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Trying to solve the "liar paradox"


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The idea usually goes: take some sentence that's just floating about, that merely reads, "This sentence is false." Like there's a piece of paper and this is all that's on it, and the paper is fluttering around in the wind. But so anyway, this self-floating self-false sentence, is it true or false? If it's true, it is what it says it is. But it says that it's false, wherefore... Next, if it's false, then it is what it says it is, and when a sentence is what it says it is, it's true... Or is it? [The following I have also, with some edits upward here, posted on Philosophy Stack Exchange.]

I noticed something about the liar paradox, when the liar sentences are taken as questions. Let’s start with the liar index, “This sentence is false.” Allow this to be questioned: “Is this sentence false?” Now, if this is a yes-or-no question, then saying, “No,” amounts to reinterpreting the sentence as, “No, this sentence is not false; it’s true.” But if the sentence now says that it is true, and since that sentence would not be false if it were true, it follows that the transformed sentence in question does not have a contradictory truth-value. The liar sentences are transcendentally disjunctive with honest sentences. In other words, the liar index is logically interchangeable with the honest index, as such (more on this in a little).

Next, take the recursive liar, “L: L is false.” As a question this would be, “L: L is false?” However, L in that case would be a question, and questions are not truth-apt as such. The recursive liar lacks an erotetic value.

Another example would be the liar loop, “The next sentence is true; the previous sentence is false.” But if either of the sentences is taken as a question, the loop halts: “The next sentence is true; is the previous sentence false?” doesn’t work since “is the previous sentence false?” is a question, so “The next sentence” does not refer to something that is truth-apt, now.

Let’s also consider the liar imperative. This comes from the epistemic-imperative theory of erotetic logic, where a question is converted into an epistemic imperative. Here, “L: Is L false?” = “L: Let me know whether L is false.” But L in this event is an imperative: again, a kind of sentence that is neither true nor the opposite of true (false), but just “not true” in the abstract. (I think there's a related analysis of, "Don't comply with this imperative," which is complied with if and only if it is not complied with, vs., "Comply with the imperative, 'Do x'" being reducible just to, "Do x" in the same way that, "It is truth that X," reduces to the straight assertion that X. But I can't remember how to "do it" (solve the problem) right now. It shows up in Hofstadter's GEB as, "I wish that this wish would not be granted," for whatever that's worth.)

I wrote an essay on the topic once where I covered a few more examples IIRC and they all go pretty much the same way. For example, trying to work out the liar disquotation ("This sentence is false," is true if and only if this sentence is false) can be used to exactly describe how saying "no" to the liar index converts it into the honest one (i.e. how (false:not true):no "goes to" true:true; it's a kind of subtle double-negation elimination that ends up with: "This sentence is false," is false if and only if, "This sentence is true," is true).

Tarski’s model of levels of truth-predication therefore holds as a two-place relation, between questions and answers in general. That is, since the liar sentences are “unquestionable,” if we accepted them (believed them), we would be using them as unquestionable axioms. The problem, then, is that the liar sentences are not answers to questions, so they have no erotetic form of truth, which would be the alternative truth-predicate of a t1/t2/...-predicate model.

NOTE: I am not arguing that the liar sentences are “meaningless.” The liar index is certainly not meaningless: it has an entirely regular use in natural language, as in saying the sentence while pointing at some other false sentence to which “This” actually refers, e.g. pointing at a written token of, “2 + 2 = 22,” and saying, “This sentence is false.” (Moreover, the existence of this natural usage is what allows us to commit to an equivalent "This" in the interior and exterior of the liar disquotation, instead of saying, "'This sentence is false,' is true if and only if that sentence is false." This haecceitic quasi-copying is allowable on the ground that the liar indexicals have to be adjudged only as sentence-tokens: the sentence-type is truth-inapt, since it does not actually refer enough to be evaluated as such.)

My last question, then, is: is this the solution to the liar paradox?

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@Ripheus23 Out of curiosity, what is your take on Quine's approach to this? Quine's Paradox.


