"This sentence is false" - impossible premise
This is probably hard to believe but I do not have the intuitions necessary to see the mysteries of some paradoxes. For example, the liar paradox this sentence is false simply appears meaningless to me and I do not enter the logic of: If 'This sentence is false. is true, then since it is stating that the sentence is false, if it is actually true that would mean that it is false, and so on.
Language conveys information and I cant extract relevant information from this sentence, this is why I do not understand why people manage to reason logically with it.
This is how I visualize the information contained in the following sentences. If the set is correct, it is green, if it is false, it is red.
The first one is the sentence The sun is yellow, and the second one is this sentence is false.
To me, the second one is simply meaningless because the sentence conveys information for an empty set and attributes a truth value to it, which isnt possible since it is empty.
Now, from my understanding, the paradox is misleading because of its grammar. This sentence is implies that the set contains something, while it doesnt. And Im imagining people see it like this:
They see true or false as both an element of the set and the validity of the set. So, if the set is valid, it needs to have the true element in it, and if it is not valid, it also needs to have it.
If you find this paradox mind boggling, does this visualization make sense?
To me, this paradox is actually a problem that contains an impossible premise, which is that the validity of a set is also an element of the set.
I found this article that also points out a problem in truth attribution:
https://link.springer.com/article/10.1007/s10516-023-09666-2
The crocodile paradox also contains an impossible premise, which is a condition implied by the crocodile that if he eats the child, he will give it back alive.
To me, this shows how much we want to keep our intuitions, as if there were some holy concepts, instead of questioning and dismissing them.
Language conveys information and I cant extract relevant information from this sentence, this is why I do not understand why people manage to reason logically with it.
This is how I visualize the information contained in the following sentences. If the set is correct, it is green, if it is false, it is red.
The first one is the sentence The sun is yellow, and the second one is this sentence is false.
To me, the second one is simply meaningless because the sentence conveys information for an empty set and attributes a truth value to it, which isnt possible since it is empty.
Now, from my understanding, the paradox is misleading because of its grammar. This sentence is implies that the set contains something, while it doesnt. And Im imagining people see it like this:
They see true or false as both an element of the set and the validity of the set. So, if the set is valid, it needs to have the true element in it, and if it is not valid, it also needs to have it.
If you find this paradox mind boggling, does this visualization make sense?
To me, this paradox is actually a problem that contains an impossible premise, which is that the validity of a set is also an element of the set.
I found this article that also points out a problem in truth attribution:
https://link.springer.com/article/10.1007/s10516-023-09666-2
The crocodile paradox also contains an impossible premise, which is a condition implied by the crocodile that if he eats the child, he will give it back alive.
To me, this shows how much we want to keep our intuitions, as if there were some holy concepts, instead of questioning and dismissing them.
Comments (84)
I agree. That is what comes of attempting to abstract logical form from content. There is a formalization in set theory involving the set of sets that are not members of themselves (normal, versus abnormal sets). Essentially, this recognizes exactly the real language constraint that a claim be about something.
Your approach seems to be the same as that of Kripke. See here.
Liar sentences are "ungrounded". Them being true or false isn't meaningful.
I think we can show this by considering the complement of a liar sentence:
1. This sentence is true
If (1) is true then there is no paradox. If (1) is not true then there is no paradox. But is (1) true or not true?
1. This sentence does not correspond to a fact.
Does (1) correspond to a fact?
After that we have a mention of Priest and dialetheia.
That seems related to an intuition that I always had about "This sentence is not true". The sentence starts with a reference to something that has not even been finished yet, for it is only when you say "false." that the sentence is complete and thus can be evaluated. But the end of the sentence itself includes an evaluation to something that has not even brought into existence yet, what OP illustrates with an empty set. Thus we end up in a loop of "if this is true, then it is false, but if it is false, then it is true, but if it is true...". It feels as though "sentence is not true" is sentence A and everytime we try to evaluate it we in fact create a new sentence A1, then A1.1, then A1.1.1, and so on.
