As was pointed out before, not all self-referential sentences are problematic:

This sentence has five words

Hence, self-reference is not the whole of the problem.

Ah. You lost me in referring to a reference to a reference to another reply... it wasn't so obvious.

So what do we conclude here?

It's not the statement itself you need to deny, but the presupposition. — Srap Tasmaner

Which statement and which presupposition?

In the case of "this sentence has five words" it is not the semantic content, but the structure, of the sentence which is being referenced. The structure of a sentence is independent of its meaning, it is a kind of "externality", a state of affairs, that can be coherently pointed to. When I referred to self-referentiality, I specifically had meaning in mind.

OK:

This sentence is true.

This sentence is about itself.

This sentence is a sentence.

This sentence is not that sentence.

Need I go on?

Try:
This sentence is not about itself.

This sentence is not a sentence.

And lest you think the problem is "not",

This sentence is that sentence.

"This sentence is true" is nonsense; it is like saying "This car is true".

"This sentence is about itself" just says that the sentence refers to itself, which seems unproblematic.

"This sentence is a sentence" again refers to something grammatical, beyond mere semantics, and thus is not self-referential in the same way as the "Liar".

"This sentence is not that sentence" again refers outside itself, so is unproblematic.

As I see it the problem with the "liar" sentence is its self-referentiality. — Janus

"This sentence is about itself" just says that the sentence refers to itself, which seems unproblematic. — Janus

Hmmm.

OK, but I never said that self-referentiality per se must lead to contradiction, did I?

Also I would make the further point that the "This sentence is about itself" is unproblematic because it doesn't really say anything about itself; it is kind of a non-statement.

Perhaps then it is the combination of mere self-referentiality with a truth claim (which to be coherent requires pointing to something else) that leads to contradiction and/or meaninglessness.

Perhaps then it is the combination of mere self-referentiality with a truth claim (which to be coherent requires pointing to something else) that leads to contradiction and/or meaninglessness. — Janus

That, in a nutshell, is Kripke's response.

https://plato.stanford.edu/entries/truth-axiomatic/#KripTheo

Cheers, will have a read. :cool: And thanks, your questions have led me to see this issue in more comprehensive and explicit ways. :smile:

Edit: I took a look at the linked article, but unfortunately it was so technical I lacked the background to unravel it without significant effort.

This sentence is true" is nonsense; it is like saying "This car is true". — Janus

Janus I'm not splitting hairs but I'm not sure that's exactly correct. Saying " this sentence is true" is actually true , I think, because it's a statement about itself.

Similarly, " this statement is a lie" is referencing the statement itself too. And so if the statement is true, then it is false; and if it's false, then it's true.

Other examples are:

Socrates: 'What Plato is about to say is false.'
Plato : 'Socrates has just spoken truly.'

And:

"Janus cannot prove this statement to be true."

It basically means that there will always exist certain true statements that cannot be proved to be true.

Janus I'm not splitting hairs but I'm not sure that's exactly correct. Saying " this sentence is true" is actually true , I think, because it's a statement about itself. — 3017amen

I don't see what you think is true here.

Socrates: 'What Plato is about to say is false.'
Plato : 'Socrates has just spoken truly.' — 3017amen

Here you have two different sentences which are referring to each other; which seems to be just a more elaborate form of self-referentiality, so I don't see why the same would not apply as with the "Liar".

"Janus cannot prove this statement to be true." — 3017amen

Can't see a problem with that one, either.

It basically means that there will always exist certain true statements that cannot be proved to be true. — 3017amen

That's an entirely different matter. No empirical statement that happens to be true can be proven (in the deductive sense) to be true. How would you prove that water boils at 100 degrees C, for example?

Which statement and which presupposition? — Banno

The idea would be something like this:

The Liar does not allow us to accept or reject it, and that's somewhat reminiscent of what goes wrong with

'The present king of France is bald.'

