1. Assume (Li) is true
2. Then '(Li) is not true' is true (substitution). — Kornelius

Nope.

The liars sentence really doesn't work.

If I say, "This sentence is false." What do I mean? That's not even proper English. Its like saying, "This sentence is run" is true. The sentence doesn't convey any truth to begin with so we can't label it true or false. Do something like, "The existence of this sentence is false", and that makes more sense.

The problem isn't truth. Its applying "truth" to something that doesn't make any sense to begin with.

the science of Linguistics uses truth-conditional semantics for natural languages. Since liar sentences can be formed in natural languages, then the linguist must provide a semantics for these sentences (on the assumption they are meaningful). But we cannot give such a semantics for such sentences, despite their being meaningful. — Kornelius

We're skipping a step though, aren't we? Even if we're not going to reach for a separate criterion of meaningfulness (not relying on truth conditions), we have to argue for the meaningfulness of gaps (won't take a truth value) and gluts (takes too many, like the Liar).

proper mass and relativistic mass. — Kornelius

Small point: relativistic mass a convenient fiction; there's only rest mass, here:
https://www.youtube.com/watch?v=LTJauaefTZM

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.

If P is true, then the proposition ‘P is not true’ is NOT true.

Here are the lyrics to a song I loved as a teenager:

https://genius.com/Avril-lavigne-sk8er-boi-lyrics

The lyrics are loopy in the sense that she is singing about a situation ("a song she wrote about a skater boy") but the song she wrote is the song... eh I can't do this. It's just loopy! I too am interested in what we can gather from loopy logic

What makes it a paradox is, as you say, it appears analogous to many other sentences that are just fine, but when we try to assign it a truth value, something goes wrong; we have two ways of deciding whether the Liar is meaningful and they give different answers.

I suggested we need an argument for why we should consider it meaningful. (I was thinking of the analogous efforts in support of treating questions, commands and the like -- over on the 'gap' side' -- as meaningful, despite starting from a model clearly designed for simple indicative statements.)

You suggest we need an argument for why we shouldn't treat it as meaningful.

We have an extra layer here now. There are arguments (M) that the Liar is or isn't meaningful; there are also arguments (A) that the Liar should or shouldn't be assumed to be meaningful, so that a convincing argument is required to overcome this assumption. A convincing M-argument would allow you to ignore the A-arguments, but we already know we're in paradox land and the Liar comes equipped with arguments on both sides. I'm not much moved by the 'apparently meaningful' argument, but I have to admit that many people are, you among them, so there's some reason to think the arguments against meaningfulness aren't that strong, or that the arguments for meaningfulness are just as strong.

So what about the A-arguments? Can we imagine a way to decide whether to assume meaningfulness or not without rehearsing the arguments for and against meaningfulness? What would we need for such a decision?

There's a common denominator between Moore's paradox and the liar. Do you see it too?

It's hard not to see it, but splitting the Liar in half allows you to assign truth values, so you can end up with a sentence that is arguably true but cannot be asserted in the first person in the present tense. (Moore had a nose for the problems indexicals raise.)

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? — Kornelius

If you refer to some other sentence P, then there is the assumption that P can be either true, or false. For it to have that possibility, it must make a claim against reality. If you say, "That sentence is false," and "that" sentence is, "This sentence is false", its still just nonsense.

This sentence is false, does not make any sense. False in what way? Its like me saying, "This true is false". That's a contradiction, and not meaningful in any way.

As a general rule of thumb, if you run into a contradiction, it means you're doing something illogical. I think its easy enough to see that such a sentence is illogical. But feel free to show me otherwise. How can you make such a sentence have actual meaning? How is it not simply saying, "This true is false"?

I was thinking more along the lines of what one would say if they knowingly believed a falsehood, which of course cannot happen.

If you refer to some other sentence P, then there is the assumption that P can be either true, or false. For it to have that possibility, it must make a claim against reality. If you say, "That sentence is false," and "that" sentence is, "This sentence is false", its still just nonsense. — Philosophim

Why not? You are just restating your position, but you are not giving reasons for it. What the liar sentence claims is the truth value of a sentence, which all natural and many formal languages are equipped to do.

