Which is what? Certainly not self-reference because, as you mention, there are self-referential sentences which don't pose a problem. Something else about the liar paradox (in conjunction with self-reference) causes the problem, but it isn't a given that this "something else" is the same thing for both "this sentence is false" and "this sentence is meaningless" (or "this sentence is either false or meaningless.").

It is the same thing, specifically asserting its own untruth (or if you want to go deeper, both versions of the paradox I gave make use of what Graham Priest calls the "Enclosure Schema"). Both "false" and "meaningless" are untrue, and it is that untruth which the sentence asserts of itself, which causes the paradox.

I'm not saying that the T-schema defines itself as a contradiction. I'm saying that the T-schema defines T(x) as x and that this definition of T(x) is inconsistent with the liar paradox's definition of x as ¬T(x).

What you said earlier was:

The T-schema defines T(x) as x. The liar paradox defines x as ¬T(x). These are contradictory definitions.

This is a mistake, The T-schema (if accepted) just says the to predicate truth on 'x' is just to say that 'x' is the case; it's a biconditional, not equality, so they have the same truth-value. The liar simply takes a proposition which self-refers regarding some property of itself, and that entails the contradiction. Just take a look at the fuller argument I gave earlier. The proposition "L" is defined as asserting it's own untruth, but never is "L" defined as asserting its own truth.

It can't both be the case that T(x) means x and that x means ¬T(x).

T(x) has the same truth-value as 'x' under the T-schema, so either one an derive the other. The Liar in the format is that "~T(x)<=>x". I think if we went with your response we couldn't even accept Tarski's undefinability theorem.

But truth (and falsity) is just another predicate, I don't see the principle difference between self-referring to that and self-referring to some other property (save for the contradiction it entails). Unless we're going to stipulate that referring to interpretations is a no-no.

But truth (and falsity) is just another predicate, — MindForged

Being true is not like being red.

...Noted, but the problem is that if referring to its own interpretation were actually inherently problematic then the following ought to produce a similar paradox (but clearly it doesn't):

"This sentence is true".

It's odd I guess, since nothing really happens if you attribute truth or falsity to it.

So we note that having a statement refer to its own interpretation can produce problems, and disbar that process by introducing a meta language as Tarski, or statements that do not have a truth value assigned to them, as Kripke.

Ah so you were objecting to the T-schema, in which case my formulation of the argument for the paradox isn't available. I think Tarksi's hierarchy of metalanguages has its own problems, but Kripke's solution has a more straightforward issue with the Revenge paradoxes which call into question if it really solves anything. Namely:

"This sentence is valueless"/"This sentence is neither true nor false"/"This sentence is either false or neither".

It seemed to me that Michael sorted the issue with T-sentences. I didn't follow your objection.

I'd forgotten about the objections to Tarski's work. Isn't there a reply to these objections in terms of ruling out such sentences as ill-formed?

Michael said:

So T(x) ≔ x and x ≔ ¬T(x). Therefore, T(x) ≔ ¬T(x). Your definitions are contradictory.

My objection was that he was incorrect, because that's not how they were defined. I took the T-schema and so said that "True(x)" has the same truth-value as just asserting that "x". And so the Liar (as per Tarski) can be expressed as "~True(x) <=> x" (The truth of x's negation has the same truth value as asserting that x)
I mean, if "~True(x) <=>x" then "x <=> ~True(x)" (it's a biconditional after all)

Not as Michael rendered: "True(x) <=> ~True(x) (that's not how the T-schema is defined, x and True(x) are inter-derivable).

Certainly not self-reference because, as you mention, there are self-referential sentences which don't pose a problem. Something else about the liar paradox (in conjunction with self-reference) causes the problem, but it isn't a given that this "something else" is the same thing for both "this sentence is false" and "this sentence is meaningless" (or "this sentence is either false or meaningless."). — Michael

I agree. It is not the self-reference alone that is the problem.

My interpretation is that the problem is that the sentence refers to its own meaning, as assessing Truth requires first assessing meaning. So one has to work out what it means before one can work out what it means - there's the problem right there.

