Uncanny has the link to knowing that I wanted. — Banno

Vagueness is better as it speaks to what we don't know. And even that to which may be unknowable or undifferentiated.

Uncanny just means strange and unsettling. Are you thinking it is a term logicians use?

You have at least two inferences here. — Kornelius

These are equivalences, not inferences. They are just a different way to say the same thing. To say « P » or to say « P is true » is to say the same thing.

To take a trivial example, « I like popcorn » and « truly, I like popcorn » are saying the same thing. There are no inferences between one and the other. Seems to me this Kevin Scharp you are reading is not very sharp.

I do not understand your point — Srap Tasmaner

My bad. It's probably because I don't get you.

Anyway, here's what I think: To believe that we can assign a truth value to the liar sentence, it must be the case that we've made the assumption that it's a proposition. Thus:

L = This sentence is false (the liar sentence)

P = L is a proposition

Since I don't see any problems in the argument I presented, it follows that your objection must revolve around L and P. Can you proceed from there and please keep it simple for my sake. Thanks.

To the eager child everything is true, until it's not. To the aged skeptic everything is false, until it's not. Both can believe and share mistruths without ever once telling a lie. Ironic how much the two have in common.

No, I definitely did not mean vague.

(1) If P is true then P (T-out)
(2) If P then P is true (T-in, also known as semantic ascent). — Kornelius

How does Scharp use these to solve the paradox?

The liar paradox was discussed in logic during the Middle Ages. They hated it lol

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.' — Srap Tasmaner

I don't think so. On the surface, the meaning is clear: First, the sentence asserts that something (presumably a proposition) is false. That's not a problem, we generally understand such assertions. The proposition, rather than being quoted, is instead indicated - also not necessarily a problem. For example, "The second sentence on page 23 is false" would be a perfectly meaningful thing to say (but what if the sentence indicated happens to be that very sentence?) It is only once we trace the logical implications that we realize that something is wrong, but in order to be able to do that, didn't we have to first understand what the sentence is saying?

Since I don't see any problems in the argument I presented, it follows that your objection must revolve around L and P. — TheMadFool

It's not just my objection; it's the revenge pattern: if you hope to avoid the Liar by using some other semantic predicate also available in your language, we'll just construct a new Liar-like sentence that is just as bad as the original.

'... is a proposition' is such a predicate: you absolutely can use it to block the Liar, just as you did, but it's no help with the revenge sentence, 'This sentence is not true or is not a proposition.' You might as well not have bothered.

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. — Srap Tasmaner

But my sentences are not as convoluted as the above.

L = This sentence is false (liar sentence)

P = L is a proposition

Nothing can done to the liar sentence so that's that but there's no issue, as far as I can tell, with P - it looks perfectly ok.

What to do when it comes to natural languages -- there's the rub. — Srap Tasmaner

Thoughts? — Kornelius

1. Assume (Li) is true
2. Then '(Li) is not true' is true (substitution).
3. Then (Li) is not true (T-out)
4. Then (Li) is true and (Li) is not true (from 1 and 3. Contradiction)
5. Then (Li) is not true (reductio 1-4)
6. Then '(Li) is not true' is true (T-in)
7. Then (Li) is true (substitution)
8. Then (Li) is true and (Li) is not true (from 5 and 7. Contradiction). — Kornelius

What blocks:

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

?

Why can't it just be false?

It is only once we trace the logical implications that we realize that something is wrong, but in order to be able to do that, didn't we have to first understand what the sentence is saying? — SophistiCat

On the one hand, sure, you can imagine having to explain to someone who doesn't get it what the problem here is, and that might feel like guiding them through a formal deduction.

On the other hand, the Liar is introduced as a premise, but then behaves like an inference rule. ('Given me, you may introduce not-me.') You have to understand the inference rule to use it, that is true. But that's understanding-how, not understanding-that. Inference rules are deliberately empty, have no 'that' content. They don't say anything themselves; it's premises that actually say stuff.

When we trace the logical implications of the Liar, we stop at contradiction, as @Kornelius did, because that's one of our things, a proof-doing practice. But I've also demonstrated it to people by starting at true and then saying, 'But if it's true, then it's false, but if it's false then it's true, but if it's true ...' and doing the same thing starting with false. You never 'finish' so you never get to a point where you've figured out what it says. The completion of the statement is always deferred.

I want to follow this but I can't.

I want to follow this but I can't. — Srap Tasmaner

Question motivated by the following insight: for everything else which is both True and False, the conjunction of True and False is simply False. Like "tautology AND contradiction" evaluates to "contradiction".

I'm pretty sure I'm being naive in some way, since evaluating the Liar as simply false is both an obvious approach and undermines a whole literature, but I'm sufficiently lazy that I'm hoping someone in thread can tell me precisely why I'm wrong.

On the other hand, the Liar is introduced as a premise, but then behaves like an inference rule. ('Given me, you may introduce not-me.') You have to understand the inference rule to use it, that is true. But that's understanding-how, not understanding-that. Inference rules are deliberately empty, have no 'that' content. They don't say anything themselves; it's premises that actually say stuff. — Srap Tasmaner

Well, exactly, there is nothing wrong with the syntax of the sentence, it is only its semantic content that seems to be a problem. But in order to come to this conclusion you have to parse and interpret the sentence, which means that the sentence is ipso facto meaningful, at least on some level.

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?

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. — Kornelius

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

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".

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

I almost gave 'Its syntax is fine' as an example of a rule for assuming a sentence is meaningful! But--

That's kinda like saying the semantic problems of a sentence are semantic problems.

