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

William B. Ewald wewald at law.upenn.edu
Sun May 22 10:27:59 EDT 2016

```Dear Richard,

Bill Tait's remark is basically correct (though the history is very complicated). Here is a rough overview:

CS Peirce was the first to talk about "first-order" logic, back in the 1880s: he clearly distinguished between quantification over individuals and over properties. But he did not have anything like the conception of a formal system, or of a model theory (except in a very rudimentary form).

Weyl and Loewenheim, in the years from 1910-1915, both (independently) introduce something that *in retrospect* can be seen as first-order logic: but you have to work hard to extract the idea from their writings.

Hilbert, in the 1917/18 lectures (which were not published until 1928, as Hilbert & Ackermann), develops a sequence of logical calculi, of gradually increasing strength. (He starts with propositional logic, then monadic logic, then polyadic first-order logic, then higher-order, with the theory of types, which is what he is most interested in. Wilfried and I have a long introductory note on the historical background to those lectures.) He is the first to isolate the system, and the first to indicate a model-theoretic approach. (He does not explicitly formulate the completeness question, though he--and even more clearly, Bernays--had raised and settled it for propositional logic.) But Hilbert does not treat FOL as special (in contrast to the other calculi), and certainly does not claim that it is *the* one, true logic. That idea (I believe) first is promoted by Quine, in the early 1940s. By that time, the work of Loewenheim-Skolem, Goedel, Church and Tarski was known, and had made clearer than it was to Hilbert in 1917 just what was involved in the distinction between first-order and second order logic.

In other words, there is no single moment at which "the" distinction appears--rather, a gradual process extending over roughly half a century.

All best,

Bill

William Ewald
Professor of Law and Philosophy
The Law School
University of Pennsylvania
3501 Sansom St.
Philadelphia, PA 19104

(215) 898-9135

________________________________
From: WILLIAM TAIT <williamtait at mac.com>
Sent: Friday, May 20, 2016 8:38 PM
To: Foundations of Mathematics
Subject: Re: [FOM] First- Vs Second-Order Logic: Origins of the Distinction?

I would think that Hilbert's 1917-8 lectures (in the Ewald/ Sieg edition of his lectures on logic and foundations) is the first place in which our distinction between first and higher order logic is made. As opposed to Frege, Richard, Hilbert had the clear model-theoretic conception of logic.

Best, Bill

Sent from my iPad

> On May 19, 2016, at 9:59 PM, Alasdair Urquhart <urquhart at cs.toronto.edu> wrote:
>
> 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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