The philosopher Rudolf Carnap, although not himself an originator of mathematical advances in logic, was much involved in the development of the subject. He was the most important and deepest philosopher of the Vienna Circle of logical positivists, or, to use the label Carnap later preferred, logical empiricists. It was Carnap who gave the most fully developed and sophisticated form to the linguistic doctrine of logical and mathematical truth: the view that the truths of mathematics and logic do not describe (...) some Platonistic realm, but rather are artifacts of the way we establish a language in which to speak of the factual, empirical world, fallouts of the representational capacity of language. Carnap was also the thinker who, after Russell, most emphasized the importance of modern logic, and the distinctive advances it enables in the foundations of mathematics, to contemporary philosophy. It was through Carnap's urgings, abetted by Hans Hahn, once Carnap arrived in Vienna as Privatdozent in philosophy in 1926, that the Vienna Circle began to take logic seriously and that positivist philosophy began to grapple with the question of how an account of mathematics compatible with empiricism can be given.A particular facet of Carnap's influence is not widely appreciated: it was Carnap who introduced Kurt Gödel to logic, in the serious sense. Although Gödel seems to have attended a course of Schlick's on philosophy of mathematics in 1925–26, his second year at the University, he did not at that time pursue logic further, nor did the seminar leave much of a trace on him. In the early summer of 1928, however, Carnap gave two lectures to the Circle which Gödel attended, or so I surmise. At these occasions, Carnap presented material from his manuscript treatise, Untersuchungen zur allgemeinen Axiomatik, that is, “Investigations into general axiomatics”, which dealt with questions of consistency, completeness and categoricity. Carnap later circulated this material to various people including Gödel. (shrink)
This text provides a straightforward, lively but rigorous, introduction to truth-functional and predicate logic, complete with lucid examples and incisive exercises, for which Warren Goldfarb is renowned.
What we call the Hilbert‐Bernays (HB) Theorem establishes that for any satisfiable first‐order quantificational schema S, there are expressions of elementary arithmetic that yield a true sentence of arithmetic when they are substituted for the predicate letters in S. Our goals here are, first, to explain and defend W. V. Quine's claim that the HB theorem licenses us to define the first‐order logical validity of a schema in terms of predicate substitution; second, to clarify the theorem by sketching an accessible (...) and illuminating new proof of it; and, third, to explain how Quine's substitutional definition of logical notions can be modified and extended in ways that make it more attractive to contemporary logicians. (shrink)
Frege and Russell were the most significant influences on the young Wittgenstein, but the relative weight of their impacts is less clear. Some interpreters have claimed for Frege an influence far surpassing that of Russell. I cast doubt on this claim, by reviewing the evidence we have of Wittgenstein's pre‐Tractatus understanding of Frege. Wittgenstein did eventually come to some views more like Frege's than Russell's; I suggest it was his own thinking rather than direct influence from Frege that led him (...) in this direction. (shrink)