
So since "p", we can now also write "~p". And since we can write "~p", we can write "~~p", and so "~~~p" and so on. The process is iterative.
We can do the same for the other symbols. We get the formation rules:
If ? and ? are wff, then
You don't have to watch. Go do something else.
I asked why we should watch, and you answered that I don't have to.
That is not logical, Captain.
But I'll take your "suggestion" to go away. Best of luck with your new blog.
Our p's and q's are standing in for propositions or sentences. Propositions and sentences are the sort of thing that can be either true or false, and we can assign a T or an F to them, to show this:
and assign values to the other symbols,
and generalise this for any WFF
So colloquially, if some proposition is true, then it's negation is false and if some proposition is false, its negation is true.
Thanks. I'm kinda hoping that other folk might butt in and add stuff.
I'm. using https://www.tablesgenerator.com/text_tables# to make generating tables a bit easier.
Some notations just us a dot "." for "&", or "?".
Here we see the similarity to the English connective, "and".
and
This differs somewhat from one common colloquial use. In colloquial English if you are offered the chocolate cake or the cheesecake you would be expected to chose one or the other, but not both. Here, you can have both. It's called an inclusive OR, the alternative being an exclusive OR.
The "?" is called a hook, and is sometimes represented by a ?. Colloquially it is "if ? then ?".
The Laws of Thought
1. Identity A = A
2. The law of the excluded middle p v ~p
3. The law of noncontradiction ~(p & ~p)
---
The simplest argument form modus ponens (abbrev. MP)
1. p [math]\to[/math] q
2. p
Ergo
3. q
---
Formal fallacies associated with modus ponens
Denying the antecedent/Inverse fallacy
1. p [math]\to[/math] q
2. ~p
Ergo
3. ~q
Affirming the consequent/Converse fallacy
1. p [math]\to[/math] q
3. q
Ergo
4. p
Cheers.
The law of identify holds between individuals, and as mentioned earlier propositional calculus deals in whole propositions. SO strictly the law of identity is a part of predicate clacualus rather then propositional calculus.
Here's the truth table for excluded middle:
and for noncontradiction
Notice that it is the negation of (p & ~p), a contradiction?
The contradiction has "F in each row in the truth table. The tautology has T.
So the first way we have of proving a theorem is looking at its truth table.
Noted! I don't want to go into the formal aspects of the law of identity and I guess you share that sentiment. Let's leave it at the level of intuitive understanding, that a thing is identical to itself, for the moment.
Quoting Banno
The point to truth tables is that given a set of rules on how logical connectives (&, v, ~, [math]\to[/math]) function (quite like mathematical operations),
a) What will be the final truth value of compound statements like A & C or B v T or E [math]\to[/math] I or ~W or more complex compound statements.
b) Evaluate for validity of an argument: Is there a possible world in which, for a given argument form, all the premises are true and the conclusion false? If there is, the argument is invalid and if there's none, the argument is valid.
c) Check for consistency. Is there a possible world in which all the propositions are true? If yes, the set of propositions in question are consistent; if no, inconsistent (we can derive a contradiction via conjunction).
---
To pick up where I left off (for those interested).
Argument form: Modus tollens (abbrev. MT)
1. p [math]\to[/math] q
2. ~q
Ergo
3. ~p
But natural deduction is more common and more direct, so we might start there.
Solid copy!
Argument form: Disjunctive syllogism (abbrev. DS)
1. p v q
2. ~p
Ergo
3. q
1. p [math]\to[/math] q
2. q [math]\to[/math] r
Ergo
3 p [math]\to[/math] r
Sure, but they have an additional role in showing which wff are tautologies, which are contradictions and which are neither.
If the column for some wff is all T's, it's a tautology; it's never false. If they are all F's it's a contradiction. If it's a mix, then it is contingent - it depends on the value of each proposition.
Quoting Agent Smith
Te use of "possible world" is problematic here. Possible worlds are used in modal logic, which again comes after (builds on) predicate logic, which is turn builds on propositional logic. Small steps.
Consider the table for ( p & q):
Whether ( p & q) is true or not depends on the values of p and of q. As the table shows, in the first row, if both o and q are true then (p & q) will be true. But, as the remaining rows show, if either or both are false, the conjunction ( p & q) will also be false.
But consider (p & ~p)
In this case, whether P is true or false makes no difference, the conjunction will aways be false. Now look at (p v ~p)
in this case, regardless of whether p is true or false, (p v ~p) is true.
Nothing to do with possible worlds.
I thought each truth value assignment specified a possible world. So given a proposition p, p is T is one world and p is F is another, it being impossible for p to be T and F in one world (LNC).
T = True
F = False
LNC = The law of noncontradiction
will always be true; (p v ~p).
But
will never be true; (p & ~p).
And
might be either; (p & q).
We don't ned talk of possible world yet. Only of different propositions and their relations.
So you have been doing a rough form of natural deduction in your posts.
In natural deduction, any well formed formula can be take and an assumption. What we do then is to use a few rules of deduction to work out the consequences of those assumptions.
To this we add modus ponens, as you described it:
We might set it out as
1. p (A)
2. p?q (A)
---------
3. q. (1,2,MP)
It's the same, just adding in brackets the justification for the deduction. (A) is an assumption, (1,2,MP) says we deduce line 3 from lines 1 and 2 by modus ponens.
Some systems write MPP for modus ponendo ponens. You might be interested in the Latin.
The line just equates to "ergo".
1. p?q (A)
2. q?r (A)
3. p (A)
the conclusion?
[hide="Reveal"]4. q (1,3,MP)
5. r (2,4,MP)[/hide]
Or
1. p?(q?r) (A)
2. p?q (A)
3. p (A)
[hide="Reveal"]4. q?r (1,3,MP)
[hide="Reveal"]5. q (2,3,MP)
[hide="Reveal"]6. r (4,5, MP)[/hide][/hide][/hide]
All T's down the sixth column, which is the conclusion, hence MP is a tautology.
Ugly, but effective.
:blush:
I don't wanna derail a good thread.
Gracias for the explanation. :up: I just looked up natural deduction on Wikipedia & Stanford Encyclopedia of Philosophy. It goes into the details of the system, more than I can take on at the moment. To beginners like myself, natural deduction is a system developed to imitate how people actually reason without compromising on the rigor of more formal systems; hence the natural in the name.
I need feedback.
Yet natural deduction is as powerful and valid as an axiomatic system.
Modus Tollens (MT) allows the following derivation:
1. p?q (A)
2. ~q. (A)
____
3. ~p (1,2 MT)
And double negation
1. ~~p
____
2. p (1, DN)
If we adopt the convention of ? as "Therefore", and Greek letters ? and ? (phi and psi) for any wff, we can list the derivations rules as follows:
[b]
A: Any wff may be assumed
MP: ???, ?, ??
MT: ???, ~?, ?~?
