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?

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.

I think it may help the discussion to distinguish between private property and personal property.

@ChatteringMonkey even if we accept the view of human nature you briefly alluded to, it isn't clear which type of property you mean. I think most of what you say is an argument for limited personal property but is not an argument that extends to ownership over means of production, for example.

@JerseyFlight Would it be correct to say that, on your view, Marx was a critic of ownership rights over means of production, but not over personal property (like a home, a television, etc.)?

I also really think we should bring out the criticism of the concept of 'human nature' into full focus. This is a concept that many people rely on when arguing, and it is not one that never is clearly defined. It is used to suit one's argument, usually.

Since there was only one premise... — Srap Tasmaner

That assumption is discharged at step 5. in the proof. The reasoning from step 5 to 8 is what we get when we have (Li) is false as a theorem. We proved it outright, it is not within the scope of a premise/assumption. I actually like natural deduction systems for this reason! :grin:

This is just to say that if we accept that (Li) is false, then we're accepting a contradiction because it implies a contradiction. We can outright prove that (Li) is false (step (5)) and a contradiction follows outright. The contradiction is not within the scope of step 1. since it was already discharged.

As far as I know, the proof outlined here is intuitionistically valid but I might be wrong. I am quite certain it is though, we never made use double negation elimination or LEM.

The usual one: if P entails a contradiction, P is false.* If you want, you could say the problem is that both the Liar and its negation are false, which in turn violates the LEM. — Srap Tasmaner

Hmm, the issue is that both (Li) and not-(Li) entail a contradiction. So the simple reductio argument to conclude not-(Li) won't suffice. Since not-(Li) also entails a contradiction, then it is not the case that not-(Li). In other words, by your own reasoning, it is false that (Li) is false. But notice that the rule you mention here doesn't get us from [(Li) is true and (Li) is not true] is false to (Li) is not true. If a conjunction is false, either of its conjuncts could be false, so the inference here is invalid. The reductio, however, works both ways and is actually formalized in the proof I outlined in the OP.

Your last suggestion is a viable one (though not necessarily one I agree with). We could think that the Liar Paradox shows that LEM is not valid. However, the reasoning in the Liar paradox is actually intuitionistically valid, so we need more than this. I am no expert on intuitionistic logic and less so on minimal logics. From the literature, I know that only very minimal logics would not allow the Liar reasoning to go through.

* It's not uncommon to just define ¬P¬P as P→⊥P→⊥. Even in systems that don't, this is an introduction rule for ¬¬. — Srap Tasmaner

Right, but in the Liar we also have $\neg P \rightarrow \bot$ so we cannot simply conclude $\neg P$, no?

Something like this; if you conclude that it's true and false, the conjunction of a truth and a falsehood is a false, so it's false. IE, it appears to be an instance of "X and not-X". — fdrake

Ok, I see (I hope!). This is just the denial of dialetheia: there are no true contradictions. This seems right, but it doesn't follow from this that (Li) is false.

What we have is [(Li) is true and (Li) is not true] is not true.

Agreed. But by what rule of inference would it follow that (Li) is not true? That makes [(Li) is true and (Li) is not true] false just as much as (Li) is true.

But maybe I'm not following your suggestion?

EDIT:
Rather, what rule of inference would establish (Li) is false only

Anyway, for the assumption angle, I was trying to think of something like that principle -- the name escapes me, has to do with entropy -- that there are always more ways of being wrong than right. Not exactly that, but something like it. A justification for assuming that a sentence, even a grammatical sentence, is nonsense until it is demonstrated that you can assign a truth value to it. — Srap Tasmaner

Hmm, I am not sure about this principle either. Perhaps you meant Russell's justification of the principle of simplicity, i.e. do not multiply entities beyond necessity because this increases the chances you are wrong. So maybe the idea is that we should multiple meanings beyond necessity. But I am not sure that makes it less likely we may be wrong. Saying that sentence P is meaningless when it is meaningful is a mistake (and conversely). So I am not sure how one assumption would be more conservative than the other in this sense.

Still, I do think you are right in pressing against the view I was arguing here, namely that we should assume it is meaningful. I want to clarify that I am not simply assuming this, but I think the sentence is intuitively meaningful. This is not an argument to its being meaningful, but it just raises the bar of proof for those who deem is meaningless.

