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

Alasdair Urquhart urquhart at cs.toronto.edu
Thu May 19 22:59:11 EDT 2016

Greg Moore's article "The Emergence of First-Order Logic" is a good place 
to start.  It's in "History and Philosophy of Modern Mathematics" edited
by Aspray and Kitcher.

Moore emphasizes (and I agree with him) that the appearance of Gödel's 
incompleteness theorem was a key event in separating first- and second-order logic.

On Thu, 19 May 2016, Richard Heck 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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