The Liar Paradox | Attic Philosophy

"This sentence isn't true". Isn't it? Either way, there's a contradiction. This is the Liar Paradox, one of the hardest paradoxes to solve. In this video, we see what the problem is, why it can't easily be dismissed, and look at what the solutions might be. Make sure you watch this one first:

00:00 - Intro
01:55 - Self-reference
04:21 - The Liar in a logical language
05:04 - Neither true nor false?
06:33 - Dialethism
07:18 - Paraconsistent logics
08:24 - Logic of Paradox
09:54 - How does Dialethism help?
11:54 - Wrap-up

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[IterativeTheoryRocks]

I always thought the resolution was due to godel - that a logic - any logic of note - cannot be both complete and consistent. If we choose consistency, we and up with gaps and things like the liar sentence. If we choose completeness, then we end up with inconsistency, again, like the liar sentence.

So it’s not a question of ‘true or false’ at all. It’s an example of a sentence (of which there are an infinite number) that proves logical systems are either inconsistent or incomplete.

But hey, I’m just a computer programmer not a logician!

[alistairkentucky-david]

Great video! Is there any way to resolve this with metaphysics instead of pure logic? I tend to think that sentences are not primary truth-bearers but rather propositions. If I maintain that L is neither true nor false because it fails to express any proposition at all, it's not so clear to me how the strengthened liar retort is supposed to go. I imagine a dialogue like:

A: Strengthened Liar is neither true nor false as it fails to express a proposition
B: A-ha! But that's exactly what Strengthened Liar says! Thus we get a contradiction
A: You beg the question by assuming that it says *anything*. I say that it fails to express a proposition and so fails to say anything at all.
B: Well it certainly looks like it says something. After all, presumably it's because of what it says that you deny it expresses a proposition
A: Such is only an appearance. The illusion comes from the fact that a sentence like "This sentence is [Adjective]" is grammatically well formed. So it seems superficially as though it says something.
B: That seems wildly implausible. How could such a simple illusion generate tens of thousands of pages of philosophical literature?
A: Ok, tell me what proposition it expresses. I say you cannot mentally grasp or understand what proposition it expresses, without first grasping what sentence said proposition 'points to'. But to grasp the sentence is to understand the proposition it expresses. Such a loop can never be completed.Thus I maintain that you have no grasp of what proposition is being expressed by SL. Thus, you have no way to infer a contradiction from my denial that it says anything. (Of course, I deny that SL says anything, so your failure to grasp what proposition it expresses is no defect in you!)

[brewcoffeebox]

This commonly misunderstood paradox exposes the catastrophic problem of self-reference which forms the sand-like foundation upon which the entire apparent universe rests. That is, the inherent presupposition (or assumption) of truth.

Take any self-referential statement and it is bound to have an unfounded assumption of truth baked into its premise (“this sentence is…”) which depending on what is ultimately asserted (“this sentence is false”) creates a paradox.
Self-reference assumes the truth because it has to, there is no other option, and so it is unable to judge its own reliability without first presupposing it.

A prime example of this is the incomplete system of mathematics which hides its fatal (self-referential) flaw behind smokescreens of technical jargon it uses in order to “proof” itself true by itself which from the get go is assumed to be true (ie the “self-evident truths” or axioms of math). No amount of math however will change the fact that it is down-right impossible to prove the validity of 2 without first making the unreasoned assumption that 2 exists.

Rather than dismissing the notion of truth altogether, the incoherence of this paradox appears to place truth outside the reference of “self”. In other words, truth is not (nor can be) self-evident. What exactly does this mean? Firstly, it means that so-called objective knowledge (in and of itself) is an enigma - analogous to subjectivity (because some self said so). While objective knowledge is assumed to have a one-to-one correspondence with reality, the truth of it can only be judged from a standpoint outside of itself - that is, independent of the mediating mind. Is that even possible? Yes, because you are NOT your “self”.

There is a self reading these words that “I” call “you” and “you” call “me”. It is a caused fact existing in three dimensional space and passing through time, manifested as perception and conception. Its purpose is to generate the world-for-me (a massive collection of apparently isolated objects it calls “things”) from the “thing-in-itself” or that which representations are of. It is bound in experience to self-reference, forced to rely on tools (sense, language, thought) to describe, understand and manage the apparent world of “things”. The truth of what anything is, however, is ultimately a complete mystery, with one exception.

Beyond the self-generated world (the insatiable, thinking, wanting, not wanting self) exists the one thing-in-itself that I have direct inward access to, that I can be, that I am - consciousness - the ultimately ineffable experience in which exists no separate facts, no space, no time and, ultimately, no difference between me and the rest of the universe - the state of being ‘I’ call ‘I’. In being conscious, I experience truth independent and free of self-reference.

[chachachi-hhks]

I think the whole problem happens when we try to evaluate a sentence. We are used to working under assumption that sentences rarily change their truth status. Meaning after one evaluation of a sentence we are done. The Liar's sentence is different, it changes its truth value after each evaluation, each of its evaluations feeds the next evaluation. We run into a trouble because our logic systems aren't adapted for dealing with evaluation loops and constantly oscilating truth values of a sentence. But at each point of time the Liar's sentence is strictly either true or false.