If this is a satisfactory solution, no need for dialetheias, in this case...
Quoting Michael
Yea, all of these approaches seem connected.
This is how i look at it:
If it is true that the Sun is yellow then the first sentence is a true statement, else it is false regardless of any other sentences that may exist. If we do not know what is or is not true then in any case...
If the second sentence is referring to the first sentence:
If it is true that the first sentence is false then the second sentence is true in stating that the first sentence is false. (= True)
If it is true that the first sentence is true then the second sentence is false in stating that the first sentence is false. (= False)
If the second sentence is referring to itself:
If the sentence is true that it is false then the sentence is true that it is false. (= True)
If the sentence is false that it is false then the sentence is false that it is false. (= False)
Can't you get around that by changing the paradox to "Everything I say is a lie"? In that case, the sentence does correspond to a fact- that I am a liar.
Quoting Skalidris
The statement is unclear to be true or false. "This sentence" doesn't indicate which sentence it is describing or declaring about. From the statement, it is implied that there must another sentence before it, for the statement to be qualified to conclude "False", but it is not clear, whether it is the case, or "This sentence" means the sentence itself.
If it is the sentence before it, then it is missing, and if it is the sentence itself, then it doesn't indicate why it is false.
Therefore, if someone uttered the statement, it would beg the question, "Which sentence do you mean?"
The term "paradox" is overrated and abused. Most "paradoxes" are simply self-contradictory, self-refuting or circular statements or statements based on a false hypotheses. In short, invalid statements.
The statement in question --This sentence is false-- is a classic example of a self-contradictory statement. It's also circular. It indicates two opposite things coexisting, an impossibility: if this sentence is true, then it is also false. There's nothing more to it. It's a dog chasing its tail, a snake swallowing itself. It does not leave room for any interpretation. It just can't stand. It's not a paradox.
The word "paradox" comes from ancient Greek "para" (= besides, contrary to) + "doxa" (= opinion). Indeed, it indicates something that exists or happens which is contrary to what one expects or believes to be true or happen. For example, a paradox would be raining without any cloud in the sky. Yet, it is possible, if there are very strong winds that bring rain from some other place than where we are.
A. This is a sentence. True
B. The sentence in point A is a false sentence. False.
There ya go.
This sentence contains 36 characters
Should we break the above sentence into the below?
A. This is a sentence
B. The sentence in point A contains 36 characters
That's another way to break it down if you would like. Same idea.
Except you cant break it down that way because This sentence contains 36 characters is true but The sentence in point A contains 36 characters is false.
You didn't tag what was true and false in your breakdown, so I assumed that A was true and B was false in isolation. If your intention is that the break down accurately fits the intention of the primary sentence, it does not. My example was the breakdown of a contraction, yours is not.
Trouble is, the paradox is right there in the initial version of Principia Mathematica; that is, an "invalid" statement was implied by the formalisation of mathematics in a first order logic. It looked as if the whole edifice would collapse.
p) this sentence is false
is implicitly the same as
q) "this sentence is false" is true
and
r) this sentence
refers to the same by self-reference, so we have both
p) this sentence is false
and, via the above
s) this sentence is true
which is an ordinary contradiction, implying anything
In a way, implicity and self-reference allow unpacking a regular contradiction, which, if not much else, isn't as mystifying.
Unfortunately, I'm not knowledgeable on the subject.
But, as I said, there are real paradoxes, which are quite perplexing or structured in a way that cannot be easily refuted or explained, or even not at all. There are such factors as perspective and relativity, which alone leave certain paradoxes "open" or "unsolvable". E.g. The Ship of Theseus paradox (thought experiment).
Yes that is probably the case.
Quoting Corvus
Yes, my reaction exactly. The most intriguing thing about this paradox is that a lot of people don't mind reasoning with something that is empty of meaning... Probably because they did not check that it actually has meaning prior entering this logic loop.
