That assumption is discharged at step 5 — Kornelius

Ugh, the Liar is such a pain. I was about to slap my forehead and say you're absolutely right, but are you? I'm genuinely not sure now. The discharge step would have this form:

4.5. P → P & Q

Can we say that P has been discharged if it's on the right hand side here? That doesn't seem right.

(This is the consideration I dismissed, in so many words, above. 'It doesn't matter that (Li) is part of the expression (Li) & ~(Li).' Geez.)

Whether we can discharge (Li) this way truly doesn't matter though because (4.5) is just P → Q, which is clearly fine, and we can have that even before getting to 4:

3.5. True( Liar) → ~True( Liar )

and that's just equivalent to ~True( Liar ).

So, yes, no question, by the time we get to applying [T-in], (1) has been discharged, either by forming the contradiction, as you did, or by forming (Li)→~(Li).

And so you are, finally, absolutely right we get ~(Li) as a theorem and then (Li) as a theorem from that, so we end up proving outright (Li) & ~(Li). Thanks for setting me straight!


(Not sure which of the logical equivalences I used are intuitionist-safe, but I so don't care at the moment.)

Ugh, the Liar is such a pain. I was about to slap my forehead and say you're absolutely right, but are you? I'm genuinely not sure now. — Srap Tasmaner

Haha, that makes two of us!

This discussion has been great; it made me think about it in greater detail than I had.

But honestly, I think the proof is much easier to see if it is presented in a natural deduction system. I hadn't given much thought to the assumption's being discharged, but that would be obvious if we used the former.

(Not sure which of the logical equivalences I used are intuitionist-safe, but I so don't care at the moment.) — Srap Tasmaner

:rofl: Well, in any case, we would surely need independent arguments to adopt a non-classical logic.

Consider the following sentence (Li): (Li) is not true. — Kornelius

Something I'm missing about the Liar...

"(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?

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

Maybe something analogous to "hammers are for hammering", "this coin is worth two cents", "this note is a middle-C", etc.

I agree that a hammer has a purpose to us in the here and now because we exist. But if thinking beings cease to exist, wouldn't the hammer cease to have a purpose, and be just a collection of atoms, subject only to purely mechanical forces of nature? — Ash Abadear

Maybe something analogous to "hammers are for hammering", "this coin is worth two cents", "this note is a middle-C". — bongo fury

Not following you at all I'm afraid. Any chance of an expansion?

added a link. Also,

speech act theory, which seems to have continued an anti-abstract trend away from positing of (as entities) propositions to only sentences to only statements on particular occasions. (Yay, tokens! ... utterances, inscriptions.) — bongo fury

Thanks. So if the sentence is a token for the proposition, then is the proposition about the sentence (token) or itself?

The token is about the things it is about. The proposition is an unnecessary abstraction?

The token is about the things it is about. The proposition is an unnecessary abstraction? — bongo fury

But in the Liar, the sentence is about the sentence, so if it were to act as a token it could only be for some kind of assertion or proposition. There's no real state of affairs it's referring to. That's why I thought it was strange to give it a truth value.

Ah, sorry, I get you. Yes, the token is about itself, in such a way as to cause an explosive mess of contradictory tokens in a deductive context.

Ah, sorry, I get you. Yes, the token is about itself — bongo fury

So is a token the sort of thing that can be 'true'?

:ok: (imv)

So the first question is this: why should we think that the concept of Truth is inconsistent? — Kornelius

In a word, truth is dynamic, not static. The need to embrace truth, from our self-awareness (conscious existence), as you so well suggested, represents another paradox. The first one you mentioned from the liar's paradox (propositions of self-reference) not only represents paradox itself, but perhaps more importantly incompleteness; hence, dynamic (see Gödel and Heisenberg uncertainty/incompleteness theorem).

Thinking itself (about truth) requires the passage of time (dynamic). And is almost yet another 'dualistic', metaphysical kind of question... .

To that end, one could argue that truth is both dynamic and static, just like one could make the case for truth being both subjective and objective.

@Kornelius@Srap Tasmaner

Thanks you two!

But sentences aren't the sorts of things which can be true either. — Isaac

Sentences are precisely the things that can be true or false. The truth predicate applies to sentences (or propositions). It does not apply to any other object.

Beliefs can be true — Isaac

Not quite (I think), unless something has changed. 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 (usually expressed by a sentence). Since a belief is technically a mental state (or a propositional attitude), I don't think truth technically applies, but we say a belief is true as a shorthand for the content of the belief is true. I might be mistaken here, but I don't attach much importance to it.

It's like saying "This horse is true", I don't know what it would even mean? — Isaac

Quite right, the sentence has no meaning because it has no semantic value. It is not a well-formed sentence because truth does not apply to objects like horses.

