The Liar's Paradox

Logic & Philosophy of Mathematics
41

You seem not to have read the linked post. It offers four distinct ways of dealing with the liar, not one.

But also, at least since Frege, logicians differentiate between the use of a sentence and what renders it true. Asserting that P is performing an act with words, and is distinct from what does or does not render P true.

The Liar concerns truth conditions, not speech acts. Your move is too fast.

What counts here is that we cannot construct a coherent bivalent truth theory that assigns stable truth conditions to such sentences.

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42

I didn’t know I had to click the link in blue. I read it now. I’m close to some of the “solutions” but no explicit mention of Habermas’ performative contradiction. It’s quite interesting to see logicians/philosophers trying to solve the paradox in so many different ways.

43

Habermas borrowed the notion of a performative contradiction from Austin, Strawson and others. Strawson is probably closest to your approach. He’d say “the present King of France is bald” (Are you familiar with that example?) is not false, but does not amount to an assertion and so is without a truth value. Roughly, an exampel of the first approach.

A valid approach, but not the only one and not without issues.

44

I see, but no explicit mention of a performative contradiction.

There’s a plethora of approaches. I first discovered your, not all statements are …, a few years ago. It would’ve been a nice dodge if not implied was abdicating the LEM. I don’t understand LEM or bivalence all that well though. LEM to me is p \vee \neg p and bivalence means 2 truth values, true and false.

45

Yep. That’s the basis of classical logic.

There’s a difference between saying that there are sentences with a third truth value, and that there are sentences with no truth value at all.

And there is the other view that allows for sentences that are both truthful and false.

There are logics that conserve some form of consistency in each case.

There is no need to choose one logic amongst these and elevate it to the level of being the “one, true logic”. Indeed, doing so implies that we already have a, presumably logical, basis for making that very decision, and so involves some form of auto-aggrandising circularity - “It’s the one true logic because it says it’s the one true logic…”.

A better approach is to choose logics and models to suit our discussions. That’s not relativism. The consequence relations logic studies capture differing notions of validity and consistency that we can acknowledge and discuss.

What is clear is that paradoxes such as the Liar do not threaten the basis of logic, so much as show us the edges of what can and can’t be said clearly. They show us why we need logic as well as natural languages.

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46

What could be a 3^{rd} truth value?

Yes, paraconsistent logic and dialetheism. How does that work? The Liar is true and false.

“Some form of consistency”?? :thinking:

I want to disagree, but I don’t think my vote counts. So you mean I should adopt dialetheism/paraconsistent logic with the Liar? Ok, so I do and then the Liar is both true and false. That’s like saying snow is white and not white to me. Can we make sense of snow being white and note white?

That’s the crux of the matter. Even with my limited understanding I can see Gottlob Frege working on predicate logic and making decisions. Why is predicate logic the way it is? Why doesn’t it possess structural features, linguistically-informed structural features, that help solve the Liar? In fact I don’t know how to express the Liar in predicate logic? Is it just L = This sentence is false?

47

In Kleen logic, “unknown”. Look in to it. As for the rest, I’m happy to answer specific questions, but reading up on nonclassical logic might better suit your needs.

See the SEP article. Generally, given a sentence L in the language and a truth predicate Tr, the Liar is expressed as L ↔ ¬Tr(⌜L⌝)

In English, L if and only if L is not true.

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49

Thanks for the suggestion. Very helpful. Chatted up an AI, have made notes for future reference.

I have a problem with the term, designated. With regard to \implies (implication), given both the antecedent and consequent is U, it evaluates to U in Kleene logic but in Lukaseiwicz logic it evaluates to T. What does this mean?

I beg the OP’s pardon if this is off-topic.

50

A can of worms. Think of Kleene logic as formalising what we know, while Łukasiewicz formalises what might be the case.

Take a murder trial. The Jury is deliberating. Now the defendant either committed the murder or didn’t, but we don’t as yet know which - so we assign a “U” to “The defendant is guilty”.

What truth value might we assign to “If the defendant is guilty, then the defendant is guilty”?

Kleene logic formalises what we know, and since we don’t know if the defendant is guilty, it assigns “Unknown” to “If the defendant is guilty, then the defendant is guilty”, since we don’t know what the Jury will decide.

Łukasiewicz logic formalises structure, and to be sure, if the defendant is guilty, than it is true that the defendant is indeed guilty. So it assigns “True” to “If the defendant is guilty, then the defendant is guilty”.

In Kleene logic, if we don’t know the defendant is guilty, we don’t pretend to know the truth of “If he is guilty, he is guilty”. In Łukasiewicz’s logic, regardless of whether he is guilty or no, if he is guilty, then he is guilty.

Both are consistent three-valued logics. Neither is “wrong.” They simply have different motivations and different domains of application.

Something like that.

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