[FOM] Foundations and Frege

Michael Kremer kremer at uchicago.edu
Tue Oct 14 15:34:51 EDT 2003

I have to disagree with Ewald's point (2).

In a polemic against Boole (unpublished in his lifetime;  see his 
Posthumous Writings volume, second essay, "Boole's logical calculus and the 
Concept-Script"), Frege emphasized the way in which his Begriffsschrift 
made possible a new mode of concept-formation (essentially, abstraction of 
complex predicates from logically complex sentences) which went beyond 
Boolean combinations.  He gave this analogy:  with Boolean logic, we have 
some lines already drawn (as in a Venn  diagram) and then recombine these 
lines to carve out new areas -- but in doing this we don't really generate 
anything new because the areas outlined are really already there in the 
diagram.  With his new logic, we draw entirely new lines.  This whole idea 
only works at the level of multiple generality, etc.   In this article 
Frege gave many examples (definition of continuity, uniform continuity etc) 
showing the new expressive power of his logic, all of which essentially 
rely on multiple generality.  The Begriffsschrift itself is divided into 
three parts.  The first concerns the basics of the system.  In this section 
he explains the need for the quantifier in terms of marking distinctions in 
the scope of the variable.  The second derives logical laws all involving 
only one-place predicates (except for identity) and no multiple 
generality.  The third involves what he is really interested in:  deriving 
serious results from logic alone (in the theory of sequences).  Heavy use 
is made of multiple generality here.

It is hard to read any of this without thinking that Frege had a pretty 
good insight into the importance of this issue, even if he didn't express 
his understanding exactly as we would nowadays.

--Michael Kremer

At 09:19 AM 10/14/03 +0200, you wrote:

>Dear William,
>A few days ago you wrote, opining that Frege's contributions to 
>foundations are overvalued, but
>   His discovery of quantification
>   theory was important for the codification of logic and so ultimately
>   for foundations.
>There was an invited talk at the recent FOL75 conference in Berlin by 
>William Ewald
>entitled "Frege, Hilbert, and the Discovery of Modern Logic" whose main 
>thrust was to
>   1. Frege was the first of several independent discoverers of the 
> quantifiers; the
>   discovery of quantification was in many respects waiting to happen.
>   2. Frege didn't realise the importance of quantification as a 
> discovery; he was
>   more impressed by other features of his Begriffschrift.  In particular, 
> when arguing
>   for the superiority of his system over Boole's, the issue of multiple 
> generality and
>   relative expressiveness doesn't occur to Frege.
>   3. Frege wasn't alone in not realising this: Pierce and Peano also 
> didn't see the
>   crucial significance of the quantifier.
>   4. Frege's technical work in fact had close to no impact, both Peano 
> and Pierce were
>   far more influential.
>   5. The crucial importance of the quantifier was not commented upon 
> until far later,
>   by Quine.
>(I summarise from memory, possibly my recollection is not entirely accurate).
>If we agree with Ewald, then it looks as if Frege's technical influence on 
>the development of
>logic is rather slender.

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