There are in fact two claims here (following Russell's analysis just for the moment to make the point); because of scope issues, we can end up, or seem to end up, denying the predication when what we want to deny is the existential claim that the there is one such individual who is the present king of France. Later work calls such implicit claims 'presuppositions'. (The usual criterion for A being a presupposition of B is that B and its negation both imply A.)

Now consider a claim like this:

'Your ideas are very tall.'

Agreement is wrong, and disagreement is wrong, but not because of anything like a vacuous singular term or a definite description, none of that. So what is wrong? In predicating 'tall' of your ideas, there is a presupposition -- or anyway, something like that -- that 'tall' could reasonably be applied or not to ideas. And this is what you want to deny -- not that this predication is, say, mistaken, that sadly your ideas are only of average height. What we need to deny is that 'tall' is even on the table for ideas.

Now the Liar.

The Liar predicates falsehood of itself. Thus the Liar presupposes (or something) that it is the sort of sentence that can be true or false. Not all can. The Liar certainly looks like one that can, no question. Does that turn out to be the case? If not, what we want to deny is not the Liar's predication, but the Liar's presupposition.

(Before this approach occurred to me, I was thinking about the Liar implicitly claiming that it is possible to be a member of the class of sentences that are true iff they are false, and that it is a member, and that this necessarily empty class has a member and is not empty. I might still end up liking that more, not sure. It's more machinery.)

What we need to say of the Liar is that it's not even wrong. And the way we get to do that is by taking away its claim to have predicated anything at all, by taking away the presuppositions upon which that predication is, well, predicated.

This was in my in tray this morning:

What Colour Are Numbers?

Thanks - that was very helpful.

(The usual criterion for A being a presupposition of B is if B and its negation both imply A.) — Srap Tasmaner

So the present King of France's being bald, and his not being bald, presuppose there being a present King of France. thanks for clarifying this use.

And for any sentence A, being true, or being false, presupposes... and here I lose track.

I want to write "there being a sentence", but that would be true of "this sentence is false"; after all, if it's not a sentence, what is it? and if we are to discount it as a sentence, then why is that not just special pleading? Is there more reason to discount it than just that it is annoying?

Edit: perhaps this: what is presuposed by "this sentence is false" that is not presupposed by "this sentence is true"?

Cheers, will have a read. — Janus

You might not thank me when you get around to tackling the article. It presupposes much.

As I said I don't have the background to read that article without a great deal of effort. So, would you do me the favour of explaining what presuppositions you are referring to? Also, since every philosophical exposition or position seems to presuppose much; in regard to this article, are you implying that you don't agree with its presuppositions? If so, some explanation for your disagreement would also be helpful.

Presuppositions were introduced by @Srap Tasmaner, for whom I have hight regard; but I do not yet see how they help here - so best address those questions to Srap.

"(Li) is happy" doesn't cause any such issue. We don't spend hours working out what "This sentence is happy" could possibly mean because sentences aren't the sorts of things which can be happy.

But sentences aren't the sorts of things which can be true either. Beliefs can be true, propositions can be true, mathematical equations can be true...sentences themselves can't. It's like saying "This horse is true", I don't know what it would even mean? — Isaac

This seems to indicate an interesting approach. If the Liar were changed to "This proposition is false" we might ask "which proposition?". There is a sentence there, but is there a proposition?

OK, I thought your statement that the article presupposes much represented your own view.

Nuh. Just wordplay.

The truth predicate applies to sentences (or propositions). It does not apply to any other object. — Kornelius

So what does it mean when I say my friend Jack is a 'true gentleman'?. What am I seeking if I seek the 'truth'? How is the 'true cost' of something different from its apparent cost? What sentence am I aiming for I I try to be 'true to my cause'? When we ask "Is the story of Jesus's resurrection true?" would it matter what the actual sentences constituting that story were, or are we asking something more general?

We often speak this way, but I think what we mean is that the content of a belief is true. And the content of a belief is a proposition — Kornelius

When I yell some obscenity after stubbing my toe, what is the proposition that constitutes the belief which drove that behaviour?

