liar paradoxes, a problem with reductio proof and speech acts

Originally published on Language and Philosophy, June 20, 2013

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.

intended paradox

Originally published on Language and Philosophy, June 16, 2013

“I’m very witty!” someone wrote in a comment box in response to the criticism “You have no wit.”

“I’m very witty” might seem at first a witless and therefore unpersuasive response, unless it is sarcastic, in which case it is actually witty. If it’s sarcastic, the meaning intended to convey is that author isn’t witty, and therefore it implies that the comment itself also is not witty. The joke is, the author knows it’s not witty; yet that’s what makes it witty. So if it’s witty, it’s a lie; if it’s a lie, it’s not witty: a liar paradox.

But if the comment is merely false, then there’s no paradox — just a reply by someone who thinks he’s witty but is too dull to know he’s not witty, and hasn’t enough wit to say so wittily.

So if it’s a lie, then it is a meta-witty paradox; if an honest falsehood, it’s just stupid.

What’s interesting is that the intention or speaker’s attitude or character of mind induces the paradox, not the words alone. The paradox depends on who’s speaking, liar or dolt, wit or fool.

fun with Gödel

Originally published on Language and Philosophy, November 3, 2011

There’s an easy answer to this question, but if you replace c) 60% with 0%, then you get a liar paradox, a Gödel-type statement — it has no numerical answer and can only be evaluated outside its terms.

http://flowingdata.com/2011/10/28/best-statistics-question-ever/

Gödel, btw, once proved Anselm’s ontological argument, the argument that proves god exists. His version, so far as I can tell, removes all modality in Anselm’s argument, and I think that’s why it works. It’s a complicated version, but I think that even extremely simple, non modal versions work too, e.g., if “god” by definition is that which nothing is greater, then whatever is greatest is that thing. This version works for any model in which there is at least one object. So if nothing exists, then it doesn’t prove anything, but since something patently exists, we can safely assume that there is a greatest thing.

It’s been pointed out to me that there may be many notions of greatness. I think this is a problem in the logical semantics, not in the syntax, unless there is no such notion as “the greatest.” The only important of the many notions is the notion of “greatest of greatest things.” If that can’t be evaluated, then the designation of “god” is indeterminate, but not necessarily nonexistent. I mean, the argument is valid, so far as it goes; it just doesn’t tell you which notion of greatness is in fact the greatest.

There are a number of ways to treat this notion of greatness. Maybe the notion is inherently relative, like “good” — a good chess player might not be a good bricklayer. If so, then there is no idea of “the greatest.” But assuming there are such notions as “the good” and “the greatest,” I don’t see any contradiction within them. So one can assume there is such, without determining which.

I am fond of Anselm’s proof in part because it seems to make everyone nervous, non believers and believers alike. It makes non believers uncomfortable because they don’t want to admit they are wrong and they don’t want to believe in god. I’m not sure which is worse to them. It makes the believers nervous because the god it proves might not be the comfortable personal one they’ve been living with. They want to believe in their god, so it’s convenient that there is no logical proof of god that might be out of their control. And that is as it should be: the notion of god should be a #$%ing scary one, if you actually take it seriously for a moment. The Old Testament got that one right. People don’t take it as seriously as it should be. They should be crossing their fingers that there is no god.

If you look at the argument more closely, the character of the god that it proves is such a cipher, that it steals away all the personal comfort of the deity they want or need. That’s one reason Aquinas didn’t include it as one of his proofs. His proofs are all friendly ones like the order of the cosmos and its beauty. Nothing so cold as logical truth.

I delight in Anselm’s proof. What it says about god is utterly trivial — so I’m happy to live with it — and the argument strikes terror in everyone else. That’s almost as good as having a personal god working for you.

The logical positivists would have approved: the argument is true, but meaningless. So god exists in all his greatestness. So what? It’s no different than pantheism, which is basically indistinguishable from atheism.