It’s easy to mistake paradoxical sentences for liar paradoxes. “If this sentence is true, then it is false,” is a liar paradox. If the sentence is true, then the antecedent is true. If the antecedent is true, then the consequent must be false, the implication as a whole is false, so the sentence must be false. So if the sentence is true, then it is a contradiction and a falsehood. So the antecedent must not be true. If the sentence is false, antecedent is false, and the implication as a whole is true.
“If this sentence is false, then it is true,” however, is not a liar paradox. If it is false, then the antecedent is true and the implication fails, and the whole is false. If the sentence is true, then the antecedent is false, the implication holds, and the sentence is true. That’s not a paradox, it’s just a sentence the truth of which cannot be determined. It’s like the sentence, “This sentence is true.” Is it true or false? How could you tell?
Similarly, “The sentence I am now writing is true,” is indeterminate. “The sentence I am now writing is false” is provably a liar paradox, athough one could ask of these two sentences “true or false of what?” The deductive proof that yields a liar paradox of the latter, is a reductio: assume the sentence is true, you deduce that it is false; assume it’s false, you deduce it’s true. So if it’s true, it’s false and vice versa. But if you ask “true of what?” then you’re asking for an empirical answer — does the sentence corresponds to something, in this case to its own truth. Is truth a thing that can be pointed to? If it’s a correspondence with something, we’re stuck in an infinite recursion. So these sentences, on the one hand, lead to a questioning of the correspondence theory. But they also lead to questioning of the validity of deductive reductio argumentation, not unlike that questioning of the reductios that led Cantor to multiple levels of infinities, and the intuitionist rejection of the reductio in favor of proof by demonstration. Several directions from here: you can say these sentences don’t correspond to anything; or correspondence is not complete; or correspondence, even with its incompleteness is a better option than reductios that lead to liar paradoxes.
Some performative acts are puzzling in relation to paradoxes and lead to a question about what their propositional content is. Suppose “I’m stating in writing that 2+2=5” is true by definition, because it is in fact true that I stated it in writing, even though the secondary content of what I stated is plainly false and wrong. Generally, speech acts (“I promise to…” “I deny…” “I insist…” “I’m saying that…”) could be viewed as true by virtue of stating them regardless of their secondary content, the proposition that is promised, denied, insisted upon or said. But what about “I’m stating in writing that I’m not stating this”? If it’s true that I stated that in writing (I just did, in fact), then it’s true. But in writing that sentence, I stated that I didn’t. Is this a contradiction or is it a paradox or something else?
It might be something else. It’s simply true that I stated and wrote the sentence. The secondary content is false. So the wrting act is true, but what I asserted was wrong. I am wrong, though the primary sentence is true. Think about a promise. Suppose you’d promised something that you can’t fulfill. It’s a promise, even though it’s a foolish promise. In these speech acts, there’s a relation not only to the content of what is promised, said, denied or insisted upon, but also a relation to the actor. It’s best not to conflate them.
So “I deny that this is a denial,” might still be considered a denial, and therefore true (Is the sentence a denial that I wrote? Yes.), even though the secondary content is false. I made a statement in the form of a denial, but what I said about that statement was wrong or false.
“I deny this” and “I deny denying this” are tricky cases. If they are true, they seem to deny themselves, so false; a liar paradox. Can these be treated as above — simple true denials with false secondary content? On this view, a lot rides on the little word “that.”
That’s one view. But what if these performative acts have no content? What if “I promise I’ll stop” means nothing more than “I’ll stop” and “I’ll stop” is in effect a promise not to stop? What if “I say it’s raining out” means nothing more than “It’s raining out” and “I deny I saw her” just means “I didn’t see her”? Then “I deny denying this” means “I deny this” which in turn means simply “This sentence is false.” Does this reduction matter? Is any content lost? Is it like the truth predicate — semantically inert (“‘2+2=4’ is true” adds no content to the simple statement “2+2=4”)?
“I’m not writing now” was plainly false when I wrote it just a moment ago. But surely the following sentence is meaningfully distinct from it: “I’m now writing ‘I’m not writing now’.”
Suppose there is no difference between those two sentences. Is there then also no difference between “I assert that I’m not writing now” and “I’m not now writing”? If there’s no difference, then either “I assert” somehow not count as language, or all utterances and written statements are true. In a previous post I claimed that that’s exaclty right — nothing of logical significance can be proved about the world by speech. Speech is nothing but opinions, only accidentally truths.
This same quandary arises with denials, since they are semantically an assertion (with a negative truth predicate): “2+2=5 s false” means “I deny 2+2=5” or “(~(2+2=5)) is true.”
Is there a difference between “I did not have sexual relations with that woman, Ms. Lewinsky” and “I deny that I had sexual relations with Ms. Lewinsky.” The first is a lie — not just false, but Bill knew it was false when he said it — the latter is true even though the secondary content is false. So the denial is true, but also involves a deception, by implication. That’s not to say the statement is false, but that it shouldn’t have been made, if honesty is required.
If these have distinct truth values, then utterances don’t have an implicit assertion predicate, contrary to my previous post.
Tags: intended meaning, logic, paradox, pragmatics, semantics