DN: ~~? ??[/b]
:snicker: Well, to me, modal logic is part and parcel of propositional logic (thank goodness the OP is delimited to that; my predicate logic is rusty). Consider the definition of validity: An argument is valid if it is impossible for the premises to be true and the conclusion false. Mind you, my knowledge is limited to natural deduction as found in introductory texts on logic.
Quoting Banno
I'm afraid I won't be able to further the discussion on that point.
[quote=Dr. Lanning (I Robot)]My responses are limited. You'll have to ask the right questions.[/quote]
Propositional logic deals in whole propositions and the connections between them. The next level is predicate logic, which takes the p's and q's of propositional logic and opens them up to expose the individuals and predicates that constitute them; so p becomes f(a). Modal logic goes a step further by adding prefixes to the propositions; usually necessarily p and possibly p: ?p and ?p; but in other variations terms for obligation, for tense (in a temporal sense), and for belief.
SO better to steer clear of talking of propositional calculus as modal.
---
Can you take a look at a statement that appears on Wikipedia regarding natural deduction?
[quote=Wikipedia]A theory [natural deduction] is said to be consistent if falsehood is not provable (from no assumptions)[sup]1[/sup] and is complete if every theorem or its negation is provable using the inference rules of the logic[sup]2[/sup].[/quote]
What do 1 and 2 mean exactly? My guesstimates below:
1. Falsehood = Contradiction
2. I have no clue. I thought completeness meant every true statement is provable.
The calculus constitutes a formal language. Yep, the language will be consistent if it is not possible to derive any contradictions. It will be complete if we can derive every tautology.
Ok :grin: You see, that's an explanation that needs an explanation in my pathetic universe. Don't worry, I'll work on the gaps in my knowledge, just not now; I have a lot on my plate.
Given any sequence
1. ? (A)
.
.
.
5. ? (with whatever justification)
6. ???
CP allows us to take any assumption and put it before any deduction from that assumption. It serves to reduce the number of assumptions by one.
An example: show p?q ? ~q?~p
1. p?q (A)
2. ~q (A)
3. ~p (1,2,MT)
4. ~q?~p (2,3,CP)
Note how the assumption ~q is incorporated into the conclusion ~q?~p.
1. p v q
2. p [math]\to[/math] r
3. q [math]\to[/math] s
Ergo
4. r v s
Given any two propositions, we can join then with &. (&I)
& elimination
Similarly, given any conjunction, we can drop one of the conjuncts. (&E)
Or Introduction
(vI) Given any proposition, we can stick it in a disjunction:
Or Elimination
(vE). More complex. If ?v? and both ??? and ???, then ?
1. p & p
Ergo
2. p
---
1. p v p
Ergo
2. p
Nice. Lets' take a look at this. We can rewrite this as:
1. p&p. (A)
______
2. p (&I)
or on a single line as
p&p ? p
Where again ? is "therefore", we read "p and p therefore p"
Recall that conditional proof removes an assumption. Suppose we took a step further, and using Conditional Proof CP to remove the assumption:
1. p&p. (A)
2. p (&I)
_____
3. (p&p)?p (1,2,CP)
When we write this on a single line we no longer need the assumption (1); so
?(p&p)?p
The wff is true regardless of the assumptions made. A wff of this sort is called a theorem. A theorem is a wff that is proven with zero assumptions.
Consider:
LANGUAGE
symbols:
sentence letters P Q R ...
connectives ~ v & -> <->
left and right parentheses ( )
formulas:
a sentence letter alone is a formula
if X and Y are formulas, then so are ~(X) (XvY) (X&Y) (X->Y) (X<->Y) [we omit parentheses if no confusion results*]
nothing else is a formula
* And if we used Polish notation, then we wouldn't need parentheses at all.
BOOLEAN FUNCTIONS
A Boolean function is function* whose domain is, for some natural number k, the set of k-tuples over {0 1} and whose range is a subset of {0 1}.
* With the exception of '1' mentioned below as a 0-place function even though it is not an actual function.
There are two 0-place Boolean functions:
1
the value 'truth, sometimes represented by the constant 't'
0
the value 'falsehood', sometimes represented by the constant 'f'
There are four 1-place Boolean functions:
{<1 1>
<0 1>}
the constant function that maps any value to 'truth'
{<1 0>
<0 0>}
the constant function that maps any value to 'falsehood'
{<1 1>
<0 0>}
the identity function
{<1 0>
<0 1>}
negation
There are sixteen 2-place Boolean functions:
{<<1 1> 1>
<<1 0> 1>
<<0 1> 1>
<<0 0> 1>}
reduces to t
{<<1 1> 1>
<<1 0> 1>
<<0 1> 1>
<<0 0> 0>}
disjunction
{<<1 1> 1>
<<1 0> 1>
<<0 1> 0>
<<0 0> 1>}
converse of material implication
{<<1 1> 1>
<<1 0> 1>
<<0 1> 0>
<<0 0> 0>}
identity on the first coordinate
{<<1 1> 1>
<<1 0> 0>
<<0 1> 1>
<<0 0> 1>}
material implication
{<<1 1> 1>
<<1 0> 0>
<<0 1> 1>
<<0 0> 0>}
identity on the second coordinate
{<<1 1> 1>
<<1 0> 0>
<<0 1> 0>
<<0 0> 1>}
material equivalence
{<<1 1> 1>
<<1 0> 0>
<<0 1> 0>
<<0 0> 0>}
conjunction
{<<1 1> 0>
<<1 0> 1>
<<0 1> 1>
<<0 0> 1>}
negation of conjunction
{<<1 1> 0>
<<1 0> 1>
<<0 1> 1>
<<0 0> 0>}
negation of material equivalence
{<<1 1> 0>
<<1 0> 1>
<<0 1> 0>
<<0 0> 1>}
negation of the second coordinate
{<<1 1> 0>
<<1 0> 1>
<<0 1> 0>
<<0 0> 0>}
negation of material implication
{<<1 1> 0>
<<1 0> 0>
<<0 1> 1>
<<0 0> 1>}
negation of the first coordinate
{<<1 1> 0>
<<1 0> 0>
<<0 1> 1>
<<0 0> 0>}
negation of the converse of material implication
{<<1 1> 0>
<<1 0> 0>
<<0 1> 0>
<<0 0> 1>}
negation of disjunction
{<<1 1> 0>
<<1 0> 0>
<<0 1> 0>
<<0 0> 0>}
reduces to f
For k>2, a k-place Boolean function can be expressed as a combination of 2-place Boolean functions.
The connectives are interpreted as Boolean functions:
~ is interpreted as negation. You can see how the truth table for ~ is another way of representing this Boolean function.
v is interpreted as disjunction. You can see how the truth table for v is another way of representing this Boolean function.
& is interpreted as conjunction. You can see how the truth table for & is another way of representing this Boolean function.