This sentence is false, does not make any sense. False in what way? — Philosophim

You can state, seemingly unproblematically, "X is false" for any number of X, including X that are sentences of a language. Why does it not make sense in this case? Again, I understand why you want to reach this conclusion, but you are not giving any reasons.

I'm probably not qualified to comment but the liar paradox looks like this to me:

L = Liar sentence = This sentence is false

P = L is a proposition (i.e. it has a truth value)

1. If P Then either L Or ~L....premise
2. If L Then ~L........................the liar
3. If ~L Then L........................the liar
4. P..................................assume for reductio ad absurdum
5. L Or ~L................................1, 4 MP
6. L..................................assume for reductio ad absurdum
7. ~L........................................2, 6 MP
8. L & ~L..................................6, 7 Conj
9. ~L........................................6 to 8 reductio ad absurdum
10. ~L.............................assume for reductio ad absurdum
11. L........................................3, 10 MP
12. L & ~L..............................10, 11 Conj
13. L.....................................10 to 12 reductio ad absurdum
14. L & ~L..............................9, 13 Conj
15. ~P....................................4 to 14 reductio ad absurdum

The Liar Sentence isn't a proposition. The concept of truth, whatever it is, is not part of this argument and so, needn't be amended as Kevin Scharp recommends.

What you have here is in the form of a proof but of course it is not a proof because we're not talking about a formal system -- Tarski told us long ago that you just don't include predicates like '... is true' and '... is a proposition' in your language and you're fine. If you want them, they have to be in the next language up, the meta- to this object. What to do when it comes to natural languages -- there's the rub.

And while I sympathize with your approach, what happens with this sentence?

(L') 'This is not a proposition.'

If it's true, then it can't be true because it can't take a truth value. If it's false, then it can take a truth value and so assigning it "false" in all models is fine. So it looks like the predicate '... is a proposition' should be fine and L' is necessarily false. Cool.

But then what about this one?

(L'') 'This sentence is not true or is not a proposition.'

If it's true, it's either not true or it's not a proposition, so it's not a proposition, so it can't be true. If it's not true, it's true and a proposition.

This is the revenge paradox, and you could do the same thing with '... is meaningful' in my post above. In a sense, even formalizing truth the minimum amount that Scharp describes blocks you from formalizing other semantic predicates like '... is meaningful' or '... is a proposition' on pain of revenge. That would leave you with a formalized truth predicate you can't actually use in a formalized way. On the other hand, if you decide to give up on bivalence and roll the other semantic predicate you want into "truth", so you have three values, then you just get revenge immediately. This is the point @Kornelius is making here.

The Liar always argues its way out of any argumentative box you try to put it in. To deal with it once and for all you have to give up on formalizing your semantics even a little, formalize it differently from the way we have so far (and there are always such proposals around, some of which are nice), or protect the old semantics from having to deal with it in some other way. (Whether this last is even an option is unclear, and it's the approach I was suggesting be explored, just for funsies.)

What is the mistake in this inference? — Kornelius

If P is true, then the proposition ‘P is not true’ is NOT true.
If P = NotP (the liar sentence) then you go into a contradiction, because you postulate a contradiction to start with.

It is fundamentally the same as saying: « Let’s see what happens if we postulate that 1 is different from 1... Oh my god, arithmetics as we know them break down! Therefore arithmetics need to be replace by something else. »

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 :)

I believe you are reading too much into the inference. — Kornelius

No, I am saying: you are starting from an obvious contradiction. Li = not Li. It’s like basing arithmetics on 1=2...

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).

Li is defined as « Li is not true » which is equivalent to « not Li is true », itself equivalent to « not Li ». Ergo you defined Li as equal to not Li, an obvious contradiction.

Why should a natural language be consistent?

Natural languages include nonsense. That seems to me to be a good thing. We can formulate the liar paradox, talk about the little man who wasn't there, describe the colour of magic, even state that this piece of bread is the body of god...

Around the edge of the Map of Language, instead of "Here be Dragons", write "Here be nonsense".

I don't have Scharp's book, so I read some reviews. If he can develop a more formal schema by breaking the T-sentence into two sentences moving in opposite directions, we have an interesting formula that might show us something of use. But I'd put money on someone showing how this results in yet another bit of nonsense. Scharp has extended the map a little bit, but the dragons are still there.

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.

