[FOM] The rule of generalization in FOL, and pseudo-theorems

[Charles Parsons]

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).