-> is interpreted as (material) implication. You can see how the truth table for -> is another way of representing this Boolean function.
<-> is interpreted as (material) equivalence. You can see how the truth table for <-> is another way of representing this Boolean function.
There we mentioned only one 1-place connective and only four 2-place connectives. That's okay, because this is an adequate set to represent any other Boolean function by a combination of these connectives.
There are combinations of Boolean functions that are adequate too:
negation of conjunction ("not both")
negation of disjunction ("neither nor")
negation with disjunction
negation with converse of implication
negation with implication
negation with equivalence
negation with conjunction
negation with negation of equivalence
negation with negation of implication
negation with negation of converse of implication
In other words, these are adequate:
negation of disjunction
negation of conjunction
negation with any one of these: disjunction, conjunction, implication, negation of implication, equivalence, negation of equivalence, converse of implication, negation of converse of implication
A NATURAL DEDUCTION SYSTEM:
Comment: Unlike quantifier logic with predicate symbols of arity greater than 1, sentential logic doesn't really need a calculus, because checking for sentential validity is mechanical (for example, using truth tables). But we like to give a calculus anyway:
Notation:
P, Q and R are any formulas
G, H and J are any sets of formulas
u for union
|- for implies
RULES:
Assumption
Enter P on a line and charge that line to itself.
{P} |- P
______
Deduction
If Q is inferred from P along with possibly other lines, then infer P->Q and charge it with all lines charged to Q except the line for P.
If Gu{P} |- Q, then G |- P->Q
______
Modus Ponens
From P and P->Q, infer Q and charge it with all lines charged to P and to P->Q.
If G |- P and H |- P->Q, then GuH |- Q
______
Intuitionistic Contradiction
If a contradiction is inferred from P, along with possibly other lines, then infer ~P and charge it with all lines charged to the contradiction except the line for P.
If Gu{P} |- Q and Hu{P} |- ~Q, then GuH |- ~P
______
Classical Contradiction
If a contradiction is inferred from ~P, along with possibly other lines, then infer P and charge it with all lines charged to the contradiction except the line for ~P.
If Gu{~P} |- Q and Hu{~P} |- ~Q, then GuH |- P
______
Conjunction Introduction
From P and Q, infer P&Q and charge it with all lines charged to P and to Q.
If G |- P and H |- Q, the GuH |- P&Q
______
Conjunction Elimination
From P&Q, infer P and charge it with all lines charged to P&Q.
From P&Q, infer Q and charge it with all lines charged to P&Q.
If G |- P&Q, then G |- P
If G |- P&Q, then G |- Q
______
Disjunction Introduction
From P, infer PvQ and charge it with all lines charged to P.
From Q, infer PvQ and charge it will all lines charged to Q.
If G |- P, then G |- PvQ
If G |- Q, then G |- PvQ
______
Disjunction Elimination
From PvQ, P->R and Q->R, infer R and charge it with all lines charged to PvQ and to P->R and to Q->R.
If G |- PvQ and H |- P->R and J |- Q->R, then GuHuJ |- R
DEFINITION:
P <-> Q stands for (P->Q)&(Q->P)
For example p [math]\to[/math] q = ~p v q and (p & q) = ~(~p v ~q).
It reminds me of how subtraction is rendered as addition of the opposite like so: a - b = a + (-b) [negative numbers] The same thing can be done with division thus: [math]a \div b = a \times \frac{1}{b}[/math] [fractions].
Sorry, this was goofy:
[s]It is sometimes desirable to effect such a transformation into a form with only ORs and NOTs (disjunctive normal form) or only ANDs and NOTs (conjunctive normal form).[/s]
CNF and DNF are interesting, but not as described. It's a question of whether you have only ANDs outside and ORs inside parentheses or the other way around.
A & (~B v C) & (D v ~E) is CNF
(A & B) v (~C & D) v (~E & ~F) is DNF.
(Your question called them to mind, as getting something into a canonical form, but then I somehow didn't notice I was writing gibberish! Ah well.)
You can also get by with a single connective, if you're so inclined, the Sheffer stroke, " | ", read "not both," or NAND.
multiplication and addition in Boolean algebra
intersection and union in set theory
universal quantifier and existential quantifier in predicate logic
Each pair is a pair of duals.
Quoting TonesInDeepFreeze
True, but no one would understand us.
Anyone remember programable calculators that used reversible polish notation? I recall using one that was the size of a typewriter... er, desktop PC.
Ah, interesting. From the definition of each: ?(x)Fx? Fm v Fn v Fo... and (x)Fx?Fm&Fn&Fo...?
To read this, an input on the left gives the output on the right, so 1 gives 1 and 0 gives 1, hence always true...
Quoting TonesInDeepFreeze
...always false...
Quoting TonesInDeepFreeze
...always the same...
Quoting TonesInDeepFreeze
...always negated. And so on.
Muchas gracias. I'll read the linked article when I get the time.
Free [s]will[/s] won't? Choices offered, one/more denied. :chin:
If the domain is finite, then an existential statement is equivalent to a finite disjunction, and a universal statement is equivalent to a finite conjunction. But, in ordinary logic, there are no infinite disjunctions nor infinite conjunctions, but, for infinite domains, we can think of the quantifiers as "in a sense" representing "infinite disjunctions" and "infinite conjunctions".
Notice that:
ExS <-> ~Ax~S compares with (P v Q) <-> ~(~P & ~Q)
AxS <-> ~Ex~S compares with (P & Q) <-> ~(~P v ~Q)
Lookism? In the ideaverse? No wonder some of these memes - the aesthetically challenged ones in all likelihood - override our self-preservation instincts and make us fatally courageous.
To that end, and because it seems at first odd, i'd like to point out that an inconsistent language - or theory, if you prefer - is one in which any and every theorem can be deduced; on in which everything is true.
Take ? to be any theorem at all.
Then since we can write:
(p & ~p) ? ?
if a contradiction is true in our system, then anything is derivable.
So conversely, if just anything is derivable, the system is inconsistent.
And so, if there are things that cannot be derived, then the system is consistent...
That's an outline of one strategy for a proof of consistency.
Great! You brought up contradiction. When I first encountered contradictions in real life, back when I was in my teens & later on, it was, now I realize, incomprehensible or incomputable (that was the pre-logic phase of my life). I couldn't make sense of it at all. I reasoned to myself there's something fundamentally wrong with statements like p & ~p. It's snowing AND it's not snowing is "wrong" for the reason that the the second conjunct denies/negates the first - they cancel each other out and its as if someone who utters/writes a contradiction says nothing at all (+y + -y = 0].
[quote=Laozi]Those who know don't speak, those who speak don't know.[/quote]
"What a genuine word of God would look like?"
Art48
[quote=Banno]Silence.[/quote]
[quote=Agent Smith]??? ?????
