Matti Eklund ON HOW LOGIC BECAME FIRST-ORDER

The logical systems within which Frege, Schröder, Russell, Zermelo and other early mathematical logicians worked were all higher-order. It was not until the 1910s that first-order logic was even distinguished as a subsystem of higher-order logic. As late as in the 1920s, higherorder quantification was still quite generally allowed: in fact, it does not seem as if any major logician, among non-intuitionists, except Thoralf Skolem restricted himself to first-order logic. Proofs were sometimes allowed to be infinite and infinitely long expressions were allowed in the languages that were used. Today, however, first-order logic has gained considerable dominance. Neither higher-order quantification nor infinite expressions and proofs are standardly allowed within logic. In textbooks of logic, what is taught is standard first-order logic. Zermelo-Fraenkel set theory and Peano arithmetic are almost always formalized in first-order languages. In this paper I pose the question: how did it happen that firstorder logic became so dominant? In particular, I am interested in why higher-order elements were excluded from logic. Thoralf Skolem’s work was undeniably of great importance for this development. Skolem presented the earliest first-order axiomatizations of set theory and arithmetic. Moreover, he first proved the LöwenheimSkolem theorem for standard first-order logic: and it is in part by virtue of the fact that this theorem holds for first-order logic that firstorder logic has a neat model theory. What I wish to do in this essay, however, is to refute the currently popularly held claim that Skolem also had quite a different kind of influence on the development toward