That's why sentences like 'This sentence is written in English' or 'This sentence has ten words' are not viciously circular. They are self-referential but the reference is to the sentence's syntax, not to its semantics (meaning). So one needs to only observe the sentence's syntax, not its semantics, before one works out its semantics.

I suppose it's that the sentence's truth value is referential that is the problem. Whereas in the examples of the last paragraph the truth value is not self-referential.

Self-referentiality need not be a problem. In mathematics, we hit self-referentiality when we have 'x', whose value we want to find, appearing more than once in an equation. Sometimes we can solve that by rearranging the equation so that 'x' appears only once, on one side. We call that 'making x the subject of the equation'. For example we can solve the equation 'x = 1/x' by re-arranging it to be 'x^2=1', so that x = sqrt(1), with solutions x=1 or x=-1.

Sometimes we can't do that but we can still guess solutions or find them by numerical analysis, as with equations like 'x = sin x'. What that involves is basically guessing a solution, seeing how it goes, and then refining it if necessary.

That approach works with the sentence 'This sentence is True' (quoted by ). We can't solve it deductively because of the vicious circle on truth value, but we can guess that the answer (solution) might be 'True' and when we substitute that for 'This sentence' in the sentence, we find that we get 'True is true', which works, and which is consistent with our tentative hypothesis (guess) that the sentence is true.

But it doesn't work with 'This sentence is false' because if we guess 'false' and substitute, we get the sentence 'False is false', which is true, which contradicts our tentative hypothesis that the sentence is false.
And if we guess 'true' and substitute, we get 'true is false', which is false, which contradicts our tentative hypothesis that the sentence is true.

So we can't guess a solution either.

My analogy is that deductively working out the truth value of a sentence, by interpreting it and assessing its truth value, is analogous to making 'x' the subject of the equation and working out the value of the other side. We can do that for 'That pot is black', 'This sentence has ten words' and for 'x=1/x', but not for 'This sentence is true', 'This sentence is false' or 'x=sin x'.
And guessing a truth value and testing how it works is analogous to guessing a value of 'x' and testing how it works. We can do that for 'x = sin x' and for 'This sentence is True' and find a truth value / x value that works, but we can't do it for 'This sentence is false' because none of the values we can guess work.

1) True(L) ∨ ¬True(L) (Excluded Middle)
2) True(L)
3) L (release)
4) ¬True(L) (definition of L)
5) ¬Tr⁡ue(L) ∧ True⁡(L) (adjunction) — MindForged

Your two premises (1 & 4) are contradictory. (5) is the least of your worries (runs for bomb shelter...)

I'm confused, why are you substituting the value in for the self-reference? The liar isn't "False is false". Like the point of my "This sentence is true" example was to object to the claim that self-referential interpretations are visciously circular. "This sentence is true" does self-refer to its own truth-value but there's no problem assigning it as "true" or "false". If "This sentence is true" is itself true, then it's true. If it's false then it's false.

But that seems to suggest that we can't immediately say there's a flaw in this sort of thing, otherwise it seems a bit unprincipled (it has all the same features as the Liar). So when you say you can't guess a solution, well, the Dialetheist is liable to just say you can guess a solution, but the solution is that it's both true and false (pick whatever semantics your prefer in order to spell out how this is this case).

That's why sentences like 'This sentence is written in English' or 'This sentence has ten words' are not viciously circular. They are self-referential but the reference is to the sentence's syntax, not to its semantics (meaning). So one needs to only observe the sentence's syntax, not its semantics, before one works out its semantics. — andrewk

Well said. It's a statement without a truth value and hence not a proposition.

Well yea it's using the T-schema to pull out the Liar. If you think the unrestricted catch and release rules are correct, it entails a contradiction. Naturally if you think the argument is sound you'd better be dropping the explosion principle. :-)

8-)

It's wonderful that logic throws up these little puzzles. That it's not as crystal-clean as some pretend.