(Ati sundar: Glorious/most beautiful).
Accounts of God having answered prayers is total hogwash! That however doesn't mean we stop praying.[/quote]
Then I took an introductory course in logic and came to realize formalization (logic to logicians) meant that contradictions, their unacceptability to be precise, need to be put on firmer ground than just than intuition I outlined in the previous paragraph. This, I came to know, takes shape in the famous ex falso quodlibet (anything follows from a contradiction). How this happens is as follows:
1. p & ~p
2. p [1 Simp]
3. p v q [2 Add]
4. ~p [1 Simp]
5. q [3, 4 DS]
q here stands for any and all statements & even their negations [you mentioned inconsistency and this is it].
According to logicians, contradictions, and I quote, "trivialize the notion of truth". What this means isn't clarified in the books I read. What does it mean Banno? If p is true and ~p is also true, I would say that negation (~) is being trivialized (it doesn't matter whether its present/absent). Please help! :smile:
If contradictions are just like equations which equal zero, then the conclusion from snowing and not snowing would be "nothing". So it couldn't be the case that "anything" follows from contradiction, because "nothing" is not the same as "anything". But "nothing" is really wrong, because snowing and not snowing are statements concerning "snowing", and negating that subject does not produce nothing in the most general sense, it leaves alone everything else except "snowing", where the contradiction creates a problem.
Following on from the previous posts, if any proposition follows from a contradiction, then if the contradiction is true, any proposition is true; that is, there is no longer a difference between a true and false propositions, and so every proposition is true: truth is trivial.
I still don't get it! I need another example of triviality to help me grasp the notion of trivializing. I read the Google definition and it says trivializing is to reduce the importance or significance of something.
I've encountered the notion of the trivial in mathematics. Take Fermat's last theorem: [math]a^n = b^n + c^n[/math] for n [math]\neq[/math] 2 (where n = 2, we have Pythagoras' theorem). The trivial solution is n = 0. What does "trivial" mean in this case? Does it mean obvious and/or uninteresting? I think it means the latter but can't say for sure.
This is untrue. Propositional calculus does not even have first-order quantifiers (forall, exists) to have modal operators. And modal operators are behaviorally analogous to quantifiers (and are rightly a kind of quantifier that doubles as an operator). Zero order logics, like prop. calc, by definition do not quantify over anything.
You might be conflating model-theoretic valuations (of prop. calc) or interpretations (of FOL) with the nodes that exist in possible world semantics for modally extended versions of FOL (like FOL? + S5).
A wff is a validity just in case it is modeled by all valuations (or interpretations).
:ok:
Consistency follows from soundness. Proving soundness is not deep. We ordinarily just do induction on the length of derivations.
Quoting Banno
'inconsistent language' doesn't make any sense.
A language is not something that can be consistent or inconsistent.
It's important to understand why that is the case:
A language doesn't make assertions, perforce, it doesn't assert contradictions. Sentences made in a language do make assertions, and a theory is a set of sentences closed under derivability, so theories are consistent or inconsistent.
Informally, think of it this way, for example with a natural language:
English doesn't make assertions. English is used to make assertions.
Quoting Banno
That is wrong in two ways:
(1) It is not informative. It should be, "With an inconsistent theory, every sentence can be deduced". By definition, a theorem is a deducible sentence, so with every theory (consistent or inconsistent), of course every theorem can be deduced, because being deducible is what it means for a sentence to be a theorem.
(2) It is not the case that every sentence is true in an inconsistent theory. Sentences are not even true or false in theories. Rather, sentences are true or false (and never both) per any given model. An inconsistent theory asserts contradictions, but that doesn't make the contradictions true. Indeed contradictions are false in all models.
Quoting Banno
Same thing. Sentences (including contradictions) are not true or false in theories. Rather they are true or false in any given model. Meanwhile, a model of a theory is a model in which every theorem of the theory is true. But a contradiction is not true in any model, so an inconsistent theory has no models. There are models for the language of an inconsistent theory, but those are not models of the theory.
What we do say is, "From a contradiction, we may derive any sentence." But there is no such thing as a "contradiction that is true in a theory". Again: sentences are not true or false in a theory (sentences are true or false in models), and (in ordinary propositional logic) there is no such thing as a "true contradiction".
Also, for example, we might say an inconsistent theory is trivial, because every sentence is a theorem, so, given that a theory is inconsistent, there is no work involved in determining which sentences are theorems.
Another example, the empty set is a function, since the empty set vacuously satisfies the definition of 'is a function'. We could say it is the most "trivial" function. In a case such as the empty set being a function, we might also say "it's vacuously the case that the empty set is a function".
'Triviality' is not a deep notion in this context. It's just a way for mathematicians to point out that they recognize that certain claims are correct but quite obviously so, or that certain objects have certain properties but in a not very informative way.
Doubtless all true, and should be accounted for. In my defence this thread is not intended to be so formal but to get on with outlining what is going on.
By all means, rain on the parade, but perhaps not so heavily as no one shows up.
Quoting TonesInDeepFreeze
Some simplified detail might be fun.
Confusing fundamental concepts is not mere informality. And any outline based on such fundamental confusions cannot be other than itself more confusion.
Quoting Banno
I'm not doing any raining. I'm giving you information and explanation.
Quoting Banno
Are you suggesting that I provide more detail, or are you suggesting that you might be providing more detail?
Wanna run something by you if it's ok with you. True that if contradictions (p & ~p) are allowed, "every proposition is true" but every refers not to logically independent propositions like "some swans are not white" and "Socrates was bald" but to logically dependent propositions like "all swans are white" and "some swans are not white" (contradictions). So the argument from the principle of explosion ( ex contradictione sequitur quodlibet) is, in fact, the circular argument: contradictions are unacceptable because contradictions are unacceptable. :chin:
No, that is not the case. I explained in detail why.
Now Banno's misconception has been inherited by you.
Quoting Agent Smith
What argument are you referring to?
The proof that from a contradiction all statements are provable is not circular.
Quoting Agent Smith
Speaking of ... the above is petitio principii.
I don't know what that means.
So, if you posted it to make us even, you succeeded.
For any wff, or for that matter any sentence in a natural language, ? and ?
(? & ~? ) ??
Quoting Agent Smith
:brow: Not sure what "logically dependent" is doing here.
If (Socrates is bald and socrates is not bald) then swans are green.
Quoting Banno
Does the truth/falsity of "Socrates was bald" depend on the truth/falsity of "Some swans are not white"?,
Some nonstandard logics are motivated precisely by the wish to avoid the principle of explosion by defining implication otherwise.
[math]\begin{array} {|r|r|}\hline ? & ? & ? ? ? \\ \hline T & T & T \\ \hline T & F & F \\ \hline F & T & T \\ \hline F & F & T \\ \hline \end{array}[/math]
(? &~?) is only ever false, so we are looking at the bottom two lines. In both, ? is true.
