Topics Logic

Question Are logical laws such as the de Morgan's ones preserved under modalisations? For example, what are the truth conditions for the following sentences: Peter knows that Mary does not invite Paul and Peter. Peter knows that it is possible that Mary does not invite Paul and Peter.

Accepted:
September 3, 2009

Comments

I'm not sure precisely what is being asked here. The first sentence is true if the following is something Peter knows: Mary does not invite (both) Paul and Peter. Perhaps there is another reading under which it is true if the following are both things Peter knows: Mary does not invite Paul; Mary does not invite Peter. But this isn't likely to be a significant difference, under most accounts of knowledge attributions. Similar things can be said about the second sentence.

What I don't understand is what any of this has to do with the de Morgan laws. These say, among other things, that something of the form ~(A & B) is logically equivalent to the related thing of the form ~A v ~B. But neither of these sentences is of that form, unless we're talking about the second mentioned reading of the first one. And, in that case, yes, it certainly is equivalent.

What might be at issue is whether, e.g., "X knows: ~(A & B)" has to be equivalent to "X knows: ~A v ~B", which is to ask whether substitution of logical equivalents inside the scope of knowledge attributions is always permissible. This, unsurprisingly, is contentious. A reason to think it's not is that every logical truth is equivalent to A v ~A, and it would seem that there are logical truths that no-one knows. For example, Fermat's Last Theorem is provable in standard systems of set theory (and, it's widely believed, in Peano arithmetic, though I don't know if anyone knows whether that is so). If so, then there is a logical truth of the form: S1 & S2 & ... & Sn → FLT, where the Si are axioms of set theory and FLT is, of course, Fermat's Last Theorem. But it seems reasonable to suppose that, prior to Weil's proof, no-one knew this logical truth. But, of course, fans of "closure" have responses to such arguments, which usually go by the name: The Problem of Logical Omniscience.