Averroes wrote: ↑Sun Oct 04, 2020 5:58 pm
You have to drop the first "P" for it to express LNC.
Whoops. Yes, you are correct about that.
But now, you write:
In natural languages words refer to the outside world but in logic the propositional variables such as "P" or "Q" have no such reference! They are purely mathematical objects.
Right. And that's all the difference in the world.
To refer to mathematics as a "language" is not deceptive, but is ambiguous, because "language" is a word we use to refer to broad systems of symbols (as in "computer languages," for example), and yet in other contexts, a word we use specifically to indicate what you call "natural languages," things like English or Arabic, which are
not composed of symbols each of which stands for a fixed quantity or property, but are rather formed into chains with empirical referents.
Mathematics takes the law of identity as a given. One might say it's so intrinsic to maths that without assuming the law of identity there would be none. Fine. But in empirical situations, there are no absolute and fixed quantities implicated. Rather, there are sets of linguistic markers which have to be compared for their relative coherence -- a much less precise matter, but no less important than the mathematical operations. And it is in this second situation that the law of identity is questioned...not in maths.
So what mathematics has to do with all this? I'm still not at all certain. So far as I know, no mathematician even questions the law of identity.