Sunday, March 9, 2008

An Unnecessary Criterion of Adequacy for a Definition of "Confirmation?"

One of Carl Hempel's criteria of adequacy for a general definition of "confirmation" in his "Studies in the Logic of Confirmation" (1945) is that it must apply to sentences of any logical form. According to Hempel, a general definition of "confirmation" cannot state only what sorts of sentences confirm, say, universally quantified sentences with a single variable.

Maybe I'm missing something, but I think this is not a good criterion. As far as I can tell, as long as we define “confirmation” for, say, universally quantified sentences of single variable (with negation), then, if we accept Hempel’s “equivalence condition”*, we have a general definition of “confirmation.”

Can’t we eliminate any existential quantifier in terms of a universal quantifier and negation by means of the logical equivalence (Ex)(Fx) <=> ~(Ax)~(Fx)?

Isn’t it true that, for any n-adic predicate R, for any place pi in the predicate, for any ordered set of objects {x1, x2, …, xi-1, xi+1, …, xn} occupying the other places in the predicate, we can define a monadic predicate R`, such that R`xi <=> Rx1,x2,…xi-1,xi,xi+1,…xn?

And, at least in languages without empty names, isn’t it true that Fa <=> (Ex)(x = a & Fa)?

So, if we allow ourselves to define somewhat contrived predicates and help ourselves to languages non-free logics, isn’t every sentence logically equivalent to some sentence in universal form with one variable? This is not, of course, to say that we could actually get by in science or daily life without sentences of more complex logical forms; this is a claim only about what a definition of “confirmation” needs to be truly general. As usual, I reserve the possibility that I’m totally mistaken about the logic here.
* - The equivalence condition states that "If an observation report confirms a hypothesis H, then it also confirms every hypothesis which is logically equivalent with H."

1 comment:

Travis H. Curran said...

OOOOOOOOOOOOOOOOOOH shit look who else has a blogger.

Dogg, you are balls deep in Information. Logic and Reason are actually your testicles, swinging long and strong underneath your phallus of Philosophy. You make my auto-autobiographical poor prose fill with metaphorical penis envy. You still at Bard?

Keep on rockin' them knowledge props.