Hence regardless of the meaning of ?, if (? &~?) then ? is true.
The third and fourth rows.
I suppose it's additionally a consequence of bivalence, since every consequent must land you on row 3 or row 4 and nowhere else.
But perhaps not all that clearly. If one sets (? &~?) as true, then since (? &~?)?? where ? is any wff, every wff would be true.
yep. Fixed.
Quoting Srap Tasmaner
Rejecting bivalence presumably implies rejects (? &~?)??.
Quoting TonesInDeepFreeze
Well, you are better informed than I. If you did it might be interesting.
Not something we need to address.
I suppose I should have put my point this way: short-circuiting is, here anyway, an unofficial procedure. If the truth table is our definition of implication, then there is no option not to consider the truth-value of the consequent, even though it's unnecessary. (We can short-circuit.) So it's almost worth pointing out that every consequent gets you to row 3 or to row 4 because no third is given.
No, with utmost clarity. You can go back to the posts.
Quoting Banno
One does not do that.
Again, statements are true or false (and not both) per a given model.
If a theory has a statement of the form P & ~P, then the theory has no model.
If one says "'P & ~P' is true", then one has simply stated a falsehood. It doesn't follow from that falsehood that everything is true.
You are conflating the syntactics with the semantics.
Yes, syntactically:
P & ~P |- Q
But semantically:
The above syntactical principle doesn't provide that Q is true in any particular model. All it does (via the soundness theorem) is provide that Q is true in any model in which P & ~P is true. But there are no models in which P & ~P is true.
Quoting Banno
Please go back to my post in which I explained with exactitude why that is not the case.
Get a good book on mathematical logic to learn the notions of provability, truth in a model, entailment, etc.
I did.
Quoting TonesInDeepFreeze
That's, indeed, rather the point. We don't do that because it undermines the enterprise at hand.
Quoting TonesInDeepFreeze
I have several. Your quibbles are doubtless correct. But not helpful.
I've no intention of writing another logic text that will satisfy @TonesInDeepFreeze. End of tread, I suppose.
I wouldn't call the points @TonesInDeepFreeze has made "quibbles" but I would call them "helpful". On the other hand, I wonder how accessible any of this is to someone who has no background at all in logic.
Ah, I see you've reached the same conclusion.
I liked the word "informal" in your previous post, it's just that propositional calculus is a formal system. It's a branch of mathematics.
If you want to raise the logical literacy of the forum, perhaps it would be better to aim at that dialect called "philosophical English," a dialect spoken by people familiar with formal systems. The traditional early chapters of a logic textbook try to show how the logical constants capture some of what we mean by familiar idioms. (The exception might be Kalish and Montague, because they're not kidding.) They give students exercises in "translating" English into the symbolism that's been defined.
But many philosophers today write in a style that's more like translated and then translated back. The style most of the SEP is written in, if it's not clear what I mean. So we're not quite talking about informal reasoning here, which is interesting in its own right but different. We're talking about a kind of semi-formal style, which aims at precision and explicitness.
In this case, the exercises would be a matter of making what you say more precise and more explicit, though still English. A guide to this style could embed a certain amount of the classical logic in everyday use in philosophy, but in English, not in mathematical notation.
That was discussed, but I wish also to state for myself:
My comments here have not been mere quibbles. They are important considerations, especially for keeping clear the distinction between syntax and semantics.
The comments would be helpful for anyone who knows about the subject but can use some help with a refresher on these particular points. Or, if one doesn't know enough about the subject, then my comments may suggest learning more about the subject.
I'm curious about your take on Kalish, Montague and Mar.
They don't give as many examples and exercises as most books, but they do give some.
What do you mean by "they're not kidding"?
First, I can't speak to the later revision.
I've tried reading Montague, but it's pretty demanding, so he's still an aspiration for me. I do know a little about his views, in a general way, and his place in the history of logic and formal semantics. So I worked through a good chunk of the logic book, to see how he (and Kalish) handled classical logic, and there's a very different vibe to it from many logic textbooks.
Now I could be wrong, but this was my impression. Most logic textbooks go something like this: hey, you youngsters reason, everybody does, but I can show you a better way to do that, one that isn't hampered by the messiness and ambiguity of a language like English; we're going to show you a kind of language made just for reasoning; you'll recognize some of it from what you've been trying to do in English, and we'll show you how to take those groping attempts at reasoning in a medium not really suited for it, and instead do it in our nice clean system. (This is a vaguely Fregean conception, I guess.)
That is not what Montague seems to be up to at all. He and Kalish are obsessively precise about how the familiar English form relates to their notation -- something many logic textbooks swish by with some inadequate handwaving. (If memory serves, this is one of the things that Peter King will pillory a book for, playing it too loose with what exactly logical schemata are supposed to be, etc.)
And I think that's so because Kalish and Montague are not offering an alternative to reasoning in English -- a formal language you would translate some English into -- but an account of how the logical constants and quantifiers in English actually work. To put it plainly, I think this book presents something a lot more like a formal semantics of the logical constants and quantifiers. The result may look similar to what other textbooks are up to, because there is a formalism, but the relation of the formalism to the natural language English is quite different, and I think you can tell that it's different in the way the book is written.
So that's what I meant by "they weren't kidding." This book is not about a formal system someone invented that you might find a useful alternative to English; this book is a formal account of a subset of English, the words we use in connection with reasoning.
Does that make sense? Am I way off base?
Of course, Monatague is famous for his work in formalizing natural language. It's an interesting question how much that involvement bears on the introductory textbook 'Logic: Techniques Of Formal Reasoning'. But I would be wary of thinking that the book suggests that natural languages lie down so easily that we can just read off its sentences always unambiguously into formal sentences. (I'm not saying that you're saying that is what the book suggests.)
I would guess that the authors would acknowledge that English (for example) has different senses of the connectives. For example, I would be surprised if the authors held that "if then" is always in English the material conditional.
I agree that the book is very careful indeed in how it states things and formulates things. I always recommend the book.
Absolutely.
Quoting TonesInDeepFreeze
I don't happen to remember, but I would presume there's a way around needing to make such a claim. You can stipulate that your account applies to one way of using a word or a phrase, though there may be others, and still claim to have given an account of a subset of English usage, even if that subset doesn't take words as the joints it's carving at.
Quoting TonesInDeepFreeze
I found it really fascinating.
I originally taught myself logic out of Quine's Methods of Logic, maybe the 3rd or 4th edition, and though he's meticulous, it's meant to be more accessible than the mathematical logic book. He has that breezy style and a lingering sort of logical positivist disdain for the unscientific, which English certainly is, so you're made to feel you're learning how to think more scientifically.
Kalish and Montague doesn't feel like that at all. It's a theory of what you were actually doing some of the time. There are very precise rules about what English words go where in the schemata, because it's intended to apply to, not replace English. So my memory of it is, anyway.