Worse: doesn't this amount to saying that if a sentence has even a very severe semantic problem, such as being meaningless even if its syntax is fine, then you only find this out by looking at its semantics, and therefore it's meaningful?

But that's too general. You're right that we have to make enough sense of the Liar to understand that it empowers us to use it to infer 'I'm false' from 'I'm true' and vice versa. I can't dispute that.

But, as I tried to express above, that's pretty weird. If you think of the Liar as an introduction rule, then its semantic content is, in one sense anyway, syntactic. And that's all it has. But that's not supposed to happen. It's why we distinguish them.

I do think you are right in pressing against the view I was arguing here, namely that we should assume it is meaningful. — Kornelius

I'm not sure it amounts to anything.

@SophistiCat and I have ended up having a discussion that's hard to distinguish from an argument about whether the Liar is meaningful. And there's just no point to that, because of revenge.

Even the other idea I had in mind amounts to the same thing. (If you ask what 3+5 is, there's a single right answer and an infinite number of wrong answers. This kind of thing shows up in a few places and there's a reason I'll bet @fdrake knows.) In essence it just amounts to having a designated value, and again we know this won't help because of revenge.

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. This is a reason we need an alternative to the concept. — Kornelius

I would argue that nothing needs replacing, but an extension is required. A missing element of meaningful assertion has to be recognised.

The liar paradox arises because of self reference. The assertion lacks an object as it only involves the subject itself. There is no epistemic cut - a division in which a claim is being made about something and so the relationship is a semiotic or formal sign-based one.

A way to illustrate this is imagining a sheet of paper half red and half green. What colour is the boundary then separating the two halves? Is it red or is it green? Both or neither?

Logic collapses, the LEM can’t apply, because the boundary is not some third thing - a separate position from which the two surfaces could be described. The boundary can’t treat either as object, yet seems to treat both as object, while being in fact always subject in being the limiting part of both/either.

So an epistemic cut is a necessity. Semantics demands a world divided into subject and object. Then logic can work.

(Of course there is then the issue of whether the world actually “has” such divisions in and of themselves. Or whether the epistemic cut is something “we” add so as to render the world manageable by a logical calculus.)

Let me get back to you on this one. — Kornelius

It would be interesting to compare it to Kripke's truth, which renders the liar paradox neutral by making it underivable.

But maybe I'm not following your suggestion? — Kornelius

Nah I think you were following it fine. It's simply a case of my suggestion not being very good!

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

Hi Tim, Interesting comment. I watched that video. I had always thought that very fast particles increased in gravitational mass. But apparently not so. As you seem to know a bit about this, can I ask a question that's a bit off the subject of the OP?

I had assumed that, if you lift a weight against the force of gravity, its gravitational potential energy increases by mgh and therefore its mass increases by mgh/c^2. Is that the case? Whatever provided the energy would have a corresponding decrease in mass to compensate.

Likewise, if you accelerate a particle, you add energy, 0.5mv^2. Does its mass increase by 0.5mv^2/c^2?

Flattery will get you nowhere, alas. I have no idea. But the Fermilab videos with Don Lincoln seem to have the mark of truth to them. I'd venture a guess, but I know the subject is too tricky for guesses. What I get, though, is that the mass does not change, and is always the rest mass. I think he offer links to other videos that expand on this.

↪Neb Flattery will get you nowhere, alas. I have no idea. But the Fermilab videos with Don Lincoln seem to have the mark of truth to them. I'd venture a guess, but I know the subject is too tricky for guesses. What I get, though, is that the mass does not change, and is always the rest mass. I think he offer lins to other videos that expand on this. — tim wood

Thanks Tim. I did look at some of his other videos. Being from Fermilabs, I tend to think he's quite possibly right. I'm on a physics forum. I might ask the question there.

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? — Kornelius

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.

It's simply a case of my suggestion not being very good! — fdrake

Disagree.


* It's not uncommon to just define $\neg P$ as $P\to\bot$. Even in systems that don't, this is an introduction rule for $\neg$.

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?

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. — Kornelius

Sorry -- knew that would be confusing. It's valid because it doesn't matter that (Li) is part of the expression (Li) & ~(Li); what matters is that that's a contradiction and it's false, and therefore one of the premises from which we derived it is false. Since there was only one premise...

Here's your original presentation:

1. Assume (Li) is true
2. Then '(Li) is not true' is true (substitution).
3. Then (Li) is not true (T-out)
4. Then (Li) is true and (Li) is not true (from 1 and 3. Contradiction)
5. Then (Li) is not true (reductio 1-4) — Kornelius

What I was pointing out is just step (5) in your (or Scharp's) presentation. The inference rule I referred to is just the reductio rule.

What we really want is for steps 5-8 to be a separate argument, using [ T-in ] instead of [ T-out ] and showing that ~(Li) is also false. By continuing on past the reductio, we just rederive the premise in step 1, rederive step 4 as step 8, all of which still only gets you ~(Li) because it's the only premise.

5. Then (Li) is not true (reductio 1-4)
6. Then '(Li) is not true' is true (T-in)
7. Then (Li) is true (substitution)
8. Then (Li) is true and (Li) is not true (from 5 and 7. Contradiction). — Kornelius

This presentation only shows, with some repetition, that the Liar is false. (If the idea was to sweep up intuitionist logic along with classical logic, by avoiding appeals to the LEM and to double negation elimination, it does not appear to me successful.)

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.