[FOM] The rule of generalization in FOL, and pseudo-theorems
Charles Parsons
parsons2 at fas.harvard.edu
Mon Sep 6 15:45:56 EDT 2004
At 12:57 AM -0600 9/3/04, Richard Zach wrote:
>Dear Sandy et al.,
>
>You raise an interesting question. I don't know of a recent textbook
>calculus of first-order logic in which all theorems are sentences.
>However, I also don't think that it raises a particularly pressing
>question for the philosophy of logic. It is true that there is a
>disconnect between the definition of validity in most textbooks
>(preservation of truth) and the requirement of soundness on systems of
>derivation (preservation of validity). But that's just that: a
>disconnect. All it means is that validity is not all that must be
>required of inference rules in a logical calculus. Even when no free
>(object) variables are involved, you can see this disconnect in
>derivation systems for modal logic: the inference from A to Necessarily
>A is invalid (on the preservation of truth definition) but is sound for
>systems of normal modal logic (in the sense of preservation of
>validity).
>
Thanks to Richard for his historical information.
But unless I've missed a posting, FOMers seem to have forgotten a
once well-known formalization of first-order logic that eschews free
variable reasoning, and in which all theorems are closed. That is the
system in W. V. Quine's book _Mathematical Logic_ (1940, 2d. ed.
Harvard UP, 1951). The quantificational part of the system is laid
out on p. 88 (2d ed.), but there is a lot of discussion leading up to
it.
The rule of generalization is avoided by the following tricks. First,
one assumes as axioms all _closures_ of formulae of the form (all
x)Fx -> Fy. Second, one assumes all closures of formulae
(all x)(Fx -> Gx) -> [(all x)Fx -> (all x)Gx].
Then Quine is able to derive (*111) a generalized modus ponens, which
says that if the _closures_ of F and F -> G are theorems, then so is
the closure of G. But only the "naive" rule of modus ponens is
assumed.
A variant of Quine's axiomatization was applied to modal logic in
Kripke's well known paper "Sematical considerations on modal logic
(Acta Philosohica Fennica, 1963).
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