I hope I say all this right (I'm pretty rusty):
Df. A model M is a model of a set of sentences G iff every member of G is true in M.
Df. A set of sentences G entails a sentence S iff every model of G is a model of {S}.
Df. A system T has the soundness property iff for every set of sentences G and sentence S, if S is derivable from G then G entails S.
Thm. If all the axioms of T are logically true, and T has the soundness property then set of theorems of T is consistent. Proof outline: If the set of theorems of T were inconsistent, then there is sentence P & ~P derivable from the axioms. But the axioms are true in all models, and soundness provides that the axioms entail P & ~P, but that is impossible since P & ~ P is false in every model.
Outline of proving a system has the soundness property: Base step: Prove that all the axioms are logically true. Inductive step: Suppose we are at step n in a derivation and soundness has obtained. For step n+1, show that the rules of inference are truth preserving.
This is found in any textbook in introductory mathematical logic.
This is interesting. Sometimes I encounter people who say symbolic logic is dogmatic because it demands that 'if then' must be taken as the material conditional even though English speakers use 'if then' in other senses. And it is true that ordinarily English speakers don't have the material conditional in mind. If you asked 10000 people whether "If London is in Asia then Groucho Marx was an aviator" is true, you'd be lucky if even one person said it is true.
So, it is helpful when an intro textbook makes a disclaimer that use of the material conditional in symbolic logic is not to be construed as a claim that the material conditional captures the many everyday senses of 'if then'. And, as far as I recall, Kalish, Montague and Mar (KMM) does not make that disclaimer.
(There are some comments about looseness on page 10.)
Another thing I wish were different in the book: Truth tables are not mentioned until page 87, after the propostional calculus has all been specified. I think it's much better to explain the truth tables before specifying the proof calculus. That way, the student can see how the proof calculus is truth preserving. Otherwise, the student is first all wrapped up in a bunch of rules while the student doesn't know the motivation for those rules.
I wish there were an intro textbook just as precise as KMM and Mar, but with a more streamlined proof system. The boxes method is intuitive, and helped me a lot as a beginner. But I would like to see a textbook with a more streamlined natural deduction system that uses line accumulations instead. (I think I posted such a system earlier in this thread?)
That said, KMM is still my favorite intro symbolic logic textbook. It certainly set me up with a solid foundation.
Quine is always quite the pleasure to read. Church too (for me, the intro chapter in 'Introduction To Mathematical Logic' is the definitive primer). Those are two of my heroes. Smullyan also is a great writer. And I particularly like Boolos. And for textbooks, Enderton is great.
I think this is the first time I've ventured into the category of 'Logic & Philosophy of Mathematics'.
Its description: What are logic and mathematics? How are they related? How do they relate to human reason and to the world?
The last question grabs my attention. To answer that, for sure, a thread on the basics of logic would come in useful. I'm not so certain about mathematics...what is the 'philosophy of...'?
https://plato.stanford.edu/entries/philosophy-mathematics/
Moving on.
Quoting Srap Tasmaner
I'm interested in being logical and literate but not to the point of formal truth tables.
I've studied this at an introductory level, read the books, then gave them away. For me, the applicability and relationship to the 'real world' seemed too narrow. The kind of 'truth', and its relationships within this artificial domain and symbolic language, seem to imply a kind of universal certainty. 'Truth preserving' perhaps...but only in a set, forced way.
What do you mean by a 'dialect called 'philosophical English'?
Quoting Banno
About writing another logic text. You might like to look at this, the first out of 5 'best books' :
[the article has an excellent introduction]
Quoting The 5 Best Books on Logic
Oh, and just for @Agent Smith ( perhaps you've read it already?)
I've since realized that's an inadequate description for the category. It's also, perhaps primarily, for problems in logic itself.
That sounds interesting. So, how then to re-write?
Re:' problems in logic'... do you mean mistakes in how we apply logic? Solving logical problems?
Do logical problems even make sense?
Oh I'd have to go look, but, if memory serves, Grice defended material implication as a faithful representation of conditional reasoning in natural languages and did not join any campaign (not even Strawson's) either to reform logic or to abandon natural language for more precise pastures. And in my view, if it was Grice's view, it deserves deep consideration.
(Smoking "might" cause cancer is due to the fact ~A OR B => A --> B , which doesn't have a conjunction of events in the premise)
For resource-sensitive logical implication that is truly material in denoting conditional changes of state over time, see linear logic for expressing "If I am in the state of smoking then I might arrive at a state of cancer". It has the same form as the above rule, but the premise can only be used one when arriving at a conclusion.
The 'might' here can also be avoided by defining only one axiom of implication in which smoking is the premise. Otherwise the resulting logic expresses multiple and mutually exclusive possible outcomes of smoking, i.e possible worlds are built into the syntax.
For a programming language with native linear types, see Idris.
You have a better understanding of logic than I do, but I seem to run into more problems related to clarity. Like, introducing terms without providing a definition or conveying their meaning. I believe when terms are introduced without clear meaning they form a roadblock preventing the conversation to progress. I cant grant an argument if a premise contains a term that I dont understand. I cant even grant that the statement is propositional.
For example, most at least implicitly understand that the term Taller has relationality built in. If I say the tree is taller, then I am saying it is taller in relation to something else (e.g., than it was last year, than the surrounding trees, etc). If I introduce the term taller but in a non-relational sense (e.g., The tree is taller [full stop], you would, presumably, require a conceptual analysis to understand what that could mean. What is more, most of the time no effort is made to define terms or to convey our sense of them. Vagueness and ambiguity often go unchecked, relying instead on the assumption that our interlocutor shares our interpretations.
I enjoy the argumentative stage of discourse, but i seem to dwell mostly in the clarification stage. Do you or any other logician take a similar view? It seems necessary to be in agreement on all terms before arguing one way or another on an issue. Otherwise, how would you know whether or not you agree without a doxastic view of it? I want to give you some examples demonstrating my approach in these situations.
If asked whether or not I believe there is a God, I require you either provide a definition, or convey what you mean by the term God. If you take a pantheistic meaning of God (e.g., God just refers to the universe or cosmos), then I do believe the universe exists. If you take an abrahamic meaning of God such as the God referenced by Christian, Judaic, and Islamic faiths, with properties defined in terms such as omnipotence, omniscience, omnipresence, omnibenevolence, then I believe any propositions stating that such a God exists are false. I think such properties are mutually incompatible, and derive contradictions as demonstrated by the problem of evil. On a separate issue, I dont understand, and therefore cannot grant any statements made by moral realists if they introduce normative terms on a stance-independent construal.
One last thing, couldnt the terms proposition and statement be differentiated with regard to truth value? A proposition being a statement capable of being true or false. A statement being an utterance which expresses a complete idea (not necessarily declarative, possibly interrogative, imperative, etc).
