Liar's Paradox
Problem:
"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?
"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?
Comments (85)
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
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.
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 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'.
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.
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.
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.
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.
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
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.
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?
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").
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.
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.
Its syntax is what misleads people into believing it's truth-apt.
The Liar Paradox is a natural language sentence, not a sentence made in some formal system.
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.
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.
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.
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.
Yes, see the first reply to this discussion. ;)
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.
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.
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.
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.
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.
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.
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.
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).
And how does a proposition come to be related to either the value "true" or "false"?
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.
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.
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).
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?
A bunch of natural language words does not a contradiction make, no matter how much it may feel as though they do.
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.
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 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.
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.
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.
I like that. It's a very neat analogy.
When I'm true I'm false
Oh what should I do?
Ride a camel? Ride a horse?
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..
1. Self reference
And
2. Binary logic
If we refrain from any or both the paradox disappears.
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.
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.
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
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"
However you are still using binary logic in your clever example
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.
Math and logic aren't grounded by the world. Also, there's fictional truths, such as Harry Potter performs magic.
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.)
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!"
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.
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.
The answer to this is the same answer I gave to The Pinocchio Paradox. The stated rule ("I will return the child if and only if you correctly guess what I will do") is one that cannot be followed without exception.
So the crocodile's claim ("I will return the child if and only if you correctly guess what I will do") is false.
I agree with everything in that post. What I am saying is that the Liar sentence in natural language does not give us line 1 in that proof. The Liar sentence is not of the form 'L and not-L', and attempts to derive a sentence of that form from the Liar sentence make untenable assumptions.
I don't know how you can say that. The contradiction in the liar statement stems from following the rules of logic. If not, then why is it considered an issue?
What I'm saying is that the truth (or falsity) of that statement isn't derived from some set of axioms. You don't say "the liar sentence is true because it follows from these true sentences" because there are no true sentences from which that sentence can be derived. Liar-like statements have no truth-maker (even in principle).
Which form of the liar statement?
I am lying.
This sentence is false.
This sentence is not true.
The next sentence is true.
The previous sentence is false.
Edit: sorry, thought that was directed at me.
The third one is the "strengthened" liar paradox. Consider:
"This sentence does not express a true proposition."
If it's true, then you're back at the same contradiction.
It's not true. The sentence isn't truth-apt.
It really is a straightforward proof by contradiction. If it being either true or false leads to a contradiction then it must be neither true nor false.
That seems like cheating. Now you can just remove any paradoxical statement by saying it's neither true or false. Let's try this as a result:
"This statement is neither true or false."
Does that lack a truth value? I say no, it has a truth value if what you stated is true. And then you're right back at the liar paradox.
Surely, if it were that easy, the liar paradox wouldn't have remained a puzzle to philosophers for 2300 years!
Are these contradictions?
"this statement doesn't correspond to some obtaining state-of-affairs" doesn't correspond to some obtaining state-of-affairs.
"this statement doesn't follow from some specified set of sentences" doesn't follow from some specified set of sentences.
I don't think they are.
So why would this be a contradiction?
"this sentence isn't true" isn't true.
Of course, this could all be made clearer if we could explain what it even means for a sentence to be true. I've provided two possible answers. Are there any others you could suggest?
That's very true, but what is at issue here is the semantics, what is meant by the statement. In your examples, what is expressed, is that the thing is misnamed. The get together of people should not be called a party, the thing you are sitting on should not be called a chair. That is the meaning expressed.
Quoting Marchesk
The liar sentence is not without meaning, it is used as an example, this use indicates meaning. So people like philosophers bring it up to discuss, and this is its context, which gives it its meaning. But there's very little difference between it and other self-contradicting, self negating examples, like "the square circle". It just demonstrates that we can say things, which according to the placement of words, appear to be meaningless due to self-contradiction, and ask others what is meant by this.
That is the meaning of the liar sentence, it is used by philosophers to demonstrate that we have the capacity to use words in this way. It should be of no surprise to anyone, because we have the capacity to deceive. And this means that we can say things which are completely different from what our intentions are. So when I say "this circle is square", or "this sentence is false", what I mean by this (what my intentions are, or what I am doing with those words), is something different from what the words appear to mean, just like common forms of deception. Again, we have exposed that separation between what is meant by the author (the author's intentions, what the author is doing), and what the words, on their own, appear to say. And this simply demonstrates our ability to deceive.
One problem is thinking that whether a sentence is what I called "substantive" is simply a matter of its form. I wouldn't say that it is.
But arguably, "This sentence is not in Italian" isn't the same form as "This sentence is false." As I mentioned, the "is false" part of "This sentence is false" is just taking the truth value assignment and making it explicit/making it part of the sentence--or would-be sentence in that case, since "This sentence" isn't a sentence. So that wouldn't be the same form in that case. The same form would be "This sentence is not in Italian is true." (Or more conventionally, "It is true that this sentence is not in Italian.") Of course, this brings up possible ambiguities, at least contra conventions, over just what a sentence's form is, too.
Quoting Marchesk
Why would it hinge on the liar sentence? At any rate, if someone assigns no meaning to that sentence, then "This sentence is meaningless" is simply true. Otherwise it's false. That's no paradox.
If the sentence is meaningless, then it can't be true.
I understand why you're saying that, but it seems wrong. However, I have to think about it further to explore how I'd analyze the problem there.
[Later;] Ah--okay, so I'd say that it simply amounts to reading the sentence as saying, "This sentence isn't making any claim other than that it's making no claim." The sentence not being meaningless in that case would be it making some claim other than that.
The sentence is not meaningless, because Jaydison posted it in the op with intent, purpose, to discuss it's meaning. Therefore it must have meaning, Jaydison meant something with it. But, as I explained in my last post, what was meant by it, what Jaydison was doing with it, is something completely different from the meaning which appears from a reading of the words. What appears is some form of meaninglessness. Therefore the sentence is actually posted as a form of deception. And this can be seen as the essence of all such sentences which appear to be paradoxical, they are simply posted as a form of deception.
For example if there is not a good formal definition for contradictions does this mean proofs by contradiction fail?
You will have to forgive me, as I said I was having trouble understanding.
I was talking about this post in particular.
Quoting andrewk
I mistook it to mean there was no formal definition of a contradiction.
You also posted this and I was hoping you would get into the weeds a bit and delve into nuances you mention.
Quoting andrewk
Those contradictions are closely related to the set in Russell's Paradox of set theory, as is the Liar Sentence. Because one cannot trust a language in which contradictions can be proved (as demonstrated in Marchesky's post above), such languages are excluded from the serious consideration of higher-order logics. That is done by putting constraints on what constitutes a well-formed sentence in the language. Those constraints make it impossible to formalise the Liar Sentence in the language.
There is a pretty good introductory explanation of this in this note from Washington Uni.
Saul Kripke (1975) suggested a solution to the problem by introducing a third-value "undefined", still, it is still problematic (see "strengthened liar" or "revenge of the liar")
Alfred Tarski, on the other hand, worked on the assumption that self-reference (proved as the diagonal lemma) is inevitable and that truth is binary. He introduced the hierarchy of languages that is in order for us to talk about a language we should another language (a meta-language). The problem is Tarski aimed at introducing truth to formalized language, and claimed that we cannot introduce truth to ordinary language without resulting into a contradiction. (see 1944 Tarski)
'This sentence' is a phrase (not a proposition) that refers to the sentence 'This sentence is false'. Thus, we would get a sentences like ''This sentence is false' is false' and so on. Let us take not that we are not assigning truth to phrase 'This sentence' but to the sentence it refers to: the sentence 'this sentence is not true'.
the problem here is that it is agreeable the every instance of usage of truth takes the form 'x is true', where 'x' is any truth-bearer such as sentence, proposition etc. Thus, 'x is true' somehow serves as a basic definition of truth. However, as we can see above, there is an instance which leads us to a contradiction.
Hello,
I do not think that we have to abandon classical logic because of the "liar",
but I like playing around with non-classical logics.
One exotic logic is my favorite:
The "logic of reflection" by Ulrich Blau
which i developed and extended to an alternative logic, the "layer logic".
Link to the logic of Ulrich Blau:
https://ivv5hpp.uni-muenster.de/u/rds/blau_review.pdf
Link to "layer logic" "Trestone":
https://www.researchgate.net/post/What_do_you_think_of_layer_logic-and_the_use_of_a_new_dimension_to_come_around_contradictions
In layer logic a hierarchy of truth layers is used (similar to the hierarchy of types of Bertrand Russell).
Proposals to not have truth values any more,
but only proposals in connection with a layer have a truth value.
The liar statement here has the following form:
L:= For all n= 0,1,2,3,... "This statement is true in layer n+1 if it is not true in layer n and else it is false"
As all statements are u=undefined in layer 0,
we get for layer n=0:
"This statement is true in layer 0+1 if it is not true in layer 0 and false else"
Therfore: "This statement is true in layer 1"
For n=1: "This statement is true in layer 1+1 if it is not true in layer 1 and false else"
Therfore: "This statement is false in layer 2"
We see: The liar statement L is undefined in layer 0, true in layer 1,3,5,7,...
and false in layer 2,4,6,8,...
So the truth-value is alternating with the layers.
A classical statement would have only one truth-value in all layers 1,2,3,4,5,...,
so we can see, that L is a non-classical statement.
In layer logic it is not paradox, but an ordinary statement.
Layer logic is a little bit cumbersome,
but has amazing advantages:
Nearly all paradoxes can be solved similiar to the liar.
With layer logic a "layer set theory" can be defined,
where the diagonalization proof of Cantor is no longer valid
and where the Russell set and the set of all sets are ordinary sets.
Even natural numbers and an arithmetic can be defined.
A small set back: The prime factorisation of natural numbers could differ in layers.
But on the other hand it is probable, that the proofs of Gödel`s incompleteness theorems are valid no more.
Layer logic is a kind of a third way between classic logic and constructivistic/intuitive logic.
It is a logic with three truth-values, but the layers are the most important part.
Indirect proofs are allowed in layer logic, but only within a layer.
As in most classic proofs there are different layers involved if transformed to layer logic,
those indirect proofs are mostly not valid any more.
Unfortunately nobody up to now has seen a layer in reality,
but as long as the idea does not lead to contradictions it remains an interesting idea.
Yours
Trestone