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

[Mummert, Carl]

> Does anyone have a good reference for historical work on the emergence of the distinction between first- and second-order logic? I'm particularly interested in how first-order logic came to be seen as "really logic".​​

One paper on the topic is "The Road to Modern Logic-An Interpretation", Jose Ferreiros, Bull. Symbolic Logic Volume 7, Number 4 (2001), 441-484.  http://projecteuclid.org/euclid.bsl/1182353823

Abstract:  This paper aims to outline an analysis and interpretation of the process that led to First-Order Logic and its consolidation as a core system of modern logic. We begin with an historical overview of landmarks along the road to modern logic, and proceed to a philosophical discussion casting doubt on the possibility of a purely rational justification of the actual delimitation of First-Order-Logic. On this basis, we advance the thesis that a certain historical tradition was essential to the emergence of modern logic; this traditional context is analyzed as consisting in some guiding principles and, particularly, a set of exemplares (i.e., paradigmatic instances). Then, we proceed to interpret the historical course of development reviewed in section 1, which can broadly be described as a two-phased movement of expansion and then restriction of the scope of logical theory. We shall try to pinpoint ambivalencies in the process, and the main motives for subsequent changes. Among the latter, one may emphasize the spirit of modern axiomatics, the situation of foundational insecurity in the 1920s, the resulting desire to find systems well-behaved from a proof-theoretical point of view, and the metatheoretical results of the 1930s. Not surprisingly, the mathematical and, more specifically, the foundational context in which Firs-Order-Logic matured will be seen to have played a primary role in its shaping.

Best,

Carl