[FOM] First- Vs Second-Order Logic: Origins of the Distinction?

josef at us.es josef at us.es
Sat May 21 07:57:44 EDT 2016

Dear Richard: 

the question how first-order logic became the paradigm logical system
was the main topic of a paper of mine, 'The road to modern logic -- an
interpretation' (BSL 2001), see
http://www.math.ucla.edu/~asl/bsl/07-toc.htm. I argued that simple type
theory was taken to be the main logical system by 1930, and I analyzed
the main reasons for the move to FOL. 

Also relevant to this is another old paper, 'Notes on types, sets and
logicism, 1930-1950' (Theoria 1997), which I can send you if you'd like.
Here the main issue was to analyze reasons for the abandonment of
logicism in the 1940s. 

Concerning second-order logic, let me also indicate my current view,
which you may find provocative. I'm convinced that insistence on full
SOL is the last remnant of logicism in foundational debates. I have
argued for this in several places, a consequence being that we should
stop calling "standard" the full powerset semantics.

Best wishes, 


El 21/05/2016 02:09, fom-request at cs.nyu.edu escribió:

> ----------------------------------------------------------------------
> Message: 1
> Date: Thu, 19 May 2016 17:56:51 -0700
> From: Walt Read <walt.read at gmail.com>
> To: rgheck at brown.edu, Foundations of Mathematics <fom at cs.nyu.edu>
> Subject: Re: [FOM] First- Vs Second-Order Logic: Origins of the
> Distinction?
> Message-ID:
> <CAFm5C8DeukiDP5WU-b58Fjs4JY+9-tbfsi-+1hPv5KDFWWSKGQ at mail.gmail.com>
> Content-Type: text/plain; charset=UTF-8
> There's some discussion in Badesa's _The Birth of Model Theory_.
> On Thu, May 19, 2016 at 10:23 AM, Richard Heck <richard_heck at brown.edu> wrote: 
>> Does anyone have a good reference for historical work on the emergence of
>> the distiction between first- and second-order logic? I'm particularly
>> interested in how first-order logic came to be seen as "really logic". Quine
>> was of course famously hostile to second-order 'logic', but I am guessing
>> that there were earlier antecedents, probably emerging from work in
>> mathematical logic itself.
>> If anyone is able to sketch that story, I'd love to hear it.
>> Thanks,
>> Richard Heck
>> PS What I myself know about this concerns only the emergence of Frege's
>> awareness of the distinction. That part of the story gets told in my paper
>> "Formal Arithmetic Before Grundgesetze", section 3, which can be found on my
>> website.
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