A core question that should be addressed is, with these types of statements, are true and false the only options, or can the option, as you note in your note, the concept of something being meaningless, also apply? Can one ask: "is the smell of blue anything other than C sharp?" A question which, if some of the parts are swapped out, makes sense, but with the parts as arranged, doesn't. "Have you stopped performing activity A" is another, because both true and false assume you have been performing activity A - you have to look at the structure itself, and note that one of its assumptions is false, rendering the question itself meaningless if the person hasn't done activity A. The, as Ted-ed puts it, "Three Alien Overlords" puzzle, allows for questions to be embedded within other questions as a solution, allowing one to gain access to useful information even when one doesn't know the finer details of the embedded question. So does that also apply, and can we ask the question: "is the liar's paradox actually a logically sound and/or valid question?", in the same way one can ask: "is the question 'is the smell of blue anything other than C sharp?' a logically sound and/or valid question?"


Still the best approach to this is to consider Turing's solution to the Halting Problem (that is, proving that there is no such possible system), using the same logic as the liar paradox to show that such a machine cannot exist - the liar paradox demonstrated algorithmically. Still, I am not entirely convinced by Turing's argument, but it definitely seems to approach the same issue. If there is a solution to the Halting Problem, then it would likewise be a solution to the Liar's Paradox

(Also, complete side note, but do you think that the Plank of Carneades is a more strict form of the Famous Violinist question? That is, if you think that someone has committed murder in the Plank of Carneades, where your life is on the line, you agree that choosing to disconnect from the violinist is also murder, where only your convenience and comfort was?)

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I'm familiar with the violinist case from abortion ethics, but just from the sidelines. So I'll have to check up on the plank case...

The temptation is to think the liar sentences to be meaningless. But the index has a meaning, it's just the indexical element in the sentence type is ambiguous between an external and internal reference. Moreover, I remember one essay that said, "This sentence is false or meaningless," also generates a paradox, so I'm trying to avoid the meaninglessness solution (also I have an incomplete belief about meaning/reference/etc.).

Other weird cases:

This sentence is true or false. It seems that if it's true, then it's not false. But then it's not "true or false" but just true. But then it's not what it says it is, so it's false...

This sentence doesn't correspond to a fact/this sentence doesn't cohere with an ideal set of propositions. These might be used to gauge "theories of the truth predicate" in some way. E.g. the first seems like a liar sentence but the coherentist version just seems weird to me haha

EDIT: haha the plank case is mentioned by Kant one time, I didn't even know that was his citation. But yeah, they fall under a similar heading, I think. There's an overlap with the idea of government democide by forced famine, too, iirc.

EDIT 2: also I think I don't have a solution to the Quine case, at least not one I consciously recognize...

EDIT 3: also, loaded questions: I think their presupposed contexts are pragmatic implicature instead of semantic value. Technically, "I haven't stopped eating baby kangaroos," for example, is true if I haven't even started eating them. But no one would usually ask a question about that, unless they assumed I'd started.

Edited by Ripheus23
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  • 1 month later...

First off, I really like your analysis @Ripheus23, but I think this is one of those cases where complexity tends to distract from the solution to the problem. Let's analyze the sentence "This sentence is false".

Firstly what is the proposition that is being assigned a truth value of true or false here? Let's look at a couple different ways we could write similar sentences, that takes out the self-referential "this" so we can get a sense of what the truth statement we are trying to determine is.

The sentence is false.

Tommy's sentence is false.

These are provable/refutable assertions because the subject of the statement corresponds to a discrete object, that is not the sentence itself. The assertion contained in both is that were you to examine "the" sentence or Tommy's sentence you would find that they were false.

So the assertion of the Liar's index sentence is that were you to examine the sentence itself you would find that it is false, but there is no testable proposition. It's simply a tautological assertion that the sentence itself "is" something, namely false.

Paradoxes abound when you construct self-testing identity statements.

As far as Quine's paradox goes, it's another case where the object that is subjected to the truth value test is self contained and self referencing. When you examine what the assertion the sentence Is actually making, it is simply a self negating identity. This particular variant does away with the "this" and the "is", but the net effect is the same.

These are paradoxes because they were constructed to be paradoxes, but unlike paradoxes like Xeno's paradox they are essentially meaningless because they are self referential, so they in effect aren't saying anything outside of themselves. 

While admittedly being very fun to think about, the net effect of these paradoxes is to aid the net increase of entropy in the universe, as brains churn out heat spinning in meaningless self-recurssive cycles.

Edited by hoiditthroughthegrapevine
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@hoiditthroughthegrapevine In a sense, if each sentence is a node in a graph, and what it refers to is another node in the graph, the referral an edge, then self-referential statements which don't touch any other node are little loops ungrounded to anything else, and so are in a sense unsupportable - there needs to be some foundational element which just as lines pointing away from it. Like a leaf from a tree that has been cut off and so falls.

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