THOMAS to class

An argument due to Georg Kreisel [1969, 1971] shows that the notions "intuitively valid" and "valid in all set-theoretic structures" are extensionally equivalent.

THOMAS

Kreisel notes that

(1) D(A)Val(A)
(2) Val(A)V(A)
(3) V(A)D(A)

jointly establish:²

(4) (V(A)D(A) D(A)Val(A)).

A corollary of Kreisel's argument is that "intuitively valid" and "valid in all set-theoretic structures" have the same extension.

(5) V(A)Val(A)

Mephisto emerges from behind podium.

MEPHISTO

Wait a minute! Not so durn fast!

THOMAS

YOU again!

MEPHISTO

You say "intuitively valid" and "valid in all set-theoretic structures" have the same extension.

THOMAS

Kreisel's argument shows that.

MEPHISTO

No, it doesn't. All it shows is what follows from what. Take the negation of (5)^* as a premise, and you'll see what I mean. From the negation of (5)^* it follows that if (2)^* holds, either (1)^* or (3)^* does not.

(*) not-(5)(2) (not-(1)not-(3))

But (2)^* is vouchsafed by the meaning of its terms. So if "intuitively valid" and "valid in all set-theoretic structures" do not have the same extension, either (1)^* or (3)^* is false.

But (3)^* "is precisely the mathematical content of Gödel's completeness theorem".³ Hence if "intuitively valid" and "valid in all set-theoretic structures" are not extensionally equivalent, first-order predicate logic with identity is either unsound or incomplete.

So, Kreisel's argument doesn't establish the extensional equivalence of "intuitively valid" and "valid in all set-theoretic structures".  What Kreisel's argument does show is that if Gödel's completeness theorem holds, the extensional equivalence of "intuitively valid" and "valid in all set-theoretic structures" is a sine qua non for the SOUNDNESS of first-order predicate logic with identity:

V(A)D(A)[(D(A)Val(A))(V(A)Val(A))]

THOMAS snorts.

MEPHISTO

Thomas?

THOMAS

Yes.

MEPHISTO

Is the Law of Contraposition a crock?

Theory P

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