Is 'All sentences are false' false, true or paradoxical?

Question

Usually you read that the sentence S: 'All sentences are false' is false because if true you got a contradiction, but if false it's not since it would just mean that not all sentences are false, e.g. S is false, but other sentences are not. Therefore the main stream in philosophy considers S to be false.

But isn't S really equivalent to 'All sentences but this one are false and this sentence is (also) false'? Because since the last part of the conjunction is paradoxical (it's the so-called liar sentence) then S as a whole would be paradoxical. So my conclusion would collide with the mainstream and I wanna know if I am right or wrong.

Answer 1

"All sentences but this one are false and this sentence is (also) false" is only true if both parts separated by "and" are true. We know that "all sentences but this one are false" is definitely false - there are many true sentence, I'm sure I don't need to give any examples. Since the first part of the sentence is false, it is utterly irrelevant whether "this sentence is (also) false" is true or not - whether or not it's true, the entire statement is false, because the first part before the "and" is false. There is no paradox - "all sentences are false" is simply untrue, no matter how you choose to rephrase it.

Answer 2

Your argument is basically:

We can write sentence A as the conjunction of sentences B and C, and C is paradoxical; therefore A is paradoxical.

There are a couple different issues here: whether the conclusion follows from the premises, and whether in fact you have exhibited such a "paradoxical dissection" of the original sentence. The first is dubious, and the second is false.

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Most obviously, there's the issue of how (meta-)truth-values behave. You're implicitly claiming that "Paradoxical and False = Paradoxical," that is, that paradoxicality is "infectious." But why should this be the case?

There's either not much to say here beyond observing non-obviousness, or a long essay or discussion about the various pros and cons. I think the right thing to do here is move on; however, this question of infectiousness of paradoxicality is actually something I think is pretty interesting and might make a good separate question.

(FWIW I strongly disagree with the OP's stance here: I think that as soon as we decide to attempt to treat paradoxical sentences in some serious way, the only choice that makes sense is for "False and Paradoxical" to evaluate to False.)

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The other issue is more subtle: insufficient care has been taken with respect to (re)naming sentences. We have four sentences in question:

• (S): All sentences are false.

• (S-): All sentences but (S) are false.

• (S+): Sentence (S) is false.

• (L): (L) is false.

Note that (S+) is not quite the liar paradox (L)! Its referent is (S), but it itself is (S+). This may seem like needless pedantry, but in fact it's exactly the sort of thing we need to pin down precisely if we're going to be able to develop any meaningful theory of paradoxical sentences: we need to stop treating words like "this" so blithely. (A similar renaming issue cropped up here.)

Answer 3

"All sentences are false" is refuted by the single valid counter-example of a the
semantic tautology (thus necessarily true sentence) "cats are animals".

Answer 4

I would observe that mathematics in general is widely regarded as having a weak notion of storage as distinct from content - here, that is effectively the distinction between the articulation of the claim and the content of the claim.

A "lie" is not (at least, not by definition) a claim which asserts the falsity of its own articulation. A lie is a statement whose content does not accord with other facts - that is, a lie forms part of a contradictory system of claims, in which the lie is deemed the claim which is responsible for the contradiction.

"The moon is made of cheese" is a lie, but the articulation itself is not false, and the content actually asserts a truth (and it is a lie because it wrongly asserts that truth).

To say "the articulation of this claim is false" is merely a nonsense, because logic does not operate on claim-articulations, it operates on claim-contents, but here the claim-content is attempting to refer out to the claim-articulation which contains it. That is not true, false, or paradoxical - it is merely an invalid logical operation.

Perhaps another way of putting it (or conceiving it) is that an operand of a logical operator, cannot be the result of that same operation, because such an operation is simply uncomputable. So, "this sentence is false" involves a circular link between the sentence, and the reference to the sentence (represented as "this sentence") contained within, and it is impossible to logically evaluate.

Answer 5

'All sentences but this one are false and this sentence is (also) false'? Because since the last part of the conjunction is paradoxical (it's the so-called liar sentence) then S as a whole would be paradoxical.

At first glance, this is a conjunction of, "All sentences besides S are false," and, "S: S is false." The truth-value of a conjunction is a function of the truth-value of the conjuncts, wherefore we would initially separate our evaluation of, "All sentences besides S are false," from our evaluation of, "S: S is false." But it also looks like the conjunction illustrates part of the problem with, "S: S is false," in the first place: it's not possible to intelligibly situate it under the universally quantified, "All sentences are false." In other words, despite appearances, "S: S is false," is not really a sentence at all! (This adverts to a pragmatic syntax: a linguistic structure is a declarative sentence if and only if it is used to actually declare something; but an otherwise unspecified self-declaration of falsity is not really a declaration; therefore...)

EDIT: on the other hand, the instance of "this" might be ambiguous over the entire conjunction, where S is both the second conjunct, and the entire conjunction. Then the secondary counterargument would be that if the second conjunct could be meaningful on its own, it could be meaningful in the context of the conjunction, which seems false.

Answer 6

In classical logic, a truth value "paradoxical" is not defined, nor the result of building conjunctions of paradoxical and false statements. Saying the liars paradox is paradoxical does not assign a truth value to it, it declares the inability to assign either true or false as truth value.

So whenever you combine a paradoxical and another statement in any way, classical logic does not define the result.

Alternative to binary logic called "many-valued logic calculi" exist (most common is ternary logic using "unknown" as third value), but there probably isn't any many-valued calculus to apply here as a default, so the answer to your question could be: it depends on the calculus applied. Also i am not aware of any such calculus that would handle "paradoxical" as truth value.

If we imagine a logical calculus in which we could formulate the liars paradox and assign a third truth value "paradox", then likely such a calculus could not generally assign "false" to the conjunction of "false" and "paradoxical", as the self-referential effects could cause new paradoxes.

However, for your example, it would seem most reasonable to expect a new useful multi-valued calculus to resolve the conjunction of the false and the paradoxical parts in your example to false, for the reasons explained in the other answers, same as common ternary logic. Hence the other answers tend towards "false".

However anyone can define a "silly" calculus with many truth values, by which the conjunction of "false" and. "paradoxical" is true, false or paradoxical. It just would not be very useful.