Notice this is quite different from assuming a sentence or all sentences are meaningless until shown otherwise (or vice versa). Certain sentences are intuitively meaningful, some are not.

Most importantly, I am not saying that we need no argument as to what the meaning of (Li) is. The Liar paradox shows that we do need such an argument.

Is there an argument for thinking that (Li) is intuitively meaningless? Does it really sound like gobbly-gook? It can still be meaningless, of course.

How does Scharp use these to solve the paradox? — Banno

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

What blocks:

9. (Li) is not true (8, truth conditions of conjunction).

?

Why can't it just be false? — fdrake

Nothing.

And nothing blocks 10. (Li) is true (8, conjunction elimination).

In fact, any proposition classically follows from 8. We can prove anything.

For this reason, we cannot simply conclude that (Li) is not true, since we can conclude (Li) is true and, in fact, any proposition we wish.

But I am thinking this explanation may not be satisfactory. Did you have something else in mind?

You have at least two inferences here. You should make them explicit and state the justification for each step. In particular, your last step is Truth-Out, one of the constitutive principles mentioned in the OP. This is what allows you to make that inference. You have not justified step 1 to 2 yet.

All to say. You are attempting to reason to a contradiction. You didn't actually get to an explicit contradiction (yet-- nowhere do you have (Li) as a statement, for example. You have (Li) is true, but you need an inference move from this to (Li)) and there are other moves you may be missing, but that's not really the point.

What did you think of the Liar involving down-true?



I think you are right: there is definitely nonsense in natural languages and a lot of it. I guess the question is whether the liar sentence itself is nonsense. We can say no to that one, even if we accept that there is a lot of nonsense around.

Hmm, I think I understand your sentiment here, but (Li) is actually not the same as not-(Li). So it isn't a contradiction in this sense.

Any sentence P would be an obvious contradiction if it is in the form of, say, Q and not-Q. But (Li) is actually not in this form. To be in this form, (Li) would need to say: "This sentence is not true and it is not the case that this sentence is not true". Then I would agree with you, we just have an outright contradiction.

But this is not the case with (Li). Contradictions take on explicit syntactic forms, and you need to reason your way to this. That is the whole point of the Liar argument. It shows that reasoning from the assumption that (Li) is true leads to an outright contradiction, i.e., we can prove that (Li) and not-(Li) (which is basically what line 8 in the proof of the OP says --- to be more specific, it says something equivalent in the sense that it includes the truth-predicate, but we can get (Li) and not-(Li) by applying T-out on 8, but we don't need to since 8. itself is already a contradiction.)

To make this point a little clearer, imagine I define the following concept Down-True (T) such that:

$T(\phi)\rightarrow \phi$

but the following fails (i.e. does not hold):

$\phi\rightarrow T(\phi)$

*In fact this is what Kevin Scharp defines as descending truth.

Now I consider the following sentence:

(Li*) This sentence is not down-true

What do you make of it? Does it lead to a contradiction? What I am arguing here is that it is the constitutive principles of truth that make the liar sentence a problem, not the liar sentence itself. That is, there is something inconsistent about the constitutive principles of truth, not (Li) in itself. (Of course we could just say (Li) is fine and restrict our logic, but this also speaks against the idea that (Li) is an outright contradiction).

I will respond to the second post you made. I believe you are reading too much into the inference. The inference from 1 to 2 is a simple syntactic substitution. For this reason, it is valid.

Let "The blue dog" be defined by the symbol B. Then we have the following:

1. Assume that: B walked to the park.
2. Then: The blue dog walked to the park (substitution).

Nothing more nefarious is going on in getting 2. from 1.

Notice, too, that a contradiction is in the form $P \wedge \neg P$. The liar is not in this form. We need to reason to a contradiction. For this reason, certain minimal logic systems are endorsed as a solution to the paradox since the inferences to the contradiction will turn out to be invalid. But as far as I know, no logic system bars syntactic substitution.

Do you think there is a problem with syntactic substitution? Why would it fail in this case specifically, but work in all others? I mean maybe there is something to this, I just don't see it as harmful because I see the inference as entirely analogous to 1-2 here.