You're right to think along these lines, and indeed Kripke's solution can be understood as a formalisation of that idea. — Banno

Interesting. Is there any paper in which this is expounded?

Edit - found your link - thanks.

This seems to indicate an interesting approach. If the Liar were changed to "This proposition is false" we might ask "which proposition?". There is a sentence there, but is there a proposition? — Janus

Yeah. As has been said - more authoritatively than I put it, the ability to evaluate the truth value of a proposition seems to depend on the nature of the proposition. It does seem rather odd to me to persist in having truth as a property of certain propositions without noting the features which those propositions share and how such shared features allow truth evaluation where lack of such denies it.

It seems that the sorts of propositions which allow truth evaluation are those which are about beliefs, and propositions which do not are those which do not reference a belief. So it's far more productive to talk about the truth of the belief rather than the truth of the proposition which references it.

From an ancient version of the Truth Wiki page, since deleted:

Saul Kripke contends that a natural language can in fact contain its own truth predicate without giving rise to contradiction. He showed how to construct one as follows:

Begin with a subset of sentences of a natural language that contains no occurrences of the expression "is true" (or "is false"). So The barn is big is included in the subset, but not " The barn is big is true", nor problematic sentences such as "This sentence is false".
Define truth just for the sentences in that subset.
Then extend the definition of truth to include sentences that predicate truth or falsity of one of the original subset of sentences. So "The barn is big is true" is now included, but not either "This sentence is false" nor "' The barn is big is true' is true".
Next, define truth for all sentences that predicate truth or falsity of a member of the second set. Imagine this process repeated infinitely, so that truth is defined for The barn is big; then for " The barn is big is true"; then for "' The barn is big is true' is true", and so on.
Notice that truth never gets defined for sentences like This sentence is false, since it was not in the original subset and does not predicate truth of any sentence in the original or any subsequent set. In Kripke's terms, these are "ungrounded." Since these sentences are never assigned either truth or falsehood even if the process is carried out infinitely, Kripke's theory implies that some sentences are neither true nor false. This contradicts the Principle of bivalence: every sentence must be either true or false. Since this principle is a key premise in deriving the Liar paradox, the paradox is dissolved.[26]

As has been said - more authoritatively than I put it, the ability to evaluate the truth value of a proposition seems to depend on the nature of the proposition. — Isaac

I'm also partial to Wittgenstein's idea that we know the meaning of a proposition when we know what would necessarily be the case if it were to be true. Applying that criterion to the "Liar" (assuming for the sake of argument that it could be called a proposition) makes it looks meaningless.

Yes. I tend to go further to say that we need know how we would behave if it were the case. I can't form a belief about whether it is the case that "this sentence is false" because I don't know what I would do differently if it were.

Self-referentiality may seem like a problematic concept, but Gödel, Tarski, and Carnap have shown that it is possible to construct a self-referential sentence (or something close enough) inside arithmetic. That is, given a sufficiently strong arithmetic theory T and a coding schema "...", we have the following:

Fixed-point or diagonal theorem: If P(x) is a formula of the language of T with only "x" as its free variable, then there is a sentence F in the same language such that T proves that F is equivalent to P("F").

In other words, for any property P, if T is a sufficiently strong arithmetical or syntactic theory, then there are sentences of the language which ascribe P to themselves (or close enough). Hence, unless syntax or arithmetic is an incoherent enterprise, self-referentiality can't be the problem.

Nine months ago I was thinking about the liar and wrote the following...

"This sentence is false" is incapable of being false. It is also incapable of being true. What makes it so puzzling is the mistaken presupposition that it is even capable of being either.

Seems I agreed with some of those who are more qualified on that basic point.

I can't form a belief about whether it is the case that "this sentence is false" because I don't know what I would do differently if it were. — Isaac

And what would you do differently as a result of "This sentence is true"?

It is also, presumably, meaningless?

Let me get back to you on this one. It will take a slightly longer post. — Kornelius

Still keen.