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

Michael Lee Finney michael.finney at metachaos.net
Fri May 20 00:50:07 EDT 2016

There is a book that may have what you need

   Foundations without Foundationalism
   A Case for Second-Order Logic
   Stewart Shapiro
   ISBN: 0-19-825029-0

Part III (pg 173) is about the historical aspects. Overall, it appears
to be well done and very convincing. Shapiro also has a paper on the topic

   Do Not Claim Too Much: Second-order Logic and First-order Logic
   Philosophia Mathematica (3) Vol. 7 (1999), pp. 42-64

but it has less historical information. I know of a couple of other papers,
but they don't have historical information. Also, Jaakko Hintakka has a book

   The Principles of Mathematics Revisited
   Jaakko Hintikka
   ISBN: 0-521-62498-3

which is about his Independence Friendly logic. I think that there is some
historical information there, but it is more scattered through the book. He
basically wants more than first order logic, but not quite as much as second
order logic. But he has a lot of the same arguments as for second order logic.

If those don't have what you need, perhaps their references will have it. I
don't have those, so I can't comment on them.

Michael Lee Finney

> 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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