To be fair, I just enjoy reading about dialetheism because it gets at a lot of intuitions we take as plausible and shows that we have to either accept the consequences of those intuitions (T-schema leading to a true contradiction) or we have to drop some rule that really seems like it ought to be a correct rule. Or maybe I just like being insufferable. :)

...T-schema leading to a true contradiction... — MindForged

How's that?

I think the T-schema about as close to a definition of "...is true..." as we can get. Don't you go getting me all worked up now...

Isn't it just Tarski's undefinability theorem and why he advocated the metalanguage hierarchy? If the language has negation, self-reference and the diagonal lemma holds, you can produce the Liar sentence:

x <=> ¬True(x)

(Phew) worried me for a bit.

The liar isn't "False is false". — MindForged

The liar sentence, as usually given, is

'This sentence is false'

But if we are being excruciatingly literal-minded, it is a simple, false sentence, because a sentence is a bunch of words and 'false' is a truth value, so the two are not the same thing (a bunch of words is not a truth value), and should not be connected by the word 'is', which implies identity.

Hence we interpret it as meaning:

'there exists x that is the truth value of this sentence and x = False'

If the truth value of the sentence is 'true' then, under the axiom schema of substitution (see I9 from here), we can substitute 'true' for the words 'the truth value of this sentence', without changing the truth value of the sentence. That gives us

''True is the truth value of this sentence and True = False'

whose truth value is False, because the second conjunct is.

Hey, here's something really cool!

I just noticed that the sentence

'This sentence is True'

or more literally, per the post I just made:

'There exists x such that x is the truth value of this sentence, and x=True'

has two different solutions. It is satisfied not only by the hypothesis that x exists and is True, but also by the hypothesis that x exists and is False.

Try it and see!

That's made my day.

I think all you're really doing is denying the possibility of self-reference, because the Liar is constructed within a semantically closed language unless you work in a metalanguage, e.g. "This sentence" is true, which is not even truth-apt. That's how Tarski attempted to resolve the Liar but this sort of approach isn't in vogue anymore (for a number of reasons, so I'll just gesture at them over on the SEP).

If you are being "excruciatingly literal minded" then you wouldn't substitute the truth value in for the referent of the sentence. The truth-value is part of the sentence that's being referred to, that's the Liar. X is not defined as the value "false", X is defined as asserting that "~True(X)".

I think all you're really doing is denying the possibility of self-reference, because the Liar is constructed within a semantically closed language — MindForged

I had assumed we were discussing within that context, along with the inconsistencies and explosions that inevitably flow from that. Why do you think that involves denying the possibility of self-reference? My expansion of the sentence to the more formal version above is following how Russell expands 'The present king of France is bald' in his theory of definite descriptions, not seeking to forbid self-reference. The aim is to make explicit the implicit assertions hidden within a definite description.

The fact that my expanded sentence still contains the word 'this sentence' should be sufficient to demonstrate that the operation did not banish self-reference.

If you are being "excruciatingly literal minded" then you wouldn't substitute the truth value in for the referent of the sentence. The truth-value is part of the sentence that's being referred to, that's the Liar. — MindForged

I'm afraid I don't know what you are referring to with the words 'the Liar'. And also, I'm afraid I can't make anything of your first sentence. In my understanding, a sentence does not have a referent, it is names or symbols that have referents.

The Liar Paradox = L = This sentence is false.

The liar does not tell the truth so we must negate whatever the liar says regardless of any statement's he says. It does not matter if the content of his statement is true or false, its must be negated. So, the truth value of ...'not "This sentence is false"' is not decidable as such.

The Belgian surrealist painter René Magritte presented a somewhat similar paradox for images in his work:
The Treachery of Images

This Is Not a Pipe

The text seems to be declaring its referential superiority, except that as part of the work as a whole without the image the text would not have the same meaning, while without the text the image still is referential.

How do we make sense of this paradox? — TheMadFool

The basic issue is that the liar sentence requires an infinite recursion to ground its referents.

So it fails to state anything, not even a contradiction.

