1. The gradual development of the conception of first-order logic is an intricate subject of which the final truth has not yet been told. Meanwhile, you can have a glimpse of the problems from studies like Gregory H. Moore, “The Emergence of First-Order Logic”, in William Aspray and Philip Kitcher, editors, History and Philosophy of Modern Mathematics (Minnesota Studies in the Philosophy of Science, vol. 11, University of Minnesota Press, Minneapolis, 1988, pp.95–135). Moore’s paper is to be read with caution, however, for he is unaware of some of the most important conceptual points concerning the idea of first-order logic. For one thing, he does not even mention Henkin’s distinction between standard and non-standard interpretations of higher-order logic. Yet it is possible to reconstruct a higher-order language, with a suitable non-standard interpretation, as a many-sorted “first-order” language, as far as logic is concerned. Cf. also below, especially sec. 16.
2. Cf., e.g., Stewart Shapiro, “Second-order Languages and Mathematical Practice”, Journal of Symbolic Logic
50 (1985), pp. 714–42; Georg Kreisel, “Informal Rigor and Completeness Proofs”, in Imre Lakatos, editor, Problems in the Philosophy of Mathematics, North-Holland, Amsterdam, 1967, pp. 138–86.
3. See here Jaakko Hintikka, “Logical Form and Linguistic Theory”, in Alex George, editor, Reflections on Chomsky, Basil Blackwell, Oxford, 1989, pp. 41–57.
4. Norbert Hornstein, Logic as Grammar, The MIT Press, Cambridge, 1984; cf. Robert May, Logical Form, The MIT Press, Cambridge, 1985.
5. See note 3 above and also Jaakko Hintikka and Gabriel Sandu, On the Methodology of Linguistics: A Case Study, Basil Blackwell, Oxford, 1991.