Quoting Alkis Piskas
Yes, I agree. And I find it quite unbelievable that no discipline has managed to reach a consensus about all of these "fake paradoxes".
Quoting Alkis Piskas
The Ship of Theseus paradox looks more like a philosophical or linguistic issue than a paradox.
An ambiguous statement disguised as a paradox.
Right. That's why I added "thought experiment" in parentheses.
Curry's paradox is an interesting extension of this.
1. Let (a) be the sentence "if this sentence is true then Germany borders China"
2. If (a) is true then Germany borders China
3. Given that (2) is true, and given that (a) and (2) are materially equivalent, then (a) is true
4. Therefore, Germany borders China
In formal logic:
1. X := (X ? Y)
2. X ? X
3. X ? (X ? Y)
4. X ? Y (from 3 by contraction)
5. X (substitute 4 by 1)
6. Y (from 4 and 5)
I think the OP made a good argument. I don't think I can add anything to it.
Two statements are materially equivalent if either both are true or both are false:
1. A if and only if B
If (1) is true then "A" and "B" are materially equivalent.
So, in the above case:
A) if this sentence is true then Germany borders China
B) if (A) is true then Germany borders China
If (B) is true then (A) is true. If (B) is false then (A) is false. Therefore, (A) and (B) are materially equivalent.
Quoting Brendan Golledge
Are you referring to step 5? As it explains, it simply takes step 4 and replaces X ? Y with X, which is allowed given the definition in step 1.
Quoting Brendan Golledge
Are you referring to step 4? As it explains, it follows from step 3 given the rule of contraction.
X ? (X ? Y) entails X ? Y.
"A" is not the same as B: "if A is true, then the statement given by A is true". B as written here is true regardless of the truth value of A. I could just as well write, A: "The sky is pink" and B: "if A is true, then the sky is pink". This A is false and this B is true.
As for your formal logic, I think I am confused about whether you are asserting logic or truth. For instance, I cant tell whether you mean, "if X is true, then Y is true" (I agree with this logic) or "X IS true, and therefore Y is true" (I disagree with this because I think X is either false or meaningless).
Consider these sentences:
1. if this sentence is true then Germany borders China
2. if (2) is true then Germany borders China
Do you accept that (1) and (2) are materially equivalent?
If so then consider these sentences:
2. if (2) is true then Germany borders China
3. if (2) is true then Germany borders China
Do you accept that (2) and (3) are materially equivalent?
If so then (1) and (3) are materially equivalent.
Quoting Brendan Golledge
Hopefully this is clearer:
1. X means if X is true then Y is true (definition)
2. If X is true then X is true (law of identity)
3. If X is true then if X is true then Y is true is true (switch in the definition of X given in (1))
4. If X is true then Y is true (from 3 by contraction)
5. X is true (switch out the definition of X given in (1))
6. Y (from 4 and 5)
Although one thing to consider is that A ? B is equivalent to ¬B ? ¬A, and so these are equivalent:
1. if this sentence is true then Germany borders China
2. if Germany does not border China then this sentence is not true
(2) appears to be a more complex version of the standard liar sentence.
Quoting Michael
Quoting Michael
I did not understand number 5, because it seemed obvious to me that X was false (or I was at least very skeptical), so I did not see how substituting it into itself could turn it true. The source I read explained that step 5 is modus ponens, and given the definition (1), it works. But the paper went on further to prove that if 6 is false, then 1 must also be false. So, it is a bad definition. I hadn't worked through all the logic yet to see the paradox, but I did see that it was false.
:Quoting Michael
Quoting Brendan Golledge
So, I guess I just never accepted that the sentence was true, and that's why I did not see the paradox.
Quoting Brendan Golledge
Going over this part again, I understood the whole argument to basically be:
A
B: A -> A
Therefore, A
B is true, but we don't know anything about A without more context. I guess this is not what you wrote down formally, and I just didn't get it, because I interpreted your words to mean the A & B I wrote immediately above.
