Yes, see the first reply to this discussion. ;)

Lol, not sure how I missed that. But that aside, as I said, even Kripke said that his proposed solution was probably suspectible to a Revenge Paradox, such as the ones I listed previously.

Yes, it's true if it's false. But what does it mean for it to be true? Are you saying that it being true means that it's false (and vice versa)? So in the context of the liar paradox, "true" and "false" mean the same thing? If so then a) there is no contradiction and b) the terms "true" and "false" in the context of the liar paradox mean something other than what they mean in ordinary usage. And then you still need to explain what it actually means for the sentence to be true/false.

To say that it's true merely means that the proposition is related to (in the mathematical sense) the value "true", and to say it's false simply means it is related to the value "false". If you're asking for a theory of truth, that is a discussion independent of logical formalisms. Formal logic is, generally speaking, neutral as to the meaning of those predicates (that's why there are a number of theories of truth). Or you could just check out Tarski's work on the matter.

No, in the paradox, "truth" and "falsity" do not mean the same thing (otherwise the formalism would simply be a trivial system). As in the standard usage, truth and falsity are duals. A proposition is deemed "true" when it's "not false", and vice-versa. If we're working in, say, Classical Logic, all propositions must have a truth-value they relate to, and that value must be either "truth" or "falsity". So unless you can show specifically how the LPs aren't propositions, there's really nowhere to go. You either have to reject Classical Logic or just accept the paradox.

If that Liar Sentence is true, then it follows that it is also false — MindForged

Not necessarily. All we can say is 'it feels as though it ought to follow that it is false'. In order to convert that into an unqualified statement like 'it follows that it is false', we need to translate the statement into a formal logical proposition, since we can only make definite statements about those. But it's in the attempt to make that translation that we hit obstacles.

In other words, we can't discuss whether certain solutions to the 'problem' are valid until we have identified a problem. If the only problem is that the sentence feels unintuitive, and the things one feels like one ought to be able to deduce from it feel as though they would contradict one another, then that's not a problem of psychology, not of symbolic logic.

The evidence is the straightforward proof by contradiction. That the Liar sentence is not truth-apt is a readily established fact. Now, you may wonder what makes it so, but that's a different — SophistiCat

You didn't provide evidence. Very few people deny that the Liar Paradoxes are contradictions. The point is that, prima facie, they appear to dialetheia - a proposition which is both true & false. You haven't established that the Liars aren't truth-apt. The fact that the Liar Paradox is an unsolved problem with no standard solution speaks to the fact that you are getting way ahead of yourself.

What does the proof by contradiction have to do with anything here? If your argument is that the truth of the Liars would result in trivialism, I agree. But that makes the assumption that the Principle of Explosion should be accepted as a valid argument (Paraconsistent Logics give us another route). So this is far from a simple matter, which is the reason why it's unsolved amongst logicians.

I'm not really sure what you're talking about. Analysis of the Liar sentences, even by strictly Classical Logicians (like Timothy Williamson) use logical formalisms to provide a rigorous assessment of the Liars. The usual response by these people is not that there is a problem with the reasoning involved. After all, in standard logic, Falsity just the truth of negation (and vice-versa).
So yes, if the Liar is false, it is also true. So a better solution would be to question if it's a proper proposition at all.

It sounds like you think there is a representation of the sentence in a formal logical language. I would be very surprised to see such a thing, and suspect that it either lacks in formality or doesn't sufficiently represent what people feel the natural language version says.

Post the representation, with details of the formal language being used, and we can discuss it.

It sounds like you think there is a representation of the sentence in a formal logical language. I would be very surprised to see such a thing, and suspect that it either lacks in formality or doesn't sufficiently represent what people feel the natural language version says.

Post the representation, with details of the formal language being used, and we can discuss it.

I'm about to leave for a New Years party, so I don't have the time to get into the nitty-gritty. However, Graham Priest gives a fuller, comprehensive look at the paradox (including formal representations of it) in his book "In Contradiction" (and probably elsewhere too).

To say that it's true merely means that the proposition is related to (in the mathematical sense) the value "true", and to say it's false simply means it is related to the value "false". — MindForged

And how does a proposition come to be related to either the value "true" or "false"?

If you're asking for a theory of truth, that is a discussion independent of logical formalisms. Formal logic is, generally speaking, neutral as to the meaning of those predicates (that's why there are a number of theories of truth).

Understanding what it means to be true or false is necessary to resolve the liar paradox. The point I am making is that the meaning of "true" and "false" are such that it isn't meaningful to predicate them of the liar sentence.

But are we talking about the liar sentence as a sentence in a natural language or are we talking about something else? If the latter then you need to explain what the terms mean in this formal system, else I don't even know what we're talking about.

If we're working in, say, Classical Logic, all propositions must have a truth-value they relate to, and that value must be either "truth" or "falsity". So unless you can show specifically how the LPs aren't propositions, there's really nowhere to go. You either have to reject Classical Logic or just accept the paradox.

I did that in the first reply to this discussion. If the liar sentence being either true or false leads to a contradiction then either the liar sentence isn't a proposition or not all propositions are truth-apt. It's a simple proof by contradiction.

I know that Graham Priest is interested in the problem. He discussed it with some of us on the old forum. Although I think there are a number of great philosophical contributions that Priest makes, I think he is mistaken about the Liar Paradox.

Just post the representation and language definition tomorrow, and we can discuss that. References to books are the second quickest way to kill a thread around here (surpassed only by references to videos).

If the only problem is that the sentence feels unintuitive, and the things one feels like one ought to be able to deduce from it feel as though they would contradict one another, then that's not a problem of psychology, not of symbolic logic. — andrewk

I don't understand what feelings have to do with this. The liar's paradox presents a logical contradiction. I can't always be lying if I'm telling the truth, so that's clearly false, but if it's false, then I'm telling the truth, but then if it's true, I'm lying so ....

How can it prima facie not be a contradiction?

It's not a contradiction unless one can set the contradiction out as a deduction in a formal logical language. And my point is that, in the process of trying to do that, one encounters obstacles that lead one to realise that there is no contradiction there. There's a bunch of nuances to this that I've left out for the sake of brevity. But we cannot even begin to consider them until somebody makes the attempt to formalise the sentence.

A bunch of natural language words does not a contradiction make, no matter how much it may feel as though they do.

It's not a contradiction unless one can set the contradiction out as a deduction in a formal logical language. — andrewk

Eh? This isn't true. There's contradiction in natural language. You might have a way to formalize it, but the contradiction exists without the formalization.

A bunch of natural language words does not a contradiction make, no matter how much it may feel as though they do. — andrewk

We use the word contradiction in natural language. That politician is stating contradictory things, or you're contradicting yourself, etc. This is the first time I've heard that contradiction is only a term applicable to formal logic. The dictionary and everyday use of the word would lead one to believe otherwise. I can make a contradictory statement in natural language quite easily. The unicorn is both pink and invisible.

I agree, we do use it, and I should have included the word 'necessarily' in that sentence you quoted ('does not necessarily a contradiction make ....'). But my understanding of what it means for a natural language sentence to contain a contradiction is that it is equivalent* to a formal sentence that contains one.

I have never seen any other definition of contradiction that is sufficiently objective to enable one to determine in all cases whether the definition is satisfied.

* 'equivalent' in the sense that the user of the sentence would not object to the translation as inaccurate, if the meaning of the translated sentence were explained to them.

For p = this statement is false to be considered, isn't there an implicit assumption that p is true?
Due to the self-reference (this statement), it is also the case that p = p is false.
So, we have both p is true and p is false, which is on the form of an ordinary contradiction, p ∧ ¬p.

"I am lying" is a self-referential sentence which leads to a rather problematic situation. If the statement is true, then it is false. If it is true, then it is false. But instead of using "I am lying" as the problem statement, we should use its strengthened version which is "This sentence is not true."

Where does its complications lies? Is it about truth? about reference? or a problem as a proposition? What do you think? — Jaydison

Hadn't paid attention to this thread before. Anyway, in my view the problem is simply that "I am lying" and "This sentence is false" don't actually say anything. They function similarly to transitive verbs without objects.

"I am lying"--well, about what? There would have to be something substantive that the utterer is calling into question, something else they said.

With "This sentence is false," it's no different than if we were to say "It's true that the cat is on the mat" or "It's false that the cat is on the mat." In other words, we're making explicit the truth-value assignment by incorporating it into the proposition in question. When we realize this, however, it becomes clear that there's no proposition that we're assigning a truth-value to. We're saying "This sentence" has a truth-value. But "This sentence" isn't a proposition, it's not substantive.

Maybe there's a way to make a self-referential, paradoxical proposition that actually says something, that's substantive, but offhand I can't think of one.

They function similarly to transitive verbs without objects. — Terrapin Station

I like that. It's a very neat analogy.

When I'm false I'm true
When I'm true I'm false
Oh what should I do?
Ride a camel? Ride a horse?

I think Jaydison exposed the problem on the first go around of the thread. It is the issue of self-referencing. Self-referencing usually takes the form of definition, but here it does not. So it's like saying "this definition is not a definition", which is contradictory nonsense. We could make other nonsense statements like "this chair is not a chair", or "this table is not a table". That is the form of this sentence.

Consider "this sentence is true". That doesn't say anything, as TS says, there's no substance. It's like saying 'this sentence is a sentence", "this chair is a chair", etc..

In my humble opinion the liar's paradox is a problem when we combine two parameters...

1. Self reference
And
2. Binary logic

If we refrain from any or both the paradox disappears.

"this chair is not a chair", or "this table is not a table" — Metaphysician Undercover

But we do say things like that on occasion. For example, "This party is not a party", meaning it's a party in name only. I'm pretty sure I have said something akin to "this chair is not a chair" when being forced to sit on something uncomfortable that served as a chair. I've also said, "I'm not myself today", which would seem to be a violation of the law of identity, but clearly it's not meant to be taken in literal terms.

Consider "this sentence is true". That doesn't say anything, as TS says, there's no substance. It's like saying 'this sentence is a sentence", "this chair is a chair", etc.. — Metaphysician Undercover

SEP's counter to this is the sentence, "This sentence is not in Italian", which is not meaningless, but is in the same form as the liar sentence. And in ordinary language, people sometimes do say things like, "Now this car is a car!"

Also, there is this very big counter to the claim that the liar sentence is without meaning:

"This sentence is meaningless."

Which would be true if the liar sentence is meaningless, but then we get ourselves into another regress.

Post the representation, with details of the formal language being used, and we can discuss it. — andrewk

You requested that of MindForge, but since he hasn't gotten back to you yet, what do you think of the argument that SEP presents?

L = "This sentence is false" or "I am lying"
Q = "1 + 1 = 3" or any other false sentence

1. L and not-L from the Liar Paradox
2. L from 1
3. L or Q from 2 using the Law of Addition
4. not-L from 1
5. Q from 3 and 4

If we refrain from any or both the paradox disappears. — TheMadFool

Well, here's a clever way to remove the self reference and still end up with the liar's paradox:

Socrates: "What Plato is saying is false"
Plato: "What Socrates is saying is true"

Well done!

However you are still using binary logic in your clever example

Well, here's a clever way to remove the self reference and still end up with the liar's paradox:

Socrates: "What Plato is saying is false"
Plato: "What Socrates is saying is true" — Marchesk

I'd say they're self-referential via proxy.

Kripke's solution addresses these examples. There must be some evaluable fact about the world for the statement(s) to be "grounded", and so have a truth-value, but there isn't such a thing for the above.

There must be some evaluable fact about the world for the statement(s) to be "grounded", and so have a truth-value, but there isn't such a thing for the above. — Michael

Math and logic aren't grounded by the world. Also, there's fictional truths, such as Harry Potter performs magic.

However you are still using binary logic in your clever example — TheMadFool

Here's an even better one that dates back to ancient times:

A crocodile takes a child but promise the parent he will return the child if the parent guesses what the crocodile will do. The parent responds that the crocodile will not give the child back.

What does the crocodile do in response? How would the crocodile get around this using an alternate form of logic?

This one is grounded, btw (other than the fact that crocodiles don't talk, but you can substitute a human kidnapper.)

Star Trek version:

KIRK: "Everything Harry tells you is a lie. Remember that! Everything Harry tells you is a lie!"

HARRY: "Now listen to this carefully, Norman: I AM LYING!"

NORMAN: "You say you are lying, but if everything you say is a lie then you are telling the truth, but you cannot tell the truth because everything you say is a lie, but... you lie, you tell the truth, but you cannot for you l... Illogical! Illogical! Please explain! You are Human! Only Humans can explain their behavior! Please explain!"

I've never really understood the problem of the liar paradox, even after reading a little about it. Think of checking for truth like running a program, and a program can loop infinitely without output. Same with self-referential paradoxes. — The Great Whatever

From finally getting curious enough to read up a bit on it, seems the motivation is to be able to define a theory of truth free of contradiction. Wittgenstein was of a different opinion. He thought it better to ignore the liar paradox instead of trying to figure out a way to resolve it, which in his view, might have worse consequences. I found that interesting.

Math and logic aren't grounded by the world. Also, there's fictional truths, such as Harry Potter performs magic. — Marchesk

The evaluable facts for truth-claims about Harry Potter are the words written in the books or the statements made by J.K. Rowling.

With math and logic it's a matter of using the axioms and the rules of inference to determine what follows from what.

There's nothing like either of that for liar-like statements.