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". — Kornelius

I'm not so sure. 'Whether an ideal speaker would consider P meaningful' looks to me like a separate criterion of meaningfulness -- that is, not just asking whether P is truth-apt.

Also, '... is meaningful' feels like a weasel-predicate. That is, '... is meaningful' deliberately avoids asking, for instance, 'What does it mean?' or 'What does it say?' Ask an average person about the Liar, and you can expect them to reply, 'Well that's stupid. It doesn't say anything.' (This applies to 'This sentence is true' as well!) Surely if you find a sentence meaningful, you could say what it means. There hasn't been 2000 years of debate over what the Liar means, what it says, because it evidently fails to quite mean anything.

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. (If it's a question or a command, say, so you fail to assign a value, then we need convincing arguments to bring them in anyway. For the 'always takes both values' case of the Liar, I can't imagine what a convincing argument would be -- we do generally prefer our sentences to take at most one truth value at a time -- but if there is one, then this approach fails anyway and we're back to the meaningfulness arguments.)

My immediate thought is that it does not follow from anything you wrote that truth is an inconsistent concept. In fact, I would - and have quite successfully - argued that truth is not a concept at all. What does follow from what you've shared is that that particular accounting practice cannot take proper account of correspondence.

We can say no to that one, even if we accept that there is a lot of nonsense around. — Kornelius

You may have missed my point. So I'll rephrase:

Natural languages curls over on itself. That seems to me to be a good thing. We can formulate the liar paradox, talk about the little man who wasn't there, describe the colour of magic.

Around the edge of the Map of Language, write of "Here be Dragons" to mark the places where the linguistic topology becomes... uncanny. Indeed, it becomes so weird that even as you work out what is going on, it changes again.

A logician saves the life of a tiny space alien. The alien is very grateful and, since she's omniscient, offers the following reward: she offers to answer any question the logician might pose. Without too much thought (after all, he's a logician), he asks: "What is the best question to ask and what is the correct answer to that question?" The tiny alien pauses. Finally she replies, "The best question is the one you just asked; and the correct answer is the one I gave."

the linguistic topology becomes... uncanny. — Banno

The technical term you were searching for is "vague".

No. I chose quite deliberately. Uncanny has the link to knowing that I wanted.

What you have here is in the form of a proof but of course it is not a proof because we're not talking about a formal system -- Tarski told us long ago that you just don't include predicates like '... is true' and '... is a proposition' in your language and you're fine. If you want them, they have to be in the next language up, the meta- to this object. What to do when it comes to natural languages -- there's the rub.

And while I sympathize with your approach, what happens with this sentence?

(L') 'This is not a proposition.'

If it's true, then it can't be true because it can't take a truth value. If it's false, then it can take a truth value and so assigning it "false" in all models is fine. So it looks like the predicate '... is a proposition' should be fine and L' is necessarily false. Cool.

But then what about this one?

(L'') 'This sentence is not true or is not a proposition.'

If it's true, it's either not true or it's not a proposition, so it's not a proposition, so it can't be true. If it's not true, it's true and a proposition.

This is the revenge paradox, and you could do the same thing with '... is meaningful' in my post above. In a sense, even formalizing truth the minimum amount that Scharp describes blocks you from formalizing other semantic predicates like '... is meaningful' or '... is a proposition' on pain of revenge. That would leave you with a formalized truth predicate you can't actually use in a formalized way. On the other hand, if you decide to give up on bivalence and roll the other semantic predicate you want into "truth", so you have three values, then you just get revenge immediately. This is the point Kornelius is making here.

The Liar always argues its way out of any argumentative box you try to put it in. To deal with it once and for all you have to give up on formalizing your semantics even a little, formalize it differently from the way we have so far (and there are always such proposals around, some of which are nice), or protect the old semantics from having to deal with it in some other way. (Whether this last is even an option is unclear, and it's the approach I was suggesting be explored, just for funsies.) — Srap Tasmaner

But there's no need for anything meta, right?

L = This sentence is false (the liar sentence)

P = L is a proposition

To tell you the truth, I fielded P because 1. it is an assumption made and 2. avoids self-referential gobbeldygook.

But there's no need for anything meta, right? — TheMadFool

In a natural language, even in philosophical English, the object language and the meta-language are one. I do not understand your point.