Its not that the definition is bad, its that when we apply the normal rules of logic to some self-referential sentences then we lead to a contradiction. Its the paradox of all liar like sentences and theres no agreed upon resolution.
Is this statement false? If I've done the truth table right, then it means that the first line of the proof is wrong.
The first line is a definition, not a premise, and so not truth apt. It is simply saying this:
Let "A" mean "if A is true then B is true".
The sentence isn't truth apt, but "married bachelor" is a contradiction.
But neither "if this sentence is true then I am 30 years old" nor "if this sentence is true then I am not 30 years old" is a contradiction, or at least not obviously so.
So what should be done instead?
Also, Tarski's undefinability theorem shows that there is no definition of a truth predicate (per the standard model for the language of arithmetic) in the language of arithmetic. The proof makes use of "This sentence is false" by showing that a truth predicate would allow, via Godelization, the formation of the sentence, which would be both true and false in the model, which is impossible since, for any given model, there is no sentence that is both true and false in that model.
In 'This sentence is false', 'this sentence' is referring to 'This sentence is false'.
Not recognizing that is just waving away the problem.
Ignore that anyhow, it's just work around and epicycles. This is the true solution.
The subject matter of truth is an idealized model of the "truth" process, of the process of these acts, inferences, observations. But because it is idealized, the required assumptions are only implicit. I like to think the best model is a communication process: an environment provides a signal and the observer responds with an appropriate signal (this in itself is an idealization ignoring how or the means by which the observer validates its signal to itself, in terms of use). We also have state transitions: producing a signal implies a change of state in the observer / environment.
Maybe not a universal explanation, but I think a big feature for many notable paradoxes... the major assumption broken is the underlying assumption that for communication to occur there must be a clear object/subject divide. To talk about the environment, the environments behavior must be completely independent of the signal the observer uses - something which seems like our assumptions about objectivity in the world. The world or things exists or have truth objectively and independently regardless of the signal the observer produces to respond to it in the communication game - context independence. If we see the observer's state transitions as being induces by the environment then the world can provide a signal which induces an observer state transition to a matching signal (which contains information of what is the case) and that is that. The communication problem is finished and the observer state will be stable so long as the environment doesn't change.
But if the world is dependent on the signal then stability is broken because, like the observer changes their signal due to the environment, now the environment changes due to the observer's signal. The observer's "truth" description of the environment then induces the environment to change its signal. If the signals that the environment and observer can make are all easily distinguishable and unique to every possible new situation, then stability is lost and the environment and observer will keep changing, inducing changes in each other. Like a (idealized) mirror scenario: you hold up a signal of what you see in the mirror, but the second you hold up your signal, the mirror image has changed to you holding a signal. So to communicate what you see, you must now hold up a signal which is about you holding a signal... but that changes the mirror image again. The stability of the observer's signal is impossible because any signal they make changes what they are observing which then changes their signal and it goes on and on. It seems to me, all these paradoxes are only ever made salient when you talk about their consequences in a sequential fashion like this which never stabilizes. Obviously the mirror alludes to how self-reference is a special case of this.
The "truth" then cannot be beyond the assumptions and processes that embody how observers enact their epistemic behaviors. And once the formal or mechanical or physical scaffolding that supports that fails then so does notions of truth. Without the strong object-subject divide enforcing context-independence then it is impossible for the signals of the environment and observer to match up at any point in time (denoting stable, coherent communication). This doesn't necessarily need to be between one observer and environment (or an observer and themself) either but maybe networks of observers communicating to each other in a way that context-independence fails for all of them because the signals directly influence other signals in the network of communication.
