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

[Neil Tennant]

On Fri, 3 Sep 2004, 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.

My book Natural Logic (Edinburgh U. Press, 1978; 2nd, revd. edn 1990)
follows Prawitz's monograph Natural Deduction (Almqvist & Wiksell,
Stockholm, 1965) in allowing only sentences to occur in proofs. Hence only
sentences can be premises or conclusions of proofs. In particular, only
sentences can be theorems.

If one proves a theorem of the form F(a), then of course one extra step of
universal introduction would produce the explicitly general theorem
(x)F(x).

Neil Tennant