I had assumed we were discussing within that context, along with the inconsistencies and explosions that inevitably flow from that. Why do you think that involves denying the possibility of self-reference? My expansion of the sentence to the more formal version above is following how Russell expands 'The present king of France is bald' in his theory of definite descriptions, not seeking to forbid self-reference. The aim is to make explicit the implicit assertions hidden within a definite description.

The fact that my expanded sentence still contains the word 'this sentence' should be sufficient to demonstrate that the operation did not banish self-reference.

Ah my mistake, I must have misread something during my last response. When I read this:

I understood it as saying that "false" was not part of the sentence being referred to ("should not be connected by the word 'is' "). So if we're going with Russell's theory of descriptions (not sure I accept it), I think an initial issue is that there is not quantifier in the Liar sentence. Note the argument I gave prior was just a propositional logic (where it is usually studied IIRC). What is actually used is predication, so the recourse to quantification seems suspect to me, if not simply misrepresentative of how the paradox is formed.

It also seems like you're conjunction is a bit odd.

'there exists x that is the truth value of this sentence and x = False'

Is what is referred to by "this sentence" is the entire conjunction or just the first conjunct? (I think it's the latter) And does this actually dissolve the paradox? I mean, if 'x' is "false" it seems like the following happens:

"False is the truth value of this sentence and false=false"

Which just seems like the Liar paradox put in a conjunction. Of course the conjunction comes out as necessarily false because the Liar paradox (a contradiction) is the first conjunct.

I'm afraid I don't know what you are referring to with the words 'the Liar'. And also, I'm afraid I can't make anything of your first sentence. In my understanding, a sentence does not have a referent, it is names or symbols that have referents.

Basically I was saying that the Liar is making a predicate of falsity of itself. And I meant to say the "subject" of the sentence, not the referent.

That seems false. Just take the Liar and it gives you a grounded truth value to start with (namely falsity). Of course, that value will entail the contrary value and voila, a contradiction. It does not require infinite recursion at all; The T-schema for instance uses the diagonal lemma and so can produce the Liar.

TheMadFool — TheMadFool

It is an incomplete thought.

"This sentence is false". False about what?

"I am lying". Lying about what?

Too simple.

The only agreement among logicians seems to be that no one has a proper solution yet, so if there is a solution it must be a strange one because all the obvious responses have been tried and they failed (e.g. Kripke's solution doesn't work, Tarski's infinite hierarchy of metalanguages doesn't work, etc.) — MindForged

Thanks. Dialetheism it is then.




Thanks for the post. It's an interesting connection. Something denying itself. It happens in some mental illnesses too. I think it's called nihilistic delusion where a person basically thinks ''I don't exist''.


But, ''this'' isn't like ''I''. If we stay true to the definition of the word then ''this'' doesn't apply to itself and it should for the liar paradox to be one.

Of course we could invent a self-referential word e.g. ''thes'' and define it as such and the paradox would appear.

If one were to be as exact as possible the definition of ''this'' doesn't include self-reference. It is grammatically incorrect (I'm not a linguistic expert).

However, people do use ''this'' as you have (''this Australian needs a bath'' :D) but note that such forms of language are classified as referring to oneself in the third person. It isn't completely an instance of self-reference. People would find it odd to hear someone refer to himself in the third person.

So, I still think the liar sentence is grammatically incorrect.

However, as I mentioned above we could invent a self-referential word like ''thes'' and the liar paradox still is a problem.

What do you think?

So it fails to state anything, not even a contradiction. — Andrew M

I have a gut feeling the liar paradox is important. It must mean something. I just don't know what it is.

The T-schema for instance uses the diagonal lemma and so can produce the Liar. — MindForged

Are you sure it can do that validly? The linked page states the lemma with a premise that restricts it to first-order languages, which I expect would rule out its use in a T-schema environment which I believe is higher order.

I was going to check the proof to see if that premise is actually used, but I got tired and didn't, so I'm hoping maybe somebody else did. It would be unusual to state a premise that was not used though.