[neopalm]

Well the question is: is there a proposition P such that P <-> ¬P? Evidently, the existence of such a proposition proves a contradiction, so we're kind of forced to throw out any logic system with such a statement unless we're okay with all implications being valid (which means all statements are true and false). At least, that is the case depending on exactly what you mean by ¬ and whether you accept the principle of explosion and the law of the excluded middle.

[raydencreed]

I’m curious what your thoughts are on my approach to the liar sentence.

I always start by observing that there is a difference between a meaning we assign to a sentence and that sentence itself. So for example, if I need water, I could just say “I need water” in English, but I could also say “Necesito agua” in Spanish. Those are very clearly different sentences, but we assign them the same meaning.

The next important observation is that the meaning we assign to a sentence is the thing which carries a truth value rather than the sentence itself. We typically call sentences true or false, but that’s just us being mildly imprecise in our speech. Normally it doesn’t matter, but it can cause a problem when we look at something like the liar sentence.

The next observation is that we can’t get at the meaning we assign to a sentence directly. That’s why we have languages in the first place. To express meaning. But sometimes, our language can be a little vague, so it can be helpful to be able to rephrase a sentence in such a way which makes the meaning we have assigned to it a little more clear.

So for example, the sentence “This sentence contains five words”. With the phrase “this sentence”, we very clearly intend to refer to the entire sentence. This allows us to rephrase it as “The sentence ‘This sentence contains five words’ contains five words”. We can call this rephrasing the “meta sentence”. We know which sentence we are referring to and we know what it means for a sentence to contain five words, so the meaning of the meta sentence is clear and can easily be the same as the meaning we assigned to the original sentence.

Then, to verify it, we just count up the words and find that, indeed, there are five words contained within that sentence. Similarly, if we had started with the sentence “This sentence contains six words”, we could still clarify the meaning in the same way with a meta sentence and count up the words to, at least in this case, *refute* the meaning we have assigned to the sentence.

So for the liar sentence, the corresponding meta sentence would be “The sentence ‘This sentence is false’ is false”. We know which sentence we are referring to, but we also know that sentences don’t carry truth values, so the meaning we have assigned to this sentence would be false. And it would be *just* false, we aren’t forced to conclude that it would also be true.

If we still want to get a contradiction, we might instead try to clarify the meaning we have assigned to the liar sentence by rephrasing it as “The meaning we have assigned to this sentence is false”. But while sentences can be self-referential, meaning cannot be. One meaning we assign to a sentence can be about another meaning we assign to a different sentence. One meaning we assign to a sentence can even be about another meaning we assign to that same sentence, but it cannot be about itself. If we try to make it so, we run into circular definitions. So even though we may think we’re describing something with the phrase “the meaning we assign to this sentence”, we really aren’t.

So the appropriate response to the liar sentence is not to say that it’s meaningless, certainly not in any sort of objective sense, but that there’s no meaning we could assign to it (or any other sentence for that matter) which would be both true and false.

[neopalm]

Actually, the "this sentence contains five words" example demonstrates the point I was getting at in the comments of the "curry's paradox" video. Sure, you can make a predicate A(x) defined as "x contains 5 words" and you can use godel numbers (or godel strings in this case) to encode that, then reach a statement P that says "P contains 5 words". What you can't do is encode a predicate like "x, if you parse it, is a false proposition" into a godel string, because there is no way to express such a predicate using the available first order language on strings (if there was, all you'd need to do to get the godel string of the expression is put double quotes around it). Due to this inability to encode, reaching a predicate that reads "this sentence, if you parse it, is a false proposition" is impossible through the means of diagonalization.

[BelegaerTheGreat]

Looks like Paraconsistent Logic is the only way to go.
EDIT: Except if we decide that self-reference IS a problem because it is an endless self-reference. "This sentence has 5 words." Refers only to the amount of words it has, but "This sentence os true" refers to itself over and over again!

Still, at 9:30, why is notOther = Other? Shouldn't notOther = trueORfalse?

10:00 *IMPORTANT: THE LOGIC OF PARADOX DOESN'T SAY "a sentence can be both True and False"! IT SAYS: "a sentence can be both True and notTrue"!*

[alanbunyan]

One approach to solving this issue is to take the view that the supposed self-referentiality of the proposition in question is impossible, for the simple reason that the sentence apparently referenced by the word 'this' does not yet exist: it is IN THE PROCESS of being formed: that is to say, it consists at the moment of reference of nothing more than a grammatical subject ("this") but, as yet lacking any predicate, it has not been formed, i.e. it does not yet exist AS A SENTENCE. It could refer to another sentence which has already been formed, but never to itself. It's a bit like my trying to sell you a car that has not yet been built and asserting '"This car never breaks down", to which you would be entitled to retort that I cannot reasonably make any such claim of something that does not actually exist!

[raydencreed]

My guess is that I’m still a novice in this kind of discussion, but it seems to me that including a truth value of “both” in LP doesn’t actually invalidate the explosion principle.