In propostional logic, we consider only declarative statements.
Where do you find that explanation of the material conditional?
The material conditional is that the conditional is false when the antecedent is true and the consequent is false, and true otherwise,
That's interesting. I'd like to understand more about that.
This is my guess how (most?) English speakers think (contrary to the material conditional) about 'if then' in everyday life:
(1a) "If London is in England, then vodka is a beverage."
False, because there is no relation between the true antecedent and the true consequent.
(1b) "If London is in England, then Westminster Abbey is in England."
True, because the true antecedent implies the true consequent.
(2a) "If London is in England, then marble is soft."
False, because there is no relation between the true antecedent and the false consequent, and the consequent is false anyway.
(2b) If The Beatles recorded "Help", then Eric Clapton played on it.
False. There is a relation between the antecedent and the consequent, but the sentence is false because The Beatles having recorded "Help" doesn't imply that Eric Clapton played on it, and it's false anyway that Eric Clapton played on "Help".
(2c) "If Paris is on the moon, then I'm a monkey's uncle."
Some people will take that as true, as an idiomatic instance of ex falso quodlibet.
(3) "If New York is in Asia, then vodka is a beverage."
False, because there is no relation between the false antecedent and the true consequent.
(4) "If New York is in Asia, then marble is soft."
False, because there is no relation between the false antecedent and the false consequent, and the consequent is false anyway.
Im am aware. Only interested in statements which are truth apt (i.e., propositional). I was only offering a possible distinction between statements and propositions. Im pretty sure its more technical than the one I offered, but I do hear both terms used interchangeably.
Just out of curiosity, was your intention simply just to inform me that only declaratives are considered in prop logic? Or was your intention to launch a criticism towards some apparent error you think Ive made? Im fine with either, although if you were attempting the latter, it would make for a fine example regarding the concerns I mentioned earlier.
I said nothing to suggest that I was criticizing you. You mentioned that there are expressions other than declaratives, which is of course true. (And, one can devise systems of logic for interrogatives too.) But since the context has been propositional logic, and to answer any potential question whether propositional logic considers expressions other than declaratives, I noted that it does not.
Of course. It's important in the study of propositional logic that we understand that 'statement' is considered only in the sense of declarative statements.
There is variation though among logicians as to the meanings of 'proposition' and 'statement'. (But not so much extending to including interrogatives and such.) What is meant by 'proposition' and 'statement' may depend on the particular logician's or philosopher's framework.
Usually, we take 'sentence' to mean the syntactical object - the string of symbols.
Then it's a question whether we take 'proposition' as a synonym for 'sentence' or whether we take 'proposition' for what is expressed by a sentence.
Same for 'statement' - whether it just means a sentence or whether it means what is expressed by a sentence.
So, 'sentence', 'statement', 'proposition'. We just have to be careful what we mean in context.
/
For example, Church takes 'sentence' in the usual syntactical sense, but for him a proposition is a different abstract object. ('Introduction To Mathematical Logic, pg 26)
I presumed as much. I only mentioned a few of the alternative types of sentences which express non-propositional statements to provide an example showing that not all statements are propositions. Im sure there are more distinctions between the terms than that one, though. Im not saying you did anything to suggest a criticism, but without further context I wasnt sure. I just try to be charitable and give people the benefit of the doubt when what they say can be interpreted both positively and negatively. I wouldnt view a criticism to be necessarily negative either, I appreciate a proper critique.
Naturally.
What are your thoughts regarding my concerns with the lack of clarity here in the forum? Was I able to articulate my concerns in a clear enough manner for you to understand? Just seeking some feedback.
You asked, here's my answer:
Quoting Cartesian trigger-puppets
Of course, definitions are crucial.
But how demanding we should be must depend on context.
Since, for example, this thread is about a subject of mathematical logic, different contexts range from just philosophy about mathematics, to a blend of philosophy about mathematics and mathematics itself, to just the mathematics itself. Then there are degrees of formality, from very liberal informality to rigorous formality.
Forum-wide, usually mathematics is not the subject, but still there may be degrees of formality, from liberally speculative philosophy to more rigorous technical aspects of philosophy.
So what context do you have in mind regarding definitions?
Most informally, we know that of course we can't be bogged down by defining every word of English we use, and even if we could, we'd encounter circularity (English is not a formal language in which there are undefined primitives and then a sequence of definitions.)
For philosophy, I would agree that there should be an expectation that a poster should provide definitions for special philosophical terminology where there is a reasonable need to know the specific definitions. But there's still a limit - since we are not posting entire treatises, we don't have the time for everything.
For mathematics, in principle, every mathematical statement should be formalizable (this is called 'Hilbert's thesis'). But that's only in principle; in actual discourse, we have to be allowed informality, as long as we know, in the background of our reasoning, that could formalize it all if we had all the time and patience to do it (I nickname this 'Bourbaki's thesis'). So, yes, mathematicians, at least in principle, must be able to define all terminology down to the primitives. But, again, in a forum we don't have time to define everything down to, say, the sole primitive ('e' for epsilon, i.e. "member of") of set theory.
On the other hand, there are cranks. Cranks often talk as if they are making mathematical statements (not just philosophical statements about mathematics) as they are using mathematical terminology. But their usage is incorrect, usually ludicrously so. And they have no concept even of what a mathematical definition is, or what the specific definitions are of the terminology they use. For me, as far as definitions, that is the worst of a forum such as this; and it's not just this forum, but all over the Internet.
Quoting Cartesian trigger-puppets
I know what 'doxastic' means, but I don't know what you mean by "a doxastic view of it" in that context.
Quoting Cartesian trigger-puppets
I know what 'moral realism' and 'normative' mean, and maybe I have a bit of a sense of what 'stance-independence' means, but I don't know what is meant by 'introduce normative terms on a stance-independent construal'.
/
"It is not our business to set up prohibitions, but rather to arrive at conventions." - Rudolph Carnap
I actually dont think definitions need be a requisite, though they are useful insofar as they capture the standard meaning of a term. I try to avoid committing to a definition since it requires a semantic thesis or a theory of public meaning. Im happy to hear your sense of a term and work from that understanding so long you are able to convey your meaning (through any means: ostensively, semantic primitivity, family resemblance, interpretive dance, etc.) so that I form a concept. Of course not every term requires the authors conveyance. It is necessary when idiosyncrasies, proprietary definitions or plain gibberish is detected.
Quoting TonesInDeepFreeze
Im very impressed. I smuggled in a couple terms that Im uncertain that I understand (certainly not well enough to use them), and this is one of them (the other being Abrahamic in context with religious traditionmaybe I got that one right). You not only caught it immediately, but were intellectually honest about it. Sometimes I test if people acknowledge not understanding the term, or pretend to. I was going for doxastic attitude (an epistemic attitude held towards a proposition) such as belief, disbelief and suspended belief. I meant to say that without a concept of a term (without a concrete image or relative abstractions), I cant say whether or not I believe, disbelieve, suspend believing any statement containing it. I dont know if its proposition, or coherent, or contradictory, or vacuous.