The fact that these paradoxes can occur just reflects the mechanical or formal capabilities of the system being described and the same which underwrites any communication process - sometimes the system cannot settle in its dynamics. Our notions of truth or even the notion of communication are secondary; communication as we commonly see it assumes that there is an objective fact of the matter about a meaning of what these signals represent which is somehow beyond the formal or physical mechanics of the situation. In reality, the mechanics are all there is, just like the epistemic activities of humans is nothing more than brain behaviour, or maybe sequences of experiences - depending on how you want to view the mind. Our everyday coherent notions of truth or communication require constraints beyond what is constrained in these mechanics and so it is no surprise they sometimes fail.
I think probably another reason why things like these fail sometimes, which I won't go much further into, is something which I crudely refer to under the umbrella of factorization assumptions. which sometimes we cannot but help make, but also may make things like computation or representation easier. For instance, If we want to make statements about the world, and that information is in both the word meanings AND syntax, well they cannot be independent when it comes to any kind of truth descriptions of the world (but maybe speech isn't just about truth so I am deliberately ignoring part of the picture). However, for whatever reasons, whether trivial or sophisticated, syntax and semantics are obviously independent, which allows you to make nonsensical statements - e.g. colorless green ideas sleep furiously - which again reflect the fact that the mechanics and formal constraints on truth are much tighter than the constraints on the systems which enact or embody these things for us. Its interesting that when you look at machine learning and occasionally computational neuroscience, things like factorization assumptions that enforce statistical independence are used because it makes inference easier... however, it also amounts so something like making a deliberately false assumption about the data. You can even see this kind of thing in a slightly different way in moral statements: many statements are very coarse like "stealing is bad" but in reality we all know that whether stealing is bad depends on the situation - its different in every scenario (e.g. what if stealing was involved in some operation which was about national security) - but because there are regularities we can assume we can make the simplifying statement "stealing is bad" is independent of the situation.
If you say, "X is false", clearly that could be represented as X -> F. Then if you say that X is "this sentence," then you could write something like "X <-> (X -> F)". This is very similar to the sentence used in Curry's paradox: "X <-> (X -> Y)", where Y is any arbitrary statement.
If you do the truth table for X, Y, (X -> F), & (X -> Y), then you see that the definitions are simply false. "X <-> (X -> F)" is exactly backwards, so that "NOT X <-> (X -> F)" is a tautology. "X <-> (X -> Y)" is only true if X = T and Y = T.
Michael said earlier that a definition is not truth apt. I can see how that would be the case if you defined an entirely new variable, such as Z <-> (X -> Y). However, since you are setting X equal to itself, you can do a truth table on it.
I remember hearing that if a system contains a contradiction, then anything can be proven. So it makes sense to me that the premise in Curry's paradox contains a contradiction, hence its ability to prove any arbitrary statement.
These are two different sentences that you seem to be confusing:
1. X ? (X ? Y)
2. X ? (X ? Y)
In ordinary language, these mean:
1. "X" means "if X is true then Y is true"
2. X is true if and only if (if X is true then Y is true)
You seem to misunderstand what is happening here.
Take the English language sentence "this sentence is English". To better examine this we decide to translate it into symbolic logic. To do that we have to do something like the below:
S ? E(S)
Now take the English language sentence "this sentence is French". In symbolic logic this is:
S ? F(S)
Now take the English language sentence "this sentence is true". In symbolic logic this is:
S ? T(S)
Now take the English language sentence "this sentence is true and English". In symbolic logic this is:
S ? T(S) ? E(S)
Now take the English language sentence "this sentence is true and French". In symbolic logic this is:
S ? T(S) ? F(S)
Regardless of whether or not the right hand side is true, these are the accepted ways to translate an ordinary language (self-referential) sentence into symbolic logic.
See also here.
If X := X->Y then X <-> (X->Y).
But we don' t have the converse that if X <-> (X->Y) then X := X->Y.
So X := X->Y is not equivalent with X <-> (X->Y).
So we can't dispense the paradox by incorrectly saying that it reduces to X <-> (X -> Y).
Trouble is, that's just an idealisation.
Happy sawing.
Quoting Banno
One can deconstruct idealizations without ones deconstruction itself being an idealization.
