The Emergence of First-Order Logic

THE EMERGENCE OF FIRST-ORDER LOGIC 109
else its contradictory is so deducible; it must be admitted, however,
that our mastery of the processes of logical deduction is not yet, and
possibly never can be, sufficiently complete to justify this assertion.
(1905, 210)
Thus Huntington was convinced that any categorical axiom system is
deductively complete: every sentence expressible in the system is either
provable or disprovable. On the other hand, Veblen was aware, however
fleetingly, of the possibility that in a categorical axiom system there might
exist propositions true in the only model of the system but unprovable
in the system itself. 7 In 1931 Godel's Incompleteness Theorem would show
that this possibility was realized, in second-order logic, for every categorical
axiom system rich enough to include the arithmetic of the natural numbers.
7. Peano and Russell: Toward Principia Mathematica
In (1888) Guiseppe Peano began his work in logic by describing that
of Boole (1854) and Schroder (1877). Frege, in a letter to Peano probably
written in 1894, described Peano as a follower of Boole, but one what
had gone further than Boole by adding a symbol for generalization. 8 Peano
introduced this symbol in (1889), in the form a3x>y,.. .b for "whatever
x, y,... may be, b is deduced from a." Thus he introduced the notion
of universal quantifier (independently of Frege and Peirce), although he
did not separate it from his symbol D for implication (1889, section II).
Peano's work gave no indication of levels of logic. In particular, he
expressed "x is a positive integer" by x e TV and so felt no need to quantify
over predicates such as N. On the other hand, his Peano Postulates
were essentially a second-order axiomatization for the positive integers,
since these postulates included the Principle of Mathematical Induction.
Beginning in 1900, Peano's formal language was adopted and extended
by Bertrand Russell, and thereby achieved a longevity denied by Frege's.
When he became an advocate of Peano's logic, Russell entered an entirely
new phase of his development. In contrast to his earlier and more
traditional views, Russell now accepted Cantor's transfinite ordinal and
cardinal numbers. Then in May 1901 he discovered what became known
as Russell's Paradox and, after trying sporadically to solve it for a year,
wrote to Frege about it on 16 June 1902. Frege was devastated. Although
his original (1879) system of logic was not threatened, he realized that
the system developed in his Fundamental Laws (1893, 1903a) was in grave