Re: Question about Frege's Theorem


lugita15@xxxxxxxxx wrote:

Isn't Henkin Semantics by itself
compatible with the axiom that given any wff with a free variable x,
there exists a concept (or predicate) that is true of x iff the wff is
true of x?

No. Take the following instance of comprehension principle

EXAxAy(Xxy <--> x =/= x)

That asserts the existence of an empty relation that obtains between
nothing. But since pure Henkin semantics puts no constraints on which
subsets of the domain are in the scope of the second-order quantifiers,
some Henkin structures will have no empty relations and hence not
verify this instance of comprehension.

You might find Chapters 3 and 4 of Shapiro's book on second-order logic
help a lot here.

.