Can one put into question the notion of gods eye truth without that questioning itself being assumed to be oriented by a gods eye perspective?
This sentence is six words long.
I don't much care what god thinks. There are true sentences, contra , and, it seems, your good self.
Yes. And No.
Honestly my Big Brain project is seeing how it might be possible to unite both of those big-azz books.
I still have work to do in both, though. And they ain't epicycles, either of them.
They (the sentences which are true) are pesky, though.
I can do the same thing without equating := to <->
You can use a truth table to prove NOT X <-> (X -> F).
(X -> Y) <-> (X -> F) in the case where Y is false, so this applies to Curry's paradox as well as "this sentence is false".
Then you take your definition X := (X ->F) and substitute NOT X for the second part.
Then you get X := NOT X
Clearly, there has to be something wrong with that definition.
Yes, that's how the sentence "this sentence is not true" is translated into symbolic logic.
Yes:
|- ~X <-> (X -> F)
If Y is false then (X -> Y) <-> (X -> F) is true.
That's not Curry's paradox.
Quoting Brendan Golledge
Who does that? You? Did someone previously define?:
X := (X -> F)
Quoting Banno
Honestly, I feel like my views on the world would actually be less consistent if I didn't think that my views or the things I said did not suffer those kinds of qualities of idealization or related issues. Part of the central basis of my views is that what we do or say is at the mercy of the constraints of how our minds, brains work as computational systems. It would actually be not as coherent if I didn't think these kinds of things to my own mind and thoughts, beliefs, theories all the time. Why would I be exempt from things I apply to the rest of the academic and cognitive world in its entirety?
The idealization thing is only an issue for people with a certain kind of goal here... which I do not think I share.
Quoting Banno
Yes, when you agree to play the game in the right way. Even better when you ignore the parts where it breaks down.
For my part the notion of "objective" truth causes more problems for those of a philosophical bent than it heals. In particular, there are folk who supose that because they cannot access "objective" truth (whatever that is), that there are no truths at all.
But it ain't so.
Quoting Apustimelogist
Are you saying it is better to play the gamein the wrong way?
Why not play it in the right way, or at least, work out what the rules are...?
Quoting TonesInDeepFreeze
It seems to me that everybody is being super-pedantic about this. I am studying formal logic informally (without being in a class), so I'm not surprised if I'm not using some symbols correctly. However, the logic should still work
Quoting TonesInDeepFreeze
"X -> F" is supposed to mean, "This sentence is false." "X := (X -> F)" is supposed to mean "This sentence says, 'This sentence is false'."
Quoting TonesInDeepFreeze
I've seen in multiple sources that Curry's paradox is defined as X := (X -> Y), and some of them then change it to X <-> (X -> Y).
Quoting TonesInDeepFreeze
You yourself said that this is allowed, so I don't know why you are arguing with me about this.
I am new for formal logic, but I understand algebra just fine. If I define Y := X + 1, then it is impossible to say that Y is false, because Y has no outside definition. However, if I define X := X + 1, then this obviously involves a contradiction. It seems the same ought to apply to formal logic. I do not see how you guys can argue about this so much.
Maybe the difference is that I come from a physics background rather than a math background. If I can make the math give me the answer I want, then I think it must be right. What I'm doing here gives me the answer I want, because the truth table for "This sentence is false" shows that X <-> NOT X, which is the same answer you get by working through the paradox with human language. In Curry's paradox, the truth table gives that the sentence is self-contradictory if the assertion is false, which resolves the paradox. It seems to me that mathematicians get stuck on arbitrary definitions & distinctions, like := vs <->, even if doing so makes everything harder and nothing easier. If the proof of Curry's paradox is correct, then we get that logic is broken, because there is a paradox. However, using a truth table to check the definition shows that the definition is contradictory, and thus there is no paradox. It seems bizarre to me that people are arguing with me that I can't check the definition for consistency when doing so makes everything so simple.
