Recent Publications in Logic

Abstract

'... we doubt that anything has been gained by the formula language' (Hoppe); '... makes a strange and chilling impression with its long and short, vertical and horizontal strokes ... whether ... logic would gain considerably if it were to improve its tools and expand in this direction may justly be doubted' (MichaElis); '. . . indulges in the Japanese habit of writing vertically . .. monstrous waste of space' (Schr6der); '. . . cumbrous and inconvenient' (Venn). Frege's Begriffsschrift, now recognized as a crucially important pioneering work in the development of modern logic, was accorded a distinctly unfavourable reception by his contemporariesa fact which may usefully remind reviewers of their fallibility. Bynum's new edition' contains, besides an English translation of the Begriffsschrift, translations also of two papers in which Frege explains the motivation for his work and defends it against critics, and of the half-dozen reviews the Begriffsschrift received. Bynum has contributed a biographical sketch, an Introduction, and a comprehensive bibliography. Frege explains that his work in pursuit of his logicist programme, which was to provide arithmetic with firm foundations in the form of rigorous proofs of arithmetical theorems from purely logical axioms, was hindered by the