“If i’m not mistaken….”

“This sentence is false”

• If the sentence is false, then it’s true, but in that case it’s not the case that it’s false, so it’s false.

“Suppose Gamma is the set of all sets that aren’t members of themselves.  Is gamma a member of itself?”

• If Gamma is a member of itself, then it isn’t the case that Gamma is one of the sets that aren’t members of themselves, so it doesn’t belong in Gamma…but in that case, it’s not a member of itself, and as a set that isn’t a member of itself, then it belongs in Gamma.

Here’s something that is disanalogous, but of an interestingly similar spirit.S: “If I’m not mistaken, p.”Presumably one is mistaken if one has said something false, so if S is mistaken, then “If I’m not mistaken, p” is false. But if S is mistaken, then the antecedent, “If I’m not mistaken” is false, so “If I’m not mistaken p” is true. So if S is mistaken, then S is not mistaken, in which case (if S has said something true, which, qua not mistaken, S has) then p is true.

Now,unlike the more canonical examples cited above, S’s statement has the virtue of being true if p is true, so where Russel’s paradox and the liar can not be true or false, “If I’m not mistaken p” can at least be true. But S’s statement cannot be false, and S isn’t actually allowing for the possibility that p is false. Assuming that S is intending to project a half-hearted commitment to p, and they’ve intended to allow for the possibility that p is false (that they might be mistaken) they’ve actually said something that literally rules out the possibility of p being false.

Here’s why. In order for p to be false, S has to be mistaken, but if S is mistaken, then “If I’m not mistaken, p” is false, and if “If I’m not mistaken, p” is false, then “I’m not mistaken” is true (and p is false) but if “I’m not mistaken” is true, then S’s statement is true, AND the antecedent is true, so p has to be true. In hedging, S has literally ruled out what she intended to explicitly allow for.

It’s worth mentioning that S could be vindicated if there were some other sentence which was the intended object of “If I’m not mistaken”.  So S might actually be saying “q. If I’m not mistaken, p.”  In this case, it would seem, if q is false, then S is mistaken, then the antecedent of the conditional is false, and thus p is allowed to be false. The locution that S utters is sometimes used like this, but not always, nor (I would suspect) with the regularity that would suggest that this latter reading is the locution’s “normal” use.

Everything is fine of course. We all know what is really happening. S doesn’t mean “if I’m not mistaken about the full sentence I’m currently uttering” (which would bring S close to uttering the liar paradox’s fraternal twin), but rather S means “I’m not mistaken about p, then em>p“. This is a tautology, but that’s ok. The conditional form contributes to the pragmatics of the sentence, and the concern is not with the sentence’s logical syntactic form. But there’s still a weirdness here: the speaker is asserting “If I’m not mistaken about p…” but the speaker has never asserted p! They’ve never asserted anything whose falsity would be implied by the falsity of p. So, if they are flagging the possibility of their being mistaken, where would such a mistake inhere? Perhaps the explanation is that the speaker has a belief that p, and that is the referent of “if I’m not mistaken about p“, but what if the speaker thinks “if i’m not mistaken about p“? Is that even a possible thought?

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