In "On How Logic Became First-Order" Matti Eklund writes (p. 2/148):

It appears to be widely held today that arguments from Skolem and Kurt Gödel, both alleged proponents of the thesis that standard firstorder logic is “the logic” (or, if you like, “the true logic” or “the correct logic”; below I will say something about just what preferring first-order logic as “the” logic might come to), had notable impact on the development toward first-order logic

Furthermore, in the book "Foundations without Foundationalism" by Shapiro it is written (page xiii):

Historically, the main proponents of first-order logic were Skolem, von Neumann, Weyl, and Gödel

Where did Gödel stated that first-order logic is the "true" logic? Were did he assert his opinion on this topic? Is there a paper in which he described his opinion? If not: Then, from where do we know that he had this opinion? Are there sources (a interview for example)? Can you give me some references?

(I' m also interested in the same questions with "Gödel" replaced by "Skolem". But for most, I' m interested in Gödel)


1 Answer 1


He did not write it anywhere. The quote itself only calls Gödel and Skolem "alleged proponents", and later in the article Eklund remarks that "the (supposed) evidence that Skolem adhered to first-order logic is that Skolem held that set theory and arithmetic should be given first-order axiomatizations, whereas... evidence that Gödel adhered to first-order logic is the (alleged) fact that Godel insisted on a first-order metalanguage" (p.151), and proceeds to dispute even this limited evidence. He mentions Gödel-Zermelo correspondence of 1931 as the alleged source, and remarks that "neither Moore nor Shapiro provides us with any quotation from the Gödel-Zermelo correspondence showing it to be the case that Godel there did insist on a first-order metalanguage" (p.158).

The dominance of first order logic is based on technical results of Skolem and Gödel, rather than on philosophical arguments. Skolem proved that first order logic has nice model theory, Gödel proved that it is recursively axiomatizable and hence has a nice proof theory (unlike higher order logics), and compact (unlike infinitary logics). At the same time, first order Zermelo-Fraenkel set theory (made first order by Skolem, who modified Zermelo's comprehension axiom accordingly) proved to be more than sufficient not only for all of classical mathematics, but even for higher set theory and model theory. On the other hand, higher order set theories, like Russell's, proved to be unwieldy. In the end, Russell had to introduce the infamous "axiom of reducibility" which effectively reduces the logic of his set theory to first order.

Since 1950s Quine routinely identified first order logic with the "language of science", Eklund himself mentions him as an influence. As for "true logic" as understood in philosophy Gödel argued quite the contrary. That human reasoning is not only uncaptured by the first order logic, but by any formalized logic, and that his incompleteness results are an indication of that.

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    Long ago a +1 from me, but it just occurred to me to add the following. Re: good model/proof theory, in a precise sense first-order logic is the strongest such logic - Lindstrom showed that any logic strictly extending FOL satisfying the downward Lowenheim-Skolem property must not be compact and must have a very complicated set of validities (in particular not r.e.). So there is a somewhat precise case for FOL available here as long as we do accept contra Skolem the DLS property as desirable (compactness seems obviously a Good Thing). Commented Jul 26, 2019 at 21:16

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