Re: Second-order and Higher-order Logic

Jan Burse schrieb:
E voila, we have weak second order logic,
as a special case of general semantics. But
this works fine because phi has no
other second order variables in it. And thus
the approach I describe in my first posting,
works fine.

If weak second order logic were among
the models of general semantic we could
not do diagonalization in the general
semantics.

Because then (I the predicate for
individuals, S the predicate for sets)
would be satisfiable:

exists F(I(x) <-> S(F(x)))


But I am not so sure whether second order
logic allows this mixing of sorts, because
F : s -> i. ?

Oops, I would need F: i -> s.

But this should work. Because we can start
with a functions F : u -> u, where u is the
sort for the universe. And then later introduce
s and i, and then simply put:

exists F(
forall x(I(x)->S(F(x))) & (i)
forall x,y(F(x)=F(y) -> x=y) & (ii)
forall x(S(x)->exists y(F(y)=x))) (iii)

(i) would express the sort constraint. (ii) the
injectivity and (iii) the surjectivity of F.

Best Regards


.

Relevant Pages

  • Re: Second-order and Higher-order Logic
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  • Re: Second-order and Higher-order Logic
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