The Truth predicate perfectly well applies to things like (Li) because the latter is a sentence.

but perhaps more importantly incompleteness — 3017amen

Hmmm, I'm not seeing the immediate connection between the Liar Paradox and the incompleteness theorems, but maybe there is an interesting one. Could you elaborate?

I've been thinking, if we're going to (what we needn't) treat True(...) as a predicate applicable to sentences, then we should also be able to talk about classes. (There are of course problems with enumerating sentences, but in natural language we do use class talk that might present technical problems or resist formalization.)

The Liar, then, claims -- speaking a little loosely here -- that it is not true, which is equivalent to claiming that it is true iff it is false, claims thus to be a member of a class that is by definition empty, and claims further that said class is not empty because it itself is a member. Thus the Liar is false.

Should add: by predicating 'not true' of itself, the Liar claims to be a member of the class of sentences that are true or false, and perhaps it is this claim that turns out to be false, making the conjunction of its claims false.

For 'claims' here and there above, perhaps 'presupposes' is what we need.

'This is the prettiest square circle anyone has ever drawn.'

Hmmm, I'm not seeing the immediate connection between the Liar Paradox and the incompleteness theorems, but maybe there is an interesting one. Could you elaborate? — Kornelius

Sure great question! And welcome to the forum by the way.

Mathematician and logician Kurt Godel (and Alan Turing/Turing machine and Bertrand Russell--you can Google that if you will) explored the concepts of infinity versus finitude as it relates to mathematics (deduction).

Without getting into the technical details (which we can if you want) Godel parced the relationship between the description of mathematics and mathematics itself. Basically, he was known to have labeled mathematical propositions combining a sequence of propositions into corresponding natural numbers that form associated labels. Logical operations about mathematics were made to correspond to mathematical operations themselves. The idea linked the self-referential concept of Godel's proof by identifying the subject with the object.

Accordingly, self-referential paradoxes is the appropriate analogy. As mentioned in the OP, the liar's paradox is an unresolved paradox that of course is self referential. It's undecidable, and it's incomplete. And it's based upon a priori logico deductive reasoning.

Sentences are precisely the things that can be true or false. The truth predicate applies to sentences (or propositions). It does not apply to any other object. — Kornelius

I'll return the favor and ask you for clarification of your foregoing quote. How are you suggesting there are no undecidable propositions? (How could this be?)

by predicating 'not true' of itself, the Liar claims to be a member of the class of sentences that are true or false, and perhaps it is this claim that turns out to be false, making the conjunction of its claims false. — Srap Tasmaner

An inscriptionalist only wants to add to that that turning out to be false can just mean (for this particular claim) being rejected by the system of token-producing agents, who will judge deduction to be an inappropriate treatment for the sentence (i.e. any token of it), and will thus invalidate it in respect of its constituting a licence to print unlimited copies (of itself and other consequent sentences).The speakers will, in other words, put a brake on application of logic to this particular bit of natural language.

I'm not seeing the immediate connection between the Liar Paradox and the incompleteness theorems — Kornelius

By way of gossip, I'm almost certain Godel himself said as much, and may even have suggested he wasn't thrilled about how close the arguments were. To be googled.

"This statement is false" and "This statement is true yet unprovable" are both self-referential. Beyond that, there's not much in common.

A statement can have either truth value assigned to it; A statement becomes an assertion when someone asserts that it is true.

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

Sentences include questions, commands, and so on; Of these, only statements can be true or false.

There's no real state of affairs it's referring to. That's why I thought it was strange to give it a truth value. — Isaac

"The present King of France is bald" is another statement to which we cannot (easily) allocate a truth value. They are not uncommon. In contrast, "This sentence is false" allocates a truth value to itself. You're right to think along these lines, and indeed Kripke's solution can be understood as a formalisation of that idea.

the immediate connection between the Liar Paradox and the incompleteness theorems — Kornelius

https://en.wikipedia.org/wiki/Diagonal_lemma?wprov=sfla1

https://thephilosophyforum.com/discussion/comment/342838

"The present King of France is bald" is another statement to which we cannot (easily) allocate a truth value. — Banno

And obviously what I had in mind when I said 'presupposition'. Now consider statements that have presuppositions that aren't contingently but necessarily false. It's not the statement itself you need to deny, but the presupposition.

??

Could you rephrase the question?

Where did you say "presupposition"?

Here.

As I see it the problem with the "liar" sentence is its self-referentiality. You can coherently say "that sentence is false", no problem and every such proposition points (or purports to point) to something beyond itself. The "Liar" is an eccentric sentence in that it points, or purports to point, back to itself, and thus becomes self-contradictory.

It seems to me, to invoke the Bard, "much ado about nothing" or "full of sound and fury, signifying nothing"; and only the anally retentive will be bothered by it.

I didn't read through the whole thread, so apologies if someone has already made this point.