Curry's paradox has a very technical context. To understand it properly requires being very careful in the formulations.
Quoting Brendan Golledge
'X - > F' means "X is false".
'X := X -> F' means that X is the sentence 'X -> F'.
But no one asserted that X is the sentence 'X -> F'. Indeed, X is not the sentence 'X -> F'.
Quoting Brendan Golledge
What sources?
Quoting Brendan Golledge
I am not arguing about that.
Again, we have:
If X := X -> Y then X <-> (X -> Y)
but we do not have:
If X <-> (X -> Y) then X := X -> Y
So we don't have:
X := X -> Y iff X <-> (X -> Y)
Quoting Brendan Golledge
It depends on the context. For example, it could be in a computer programming language or something. But in this context, we would not write that. (I could explain why, but it's another subject).
Quoting Brendan Golledge
It's impossible to say Y is false (for given values of X and Y) because, as it seems, you're using 'Y' as variable ranging over numbers not sentences.
Quoting Brendan Golledge
It's not apparent what such a truth table would be for such self-referring sentence.
Quoting Brendan Golledge
The logic isn't broken. In English, we can make such utterances. But in the logic, we are not allowed to define a sentence symbol that way. (The following part I'm not well versed enough, so take it with a grain of salt.) But with such things as arithmetization, we can form certain sentences that are "self-referring". In those cases, where Curry's paradox can be performed, we find not that the logic is broken but that the particular theories in which Curry's paradox occurs are inconsistent. (Again, I'm not real clear on that, so take it with a grain of salt.)
I think that in this discussion, we're assuming that there is some context in which we can justify the definition of 'X' and we're reasoning from that assumption. Upon specifying a justifying context, we would then look for the import of the contradiction in that context.
Ah, if you guys had only participated in my thread on The Laws of Form, you would have discovered that such self contradictory sentences are formed by "re-entry" or recursive definition, and result in truth values that oscillate in time.
Logic is static, and does not deal well with time, but presumes an unchanging block of eternal truth. But sometimes the cat is on the mat, and sometimes the cat is not on the mat. Cats are fickle.
Quoting Banno
Yes, I do play the game, there's no other choice, but there's always a caveat. I think its impossible to view the world outside of some particular perspective and so in that sense I would say that our notion of objective truth is an idealization. We might say there is an objective way the world is but I don't think there is a single perspective-independent way to characterize it. If I were to say there are objective truths, I don't think I would be able to give a satisfying characterization without caveats. I don't want to conflate my belief in an objective state of the world and my ability to articulate things about them because the latter is something I cannot do.
Quoting Banno
No, I mean right way in the sense of avoiding and ignoring caveats which makes the veracity of "truths" seem obvious.
I'll not disagree with you about "objective" truth. I don't think the notion of much use. By talking to each other we can remove biases of perspective. There are true sentences about how things are. And overwhelmingly, we agree as to what is true and what false. The places we disagree tend to be either misunderstandings or differences in what one should to do about how things are.
Consider how much agreement was involved in your simply reading that paragraph.
:up:
People underestimate the usefulness of etymology and dismiss it as "etymological fallacy" after a 5 minute reading session. But given some background facts about some of those who underestimate it, it does not surprise me at all.
This is very true. However, etymology in English --and I believe other languages too-- is often complex and even useless. This is not the case with ancient Greek and Latin, however. Esp. in Greek, one can undestand the meaning of a word just by its etymology.
But what are we talking about? People ignore or even hate dictionaries in general. People don't like definitions. This is what I learned from this and othe similar places. All the more about etymology.
It is true, and that is what I meant with "background facts".
Quoting Alkis Piskas
And that is exactly when philosophy becomes affectation. Title drop.
Well said, Lionino. :up:
Yes, I think our main difference is that you want to hang on to the idea of true sentences and I do not really care for that.
There is no 'Godel's paradox'.
Anyway, as best I understand your question, the answer is 'no'.
Thank you for the clarification on Godel. I see that.
