Thursday, January 17, 2008

Rambling on Subalternation and Existential Import

It's not uncommon in logic textbooks to find long and belabored (or short and simplistic) complaints about the traditional operation of subalternation (the inference from "Y is predicated of all X" to "Y is predicated of some X") because it violates existential import. I've noted before that this is a bit silly, first (as many people have noted) because it confuses particular propositions with existential propositions -- it is, in fact, a matter open to controversy whether particular propositions are existential, and we should not assume that the modern assumption that they are is right, just on the logic textbook's ipse dixit; and second, because most of these logic textbooks go on later to teach universal instantiation as the most natural thing in the world. But universal instantiation and subalternation are so closely analogous as logical operations that they have one and only one logically significant difference: subalternation concludes to an indefinite particular (some), while universal instantiation concludes to a definite one (a particular). Thus, from (x)(Fx) you can infer Fa for some a that's an x; and from "All things are F" you can infer Some things are F. Indeed, strictly speaking, although it's not the usual way of doing it, there seems no reason whatsoever why you couldn't have a subalternation that does the same thing as universal instantiation, concluding "A certain thing is F" from "All things are F", and it would take considerable ingenuity (which I've never found anyone taking) to explain why they aren't both giving us a result we could identify as "A certain thing, let's call it a, is F". Perhaps this is what Boethius has in mind with his quidams. I don't know. I certainly wouldn't wager on it; but I still think it gives me reason to complain about how cavalier most logic textbooks are about this. If you're worried by it in subalternation you'd better have some excellent reasons for not worrying about it with UI; and these reasons are never anywhere in sight.

But setting that aside, we can see more clearly what's involved in subalternation, and what you have to assume for it to work, by putting it in terms of Sommers's Term Functor Logic, which (to distinguish it from all other possible term functor logics) I call SETL.

"All S is P" in SETL is:


The first sign indicates quantity (universal), the second quality (affirmative). "Some S is P" is:


To get from -S+P to +S+P you need to make an assumption. That assumption is:


Some S is S. Thus you have:


Which simplified is:


Which gives you:

+P+S or +S+P, as you please.

All that's pretty basic. What it shows is that all you need in order to allow subalternation is the assumption that some S is S. The real question, then, is whether "Some S is S" requires the further assumption that an S exists. That's the one and only question to be asking about subalternation.