I think it does clarify the issue to separate M-type arguments from A-type arguments. I was largely making an A-type argument. That is, I was arguing without using a specific criterion of meaning. This is just to say that I was arguing by referencing speakers' intuitions and was anticipating with the other side may say to argue it is meaningless (i.e. ungrammatical, issue with self-reference). I think that (Li) is intuitively meaningful in this sense. Most competent speakers of English will not react to (Li) as they would to the sentence "the runner runs runningly run running". (Li) is apparently meaningful, and I think we should not give up on giving a semantic value for this sentence without good arguments to establish that its apparent meaningfulness is illusory. It is in this sense that I think the burden is on the one who wishes to argue that the sentence is meaningless.

The Liar paradox shows that we cannot give a standard truth-conditional semantics for (Li) because something goes wrong with our concept of Truth. So the position this person is in is now to find an alternative semantics or fix the concept. I think the move to meaninglessness is not well-established by (Li) alone. I think the latter only establishes that the standard truth-conditional account with the standard concept of truth does not work for this sentence. You are right that there are two roads ahead: either it is meaningful or it is not, but the former position does not have the immediate burden of establishing that (Li) is meaningful, really their burden is to show how we can give a semantics for the sentence (and so explain why it is meaningful) or block the Liar paradox in another way (e.g. restricting our logic). In this sense, I think the claim that it is meaningless is counter-intuitive; the burden is on those who favour the latter view to show why it is meaningless without simply citing the Liar Paradox.

So far, the only arguments to that effect have been to claim that (Li) is analogous to obviously meaningless sentences in English (i.e. ungrammatical sentences). But (Li) is perfectly grammatical, so the analogy is a non-starter.



haha, I heard this song as a teenager too but never realized this.



Perhaps I am not following the thread here, but I think the argument you are marking is that (Li) is analogous to "The true is false". But I don't think it is. This is because the latter sentence is not grammatically correct. Truth is a property (a concept) of sentences, not a noun. But maybe I am missing something?

But you are absolutely right to ask about the way in which (Li) could be false. That, I think, is the point of the Liar Paradox. If our standard truth-conditional account doesn't work, then what do we do? Scharp's response is to say that the problem is in the concept of Truth, and he will eventually replace them with ascending truth and descending truth. (Li) then may be ascending false or descending false. I am not sure yet which one. Let me get back to you on that, but it would require some more discussion of how Scharp replaces the concept. Still, the idea here is that though the task is difficult (i.e. establish the way in which the sentence could be false or true), it does not mean that we cannot do it.

Thanks, everyone for your replies! I wasn't anticipating this many responses so quickly :)

What is the mistake in this inference?



Yes, this is another response to the liar paradox! As you say, maybe the Liar Sentence is meaningless and if that is the case it simply doesn't have truth-conditions and so no harm is done.

Still, I do think the claim that (Li) is meaningless is very counterintuitive. This is not to say that the claim is false, of course, but only that there needs to be stronger arguments for this claim. I dispute the fact that it isn't proper English, as you say. In what sense? It seems grammatical, and its constituent terms are all meaningful. Intuitively, if I say "That sentence is false", referring to some other sentence P, then I know precisely what that means. If I say "This sentence contains five words" this is perfectly meaningful (and true). So the problem is not self-reference. So if the issue isn't grammar, the meaning of constituent terms or self-reference, why think the sentence isn't meaningful? It is not analogous to the other sentences you use. For example "This sentence is run" is, of course, not meaningful because it is not grammatical. (Li) is a well-formed sentence of English.

I can be convinced otherwise, but so far I don't see a reason to accept the counterintuitive position that (Li) is not meaningful.



You are right. I know that Scharp argues against third-value semantics to solve the Liar Paradox because it falls prey to revenge paradoxes, which are essentially just Liar Paradoxes for languages with three+ value semantics. This is not to say that a three-value+ semantics isn't appropriate for natural languages, only that the Liar Paradox can be re-crafted, so we're not out of the woods.



Thanks! I honestly don't know much about this point so I will check out this video.

This works. Thanks so much!

Hey fdrake,

I am actually using the same email address. I suspected the same thing so I checked, but the original activation email from over a year ago was sent to the same address. I tried replying to that to see if someone would see it, but no luck so I posted here by creating a new account.