My question was twofold.
1. Isn't it futile to "make sense" of paradoxes?
2. Dont paradoxes expose the limitation(s) of Logic (here, in pursuit of absolutes)?
I'll presume your answer is no; and therefore, you think that we can make sense of paradoxes, or, at least that it's a worthwhile pursuit; and, either they don't expose the limitations of logic or logic is not limited in its pursuits.
If that's the case, I'm interested in understanding your reasons.
1. I don't know what you mean by ""make sense" of".
2. Frege's system was taken to be a derivation of mathematics from logic alone. Russell's paradox showed that Frege's system was inconsistent. But does that mean that there can't be a derivation of mathematics from logic alone? Whitehead and Russell offered a system that was an attempt to derive mathematics from logic alone, but their system was not logic alone. But does that mean that there can't be a derivation of mathematics from logic alone? The Godel-Rosser theorem may discourage us even more from thinking that we can derive mathematics from logic alone. So, can we derive mathematics from logic alone? I think the preponderance of philosophers of mathematics think we cannot, but there are dissenters.
I see the relevance of your point, though indirect. It remains, then, the answer to whether or not, as you put it, we can derive mathematics from logic alone, requires at the very final stage at least, a leap from logic. Some say no, dissenters say yes, in the end, a leap must get one there.
The leap is in the form of axioms.
Just to be clear, the logic system itself is not contravened. Rather, we add non-logical axioms. We add axioms that are consistent, thus true in some models, but that are not logically true, thus not true in all models.
Good point. So simple, but something I haven't thought about properly. One I'll consider more thoroughly. Especially its implications on my personal interests about human Mind and the status/role/nature of logic. Thank you.
Any time. Thank you.
The sentence does not express a statement that has a truth value, which means that it does not express a statement at all. There are countless sentences like that. One could devise countless sentences like that. A sentence can be vapid and nonsensical, and "This sentence is false" is one of them.
One need not consider vapid claims, and one need not construct logical proofs of their vapidity.
Agreed.
Quoting Lionino
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Quoting Faust Fiore
How do we know otherwise which claims are vapid or not? Proofs are welcome wherever we may find them.
This a particular type of statement: the only argument for it is that it can't be false; the only argument against it is that it can't be true. But in natural language not every sentence is true or false.There are many other possibilities, including meaningless, as this thread suggests.
Welcome to TPF, Gary. (Well, two weeks ago, but welcoming lasts for a while ... :smile:)
Quoting Gary Venter
I believe that both "formal logic" and "natural-language logic" are simply two different ways of expressing logic elements and logical schemes. The same applies to Math sets, probabilities, etc.: they can be expressed with symbols as well as with graphical scemes and also with words. It's like "1+2=3" (mathematical/numeric notation) and "one plus two equal three" (words). Both of them express the same conventional truth.
True about the logic but natural language sentences do not have to be true or false. Of course "meaningless" is one alternative, but so are "mostly right but very misleading," "ambiguous," "changed the sense of a word in the middle of the argument," etc.
I had forgotten that this was also Kripke's analysis. Thanks for reminding me. This same analysis equally applies to an expression of formal language that asserts its own unprovability.
I can still see no difference. When we are talking about facts, our statements have to be true or false. E.g. saying "George is taller than Alex" is like "a > b". But of course, as you say, natural language is much richer and has more attributes than symbolic one, e.g. ambiguous and meaningless, as you pointer out. Yet, Math can also be ambiguous, e.g. sqrt(4) can be 2 and -2. And it can be also meaningness, e.g. the expression 4 + 5 x 6, besides being ambiguous, it is ill-defined and therefore meaningless.
If you look "wider", you will find more similarities between natural and symbolic languages than the obvious ones ...
Quoting Intuitionism in the Philosophy of Mathematics
If we don't assume the Law of Excluded Middle is true, the Zermelo-Russell paradox still does not dissolve however, because it can be formulated in terms of the LNC.