Quoting TonesInDeepFreeze
Stance-independence, in the context of normative terms, is a metaethical view regarding the meaning of such terms as good, bad, proper, improper, etc. A construal is a way in which something must be in order to be understood. I dont understand normative terms in a stance-independent sense, other than what realists claim to refer to (spooky metaphysics). I understand normative terms on a stance-dependent construal (antirealist). Saying something is good on a stance-dependent understanding of the term, is to say that goodness is understood in accordance with the desires of an agent, or with a given standard. To say friends are good is to say I desire friends, and to say the heart is functioning properly is to say the heart is functioning in accordance with medical standards.
Its important to remember that on a stance-dependent construal, normative terms must be indexed to an agent or a standard. So if you ask me Is committing racist war crimes good since the Nazi desired it? Im committed to say yesbut only in the sense that Im uttering the tautology Committing racist war crimes is desired by the Nazi because its what the Nazi desirednot good in accordance to my desires.
Isn't there a problem with the 'naturalistic fallacy'? The medical standards may be too low or otherwise in error. In that case we could say without contradiction that someone's heart is functioning in accordance with medical standards but is not functioning properly. So they do not mean the same thing - if they did, it would be self-contradictory to say one and deny the other.
p [math]\to[/math] q :: ~p v q
Commutation (abbrev. Comm)
p & q :: q & p
p v q :: q v p
Distribution (abbrev. Dist)
p & (q v r) :: (p & q) v (p & r)
p v (q & r) :: (p v q) & (p v r)
Association (abbrev. Assoc)
p v (q v r) :: (p v q) v r
p & (q & r) :: p & (q & r)
Double Negation
p :: ~~p
True stands in contradistinction to false, the concepts being specifically designed to categorize propositions, perhaps with the intention of separating wheat from chaff. So if every proposition is true, we can no longer do that; in short truth/falsity lose their utility as proposition-classifying properties,
Yet you asked me to comment on your post that includes:
Quoting Cartesian trigger-puppets [emphasis added]
Quoting Cartesian trigger-puppets
Quoting Cartesian trigger-puppets [emphasis added]
Quoting Cartesian trigger-puppets [emphasis added]
So I took the time to write a post about that. Then you say the opposite, that definitions are not required. .
So I don't understand you.
Regarding definitions, Im saying one ought not be required to provide a definition for a term so long as they convey what the term means. Definitions require a semantic thesis or a theory of public meaning (to claim that the best understanding of a term is the meaning communicated by the public at the time), necessary and sufficient conditions (for example, a definition for women, in the context of biological sex, would have sufficient conditions such as: physiological or phenotypical proximity to the archetypal female human; as well as necessary conditions such as: a natural genetic predisposition to produce large gametes), the definiendum must be defined by the definiens (in the defining statement a woman is a being with physiological or phenotypical proximity to the archetypal female human, with a natural genetic predisposition to produce large gametes, being, physiological or phenotypical proximity to the archetypal female human, natural genetic predisposition to produce large gametes is the definiens defining the definiendum woman), and other such criterion. Im trying to make things simpler by just conveying my sense of the term and understanding yours.
I was being lazy and equivocated two senses of the term definition. Thank you for pointing out the equivocation. Ill edit it real quick.
Except in formal mathematics, definitions are purely syntactical.
Quoting Cartesian trigger-puppets
Indeed.
I used women because it is a notoriously difficult term to define, thus a good example to portray how complicated definitions can become. In case your wondering why, it wasnt just to be complicated; there was utility.
Quoting Cuthbert
I dont believe so, since the aim in medicine is to maximize health and well-being, without necessarily committing to views such as One ought to be healthy or Health is good despite the ubiquitous use of normative language within systems of healthcare and medical vocabulary (e.g., disease, disorder, etc.). This is because terms such as health and well-being are cashed out in relation to the desires of an agent. So, when we say Disease is bad in medicine, what that actually translates into is Disease is [generally] undesired. Furthermore, when asked what is meant by undesired there, it refers to the preferences commonly regarded, but not limited to, by the public. If probed further, one would have to regard individuals from a case by case perspective, such as: Patient x prefers not to have disease y, because disease y increases stimuli that patient x associates with pain, and patient x desires to avoid the experience of pain.
So, when statements such as The heart is functioning properly are made in medicine, what is implicitly being said is The heart is functioning in accordance with agreed upon standards for maximal health and well-being which is essentially making the following argument: If you desire health and well-being (e.g., longevity and less pain), then your heart should function in accordance with medical standards. It is up to you, the individual agent, whether or not to make the assertion I desire health and well-being, (an objective claim regarding your own psychology) or to agree with the inference Then your heart ought to function in accordance with medical standards (an empirical claim backed up by facts extracted from clinical research data).
A variant of the naturalistic fallacy occurs when explanations for terms such as Good are reduced to naturalistic properties (Moore gives examples such as Desires or [Mills] Pleasure) which identify Good with its object. This is not occurring in the examples above. What is said is not that Health and well-being ARE Good ontologically speaking, but rather that Health and well-being ARE WHAT IS MEANT by the term Good semantically speaking. Another variant of naturalistic fallacy occurs when crossing the Is/Ought divide (deriving the way things ought to be from the way things, in fact, are). This too is avoided by use of the conditional statement (if p, then q). Since the term Good refers to what is desired, then the conditional statement reads NOT as If you desire x, then x is good, but rather it reads If you desire x, then you desire x, which is a tautology we would all believe to be trivially true. The above doesnt appeal to nature (medical intervention often preventing natural occurrences such as a virus). The fallacy can and sometimes is committed upon meta-ethical investigations regarding terms such as Proper function, Malfunction, Disfunction, Disease, etc. when the individuals being asked attempt to, indeed, fallaciously explain such normative terms reductively (to say Good is some object or objective property, rather than a percept or subjective property). This problem seems to be an issue with moral realism.
An archived SEP article.
THE OPEN LOGIC TEXT
Home page.
Wish I'd seen this earlier.
(1) [math]\neg P[/math] [assume]
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( n ) [math]\bot[/math]
(n + 1). [math]\neg P \to \bot[/math][1 to n, CP]
(n + 2). [math]\neg\neg P \lor \bot[/math] [(n + 1) Imp]
(n + 3). [math]P \lor \bot[/math] [(n + 2) DN]
(n + 4). [math]\neg \bot[/math] [LNC]
(n + 5). [math]P[/math] [(n + 3), (n + 4), DS]
Pretty much, you proved P from the premise ~~P. Congratulations. And your point is?
There's another way of showing why RAA works.
Just look at the truth table.
Anyway, you didn't use RAA.
:chin: