The complication lies in the assumption that the sentence must be truth-apt. But given that it being true or it being false would lead to a contradiction, and given that the law of non-contradiction is an axiom, it follows that this assumption must be wrong. That's how a proof by contradiction works. For some reason, however, many refuse to give up the initial assumption and so proclaim it a paradox.

I'd side with Kripke's answer to this. There must be some evaluable fact that grounds a statement for it to be either true or false, and liar-like statements have nothing of this sort.

I am quite new to this paradox and as I see it, It is well stated proposition that is clear and so on. It is maybe the case that this proposition or rather, this statement lacks the ground/s for it to be either true or false. But in the case of most or all of propositions, I think, that their mechanisms are very similar to models of mathematics or so, they are the same sort. They are models of reality as we imagine it (TLP 4.01), and so does not necessarily rely on fact/s.

It is maybe the case that statements should have grounds in order to assert whether that statement is true or false. But also, it is not wrong to assume its truth and falsity.

From what I see, as I have stated above, is it is a proposition thus it can be either true or false.It is also the case that it is truth-apt. But by looking at its components:

"This sentence is not true" is *true *assumed

Could we really put two truth-predicate in one proposition?

As I see it: "This sentence" is the subj, "is not true" is the predicate, and " is true" is the truth predicate.

Is it not that the predicate of the proposition counts as a truth predicate? If the predicate is not a truth predicate, then why do it function as a truth predicate?

Clarify me if I am wrong on some of my points

It is well stated proposition — Jaydison

Under my analysis, that's where it falls apart. It is a sentence, but not a proposition. Propositions have truth values. Sentences only have truth values if they can be translated into propositions.

But that sentence is incapable of being translated into a proposition, because the attempt to formalise the subject 'this sentence' (ie to express it in symbolic logic) generates an infinite regress.

If a string of words cannot be translated into symbolic logic, it is not a proposition, regardless of how grammatical it may be, or how reasonable it may sound.

It is maybe the case that statements should have grounds in order to assert whether that statement is true or false. But also, it is not wrong to assume its truth and falsity. — Jaydison

Given that it being true or false leads to a contradiction it must be that it is wrong to assume its truth or falsity. That's how a proof by contradiction works.

Of course, you can always take the dialetheist's approach and reject the law of non-contradiction...

Thank you for clarifying my error. It seems that I am under illusion that "This sentence is not true" is a proposition as it already contains a truth-predicate (Its seems a bit weird and wrong. It is also weird to me, but I am trying to come of a better explanation)

The sentence contains a subj and pred. One thing that bothers me is the pred. "is not true". That must be a truth-predicate as it function as one.

Maybe there is a way to re-state this sentence...

Is the sentence ""This sentence is false" is true" could be restated as "This false sentence is true"?

On that quoted quote, It seems that I have not clarified that I am talking to statements in general and I do apologize.

On the other hand, it is wrong to put a truth value, or even assume, on a statement that already contain a truth-predicate. (I have stated ideas above that would bring light to what I am saying)

It is very difficult for me to reject the law of non-contradiction as it appeals to my intuition. There must be another way. If it is the case that dialetheism is the way to approach this kinds of sentences, I ll see into it that it would be etched into my mind the right way.

I do hope that you (refer to all) will be patient and understanding as I am a person who still lacks.

Also, even if I am right that "is false" is a truth-predicate and "This sentence" is stated rather differently. There is still this problem of self reference and lack of property.

The sentence contains a subj and pred. One thing that bothers me is the pred. "is not true". That must be a truth-predicate as it function as one — Jaydison

The biggest problem is not with the predicate, it is with the subject - 'this sentence'. The problem is that, when one tries to formally state the sentence, the predicate expands recursively without limit. It's like the delightful joke that, when fractal pioneer mathematician Benoit B Mandelbrot was asked what the middle initial 'B' in his name stood for, he replied 'Benoit B Mandelbrot'.

'I am lying' is something I've heard someone say. They smiled wryly. How can we deal with people who utter paradoxes yet succceed in making sense? It's so annnoying of them, as if there were something about language beyond logic.

The way I see it is that the sentence can't be true or false because, as others mentioned, it doesn't qualify as a proposition.

It is similar to the way that if I asked with, no context, whether the following statement was true or false, neither would make sense, but that hardly means it is a paradox:

"The dog jumped over the cat."

It is not a sensible question to ask because it is not a statement that can be (given the lack of context) either true or false. Likewise, "This statement is not true", which just happens to sound an awful lot like a proposition, simply isn't, by the very fact that asking whether it's true is a nonsensical question.

In my experience, when people say "I am lying" they mean something more like "What I just said was a lie". I don't think anyone says that language can be reduced to logic, but we still have to deal with the paradoxes that come up when we take a part of language and formalize it.

The Liar's Paradox carries an unusual amount of weight in philosophy of language. Even people who aren't talking about it explicitly feel obliged to set it aside in a footnote, like 'I realize this truth schema doesn't solve the liar's paradox.'

It has always seemed pretty straightforward to me. If you think of truth conditions as a kind of program that can be run to return a (truth) value, then the liar sentence will cause a crash because it will loop indefinitely trying to establish those conditions, and you'll get no output. This seems to be predicted by any ordinary semantics and to be exactly the right empirical prediction, so I've never seen what the problem was.

ask the liar if they are an elephant

I do agree. There is a complication alone on its subject. By looking at the earlier form of that paradox " I am lying". It refers to itself, thus becoming problematic.

One way of elucidating this type of self-referring sentence such as "What is this?" is by employing the context principle. On "I am lying", this type of sentences should not refer to itself for it to make sense.

"I am a female" > " I am lying" > " I am a male"
*a picture of a cat > "What is this?" > "A cat"

Thus, in statements such as "This sentence" it should refer to others and not itself. Does that satisfy it at all or not?

Attempting to clarify the ideas presented above:

I see that the sentence ""This sentence is not true" is true" as problematic as it contains 2 truth values.
*It is assumed - It is wrong to put a truth predicate on a sentence that already have a truth value in it.

The sentence "This sentence is not true" as problematic as it refer to itself, or to point out the core problem: the statement "This sentence".
- "This sentence is real" - "This sentence is square" - "This sentence is x"

*As stated above, if we could find a way to restate it to another way then the problem of contradiction would vanish. But there is still this problem of self-reference.

From what I know and read, a proposition is a statement or sentence that expresses an agreement or disagreement of its facts. Or to simply put it, a sentence that contains a truth value.

I think that the sentence already contains a truth value which is "not true". What I lack is a way to restate that sentence as its subject alone is problematic. This is still a possibility for me, there must be a way.

Maybe there some aspects of a proposition that I did miss, would you kindly tell me what those are?

They said "no" :)

I do believe that by saying "I am lying", what you really mean is "What I have just said is a lie". There is a context in that certain conversation that makes "I am lying" meaningful.

Maybe there some aspects of a proposition that I did miss, would you kindly tell me what those are? — Jaydison

I actually can't because I don't know much about it myself. :P I was just throwing out an idea that makes sense to me, and settles the apparent contradiction in my head.

Given the pedigree of the Liar's Paradox, one must be careful in seeking simple, obvious solutions without considering why those solutions haven't gained widespread acceptance amongst logicians and philosophers of logic. To quote Tim Williamson on this issue, "We'll have to be willing to tolerate some pretty strange ideas [to solve the Liar's Paradoxes] because we've tried all the non-strange ideas, and none of them worked."

So the Liar's Paradox (LP) is standardly formulated as follows:

"This [sentence/statement/proposition] is false."

Where's the problem? It's simple. If that Liar Sentence is true, then it follows that it is also false. After all, it says (about itself) that it is false. But if you say that the LP is false, then it follows that it is also true. After all, it says (about itself) that it is false. Giving it either truth-value ends in a contradiction, which people (especially in the Western philosophical tradition) find intolerable.

So a couple of people have voiced their preferred solution, so to address them in brief:

"The Liar Sentence is not Truth-Apt"

The idea here is simple. If giving the LP either the value of "true" or the value of "false" results in an inescapable contradiction, we can avoid the Paradox by saying that the LP has neither value, thus preventing the contradiction. There are many arguable problems here. Firstly, this would seem to require abandoning the Law of the Excluded Middle or else the Principle of Bivalence. Now, I've no qualms with dropping Classical Logic in favor of a Non-Classical Logic, but I get the feeling many people would not like that.
But most importantly, this does *not* actually solve the Liar's Paradox. A so-called Revenge Paradox (RP) can constructed to prove the futility here:

"This proposition is [not true/a valueless proposition]"
"This proposition is neither true nor false"
"This sentence is either false or neither true nor false"
"This sentence is ungrounded."

This RP shows that simply denying the LP a truth-value gets you nowhere. If this RP is truth-valueless (as it must be, if the original LP was valueless), then this Revenge Paradox is *both* True and Truth-Valueless (because it says, about itself, that it lacks a truth-value). And if the RP is not Truth-Valueless, the it is False and Truth-Valueless (because it says of itself, a valueless statement, that it has a truth-value). Again, the contradiction pops up. Revenge Paradoxes are the standard problem with solutions to the Liar's Paradox, and positing truth-value gaps don't have a good track record. And so solutions like Kripke's just don't seem to work. Unless I'm mistaken, Kripke himself posited that his solution would possibly be subject to a Revenge Paradox when he first wrote about his solution.

"The Liar refers to Sentences, when Only Propositions are Truth-Apt"

So the idea here is that sentences aren't the objects which possess the property of truth. But rather, that propositions are the objects which bear truth. Ignoring the debate about what actually bears truth, this seems like a dubious solution to the problem. It seems to be basically Kripke's solution: that the Liar sentences are ungrounded. If that's the case, I don't see how one escapes the Revenge Paradoxes.

"The Liar is an infinite Regress"

The ideas here seem to be a bit odd to me. The Liar doesn't loop endlessly, it can simply be taken to be a proposition which relates to 2 truth-values at the same time. That a program would loop in assessing the Liar's value doesn't mean that the Liar fails to have truth-conditions. Computational processes are (arguably) limited in a number of ways, yet we don't axe logical problems on that basis. For example, so far as I know, no computational process can demonstrate the Incompleteness of logical systems of sufficient complexity, and yet it's plainly obvious to logicians why these systems are Incomplete (see Gödel's Incompleteness theorems & Gödel encoding).

The reason why the Liar sentences are philosophically and logically interesting is because they've played a big role in a number of key areas: the foundations of modern mathematics (Russell’s Paradox), discussions about the nature of truth (Tarski's Theorem, dialetheism), and so on.



Now I personally find the Dialetheist response to these paradoxes compelling. E.g. Accept these as true contradictions, switch to a Paraconsistent Logic, and adopt truth-relational semantics. But that's quite controversial

The idea here is simple. If giving the LP either the value of "true" or the value of "false" results in an inescapable contradiction, we can avoid the Paradox by saying that the LP has neither value, thus preventing the contradiction. There are many arguable problems here. Firstly, this would seem to require abandoning the Law of the Excluded Middle or else the Principle of Bivalence. Now, I've no qualms with dropping Classical Logic in favor of a Non-Classical Logic, but I get the feeling many people would not like that. — MindForged

Do we have to abandon classical logic when we claim that the sentence "go away" is neither true nor false?

Do we have to abandon classical logic when we claim that the sentence "go away" is neither true nor false?

No, because that sentence isn't truth-apt, nor can I see how you could attach a truth predicate to it. Its just a command. The Liar sentences seem no different that other truth-apt sentences, in which case, denying it a truth-value would necessitate adopting some type of Non-Classical Logic.

No, because that sentence isn't truth-apt, nor can I see how you could attach a truth predicate to it. Its just a command. The Liar sentences seem no different that other truth-apt sentences, in which case, denying it a truth-value would necessitate adopting some type of Non-Classical Logic. — MindForged

It might seem a truth-apt sentence but the claim is that it isn't. Its syntax is misleading.

It might seem a truth-apt sentence but the claim is that it isn't. Its syntax is misleading.

How so? This is why I mentioned the Williamson quote, because the idea that there is a simple solution to this problem is vitiated by the fact that there is not a standard resolution to these paradoxes amongst logicians. If it were simply a syntactic issue, the problem wouldn't persist. And what do you think that syntactic issue is, anyway?

How so? This is why I mentioned the Williamson quote, because the idea that there is a simple solution to this problem is vitiated by the fact that there is not a standard resolution to these paradoxes amongst logicians. If it were simply a syntactic issue, the problem wouldn't persist. And what do you think that syntactic issue is, anyway? — MindForged

It's misleading because, as you say, it seems like a truth-apt sentence, being that it looks like most other truth-apt sentences, but it isn't.

And it's not that it's a syntactic issue. It's actually a semantic issue. Despite it's structure, it doesn't actually mean anything. Truth-predication is only meaningful when there's some evaluable fact about the world. Liar-like sentences don't have such a thing (much like the sentence "I am a squiloople").

It's misleading because, as you say, it seems like a truth-apt sentence, being that it looks like most other truth-apt sentences, but it isn't.

Again, I'm wondering what the evidence for this is. That it looks no different than other truth-apt sentences would support the claim that it is itself truth-apt.

And it's not that it's a syntactic issue. It's actually a semantic issue. Despite it's structure, it doesn't actually mean anything. Truth-predication is only meaningful when there's some evaluable fact about the world. Liar-like sentences don't have such a thing (much like the sentence "I am a squiloople").

I'm aware that it's a semantic issue, you were the one who said it was an issue of syntax. :P
That solution seems dubious, as many truth-apt sentences have nothing to do with the world (e.g. mathematical and logical truths). Further, that seems subject to an obvious set of Revenge Paradoxes:

"This sentence doesn't mean anything"
"This sentence doesn't involve evaluate to a fact about the world"


Hopefully the contradictions are obvious. And besides which, the idea that sentences containing an empty term (e.g. "squiloople") are somehow meaningless seems clearly false. The sentence you gave wasn't meaningless, it's just that the term "squiloople" has no apparent referent. The sentence is meaningful, I just cannot parse one of the terms. And again, the Liar Sentence does have a referent: Itself. So in don't understand how that constitutes a solution.

I'm aware that it's a semantic issue, you were the one who said it was an issue of syntax. — MindForged

Its syntax is what misleads people into believing it's truth-apt.

That solution seems dubious, as many truth-apt sentences have nothing to do with the world (e.g. mathematical and logical truths).

The Liar Paradox is a natural language sentence, not a sentence made in some formal system.

Further, that seems subject to an obvious set of Revenge Paradoxes:

"This sentence doesn't mean anything"
"This sentence doesn't involve evaluate to a fact about the world"

...

And again, the Liar Sentence does have a referent: Itself. So in don't understand how that constitutes a solution.

Sorry, I wasn't clear. I meant that it being true or false doesn't mean anything given that there's no evaluable fact in virtue of which it is either true or false.

Its syntax is what misleads people into believing it's truth-apt.

But *how so* is my question. If it has the same structure as other truth-apt sentences, clearly it's not the syntax which is the issue.

The Liar Paradox is a natural language sentence, not a sentence made in some formal system.

There has been no better luck in solving the LPs in formal languages either; the Liars can & have been articulated in formal systems too, otherwise we would reject it as a pseudo-problem. An LP is a sentence L which is true iff it isn't true (L <=> ~L). There are even purely syntactic versions, such as Russell's Paradox in naive set theory (R ∈ R <=> R ∉ R). And anyway, logicians and mathematicians (especially) almost always reason in vernacular anyway. The problem exists for both kinds of languages, which (as an aside) was the reason Tarski's suggested jettisoning natural language in favor of an artificial language to solve the LP.

Sorry, I wasn't clear. I meant that it being true or false doesn't mean anything given that there's no evaluable fact in virtue of which it is either true or false.

It doesn't mean anything empirical, but that doesn't mean it lacks a meaning. It's a sentence which is true when it is false, and vice-versa. I think you're treading down the path that Kripke went down. And speaking of Kripke, you should see some of his work on the Liar Paradoxes. There are versions of the LPs which are actually tied up in empirical facts. They are called the Contingent Liar Paradoxes.

I think you're treading down the path that Kripke went down. And speaking of Kripke, you should see some of his work on the Liar Paradoxes. — MindForged

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

It doesn't mean anything empirical, but that doesn't mean it lacks a meaning. It's a sentence which is true when it is false, and vice-versa.

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.

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.

— MindForged

It's misleading because, as you say, it seems like a truth-apt sentence, being that it looks like most other truth-apt sentences, but it isn't. — Michael

Again, I'm wondering what the evidence for this is.

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 question.