I’ve come to this conclusion because I think it’s more appropriate to express the explosion principle (and any other form of argument) purely in terms of propositions and operators.

The explosion principle, then, would be written as

(A & ~ A) > B

Then, according to the truth tables of these operators (of course, modified to include a “both” truth value in LP), assigning “both” to A and ~ A also leaves

A & ~ A

with a truth value of “both”. This again, along with assigning a value of “false” to B, also leaves the “>” in

(A & ~ A) > B

with a truth value of “both”, not just “false”. My guess is that I’ve misunderstood the modified truth tables, or that this just doesn’t matter.

[Philosophyoftruth_]

I know this is off topic, but is a masters in Philosophy worth my time, effort and money? I really like the masters course they offer at Kings. Thanks.

[wagfinpis]

I like your bow and bow next to the bow.
True/ False can be logical or narrative.
If someone is lying or has a different perception then it is narrative, if it is logical we are trying to reduce an immediate interpretation, not devine the ultimate truth.

Is true and Is false logic rule: "is true or "is false" logic can only be applied outside of/after/beyond a complete statement.

"This sentence" is not a statement.

"This sentence is false" is like "9÷0="; it is like saying there is a hole in nothing.

"This sentence is true" is like "0÷9=0"; it is like making nothing disappear.

These sentences are intuitively not logical, so you make a correlated rule or method to categorize them in a way that tags them as logically invalid or to lack any communicated logic value.
*They subjectively make grammar, but they lack any objective logic, so you correlate their logic against the grammar via a rule that represents practical observation. It's just nomenclature like particle "spin" in physics; it's based on what we can know.

[jonc]

Dialetheism is an important response to this issue - & works well.

[mathieumorier]

Logic expresses what we mean by existence. Something exists if and only if it has at least one attribute(positive attribute). If a thing has at least one attribute then we can use this attribute to predicate on that thing. 'X is y' is true only if x has the attribute y. 'Every real number has an additive inverse', and 'The speed of light in a vacuum is c' are all meaningful propositions that predicate on something that exist(abstractly or concretely). We could also construct false proposition where the subject is clearly stated and the attribute also clearly stated. 'This sentence is false' has no predicate on the subject 'this sentence'. It is like saying 'this is false' 'this rock is false', 'this cloud is false', or 'this video is false'. These sentences are badly constructed and unclear. For a proposition to make sense, the subject and the attribute(s) used to predicate on it must be cleary stated and defined.

[GeorgWilde]

If a sentence refers to a set of sentences, those sentences can in turn refer to further sentences. In order for me to care whather the original sentence might be true or not, there must be no loop in the referential chain. That is, at some point, every chain of references (sentence refering to something) must terminate pointing to some concept i can make sense of through my subjective experience.
All this nosense about liar paradox stems from from many philosophical misgivings like "analytic proposition", "truth functionality", "validity in virtue of the form". Of course truth is not contained in syntactic structure or some relationships between definitions. Truth always depends on things outside of language. Everything that is called analytic knowledge, in fact depends on a priori knowledge or even some synthetic knowledge. But the only certain knowledge is a priori.
Formal languages were created to model and encode human reasoning. We first reasoned and then we invented these devices. Blaming failure of a tool for reasoning on the reasoning itself is foolish.
"This sentence has no meaning for you." - Yeah, i still don't care.

[frankavocado]

'This sentence has five words' is verifiable. 'This sentence is false' is not. The liar paradox is a sub-class of self-referential propositions, in so far is it, in all versions (afaik), refers only to logical form and truth. In that sense it represents the limits of a-priori thought.

[Lucky]

My instinct is still to say that self-referential sentences are neither nor false but meaningless because they aren't actually claiming anything at all. In regards to the examples you gave of apparently meaningful self-referential sentences, I would say they aren't actually self-referential because what they're actually referring to is the way we've described them. e.g. "This sentence is white" isn't actually talking about the sentence itself, which is just information, but rather the chalk, ink, etc. used to write it. "This sentence contains five words" is referring to a feature of how we've chosen to express it -- e.g. I could instead write it as "This sentence contains 5 words" and it would be false since "5" is number symbol, not a word.

[jherbranson]

I think the issue with the "This sentence has five words" is that it has nothing to do with logic. If you actually think about the message that is being encoded in the English it is strictly arbitrary. You could write that exact observation in a number of different languages like Cantonese, etc. and I'd bet many of them would no longer have 5 words even though they would claim so. In fact, you can write that exact message in English and it would not have 5 words. "The statement you are currently reading now has 5 words." Is just as valid of a demonstration of the logic and it is not true. So the point is, these self referential statements are just vacuous and have no real logical meaning. If you can't effectively transmit the message during a translation then there is no real substance and the statement is not appropriate for logical consideration.

[hihoktf]

I think the true liar is "I do not exist." It is self-referential denial that should be verboten.

[ganrimmonim]

This makes me think of the wave particle duality of light. That in effect light can be (is?) Both a wave and a particle until you 'ask'.