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

[Joseph Shipman]

I agree that an important motivation for SOL is logicism, but your use of the connotationally loaded descriptors "insist" and "remnant" suggests that you think there is something wrong with this much logicism; please elaborate.

-- JS

Sent from my iPhone

> On May 21, 2016, at 7:57 AM, josef at us.es wrote:
> 
> 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,
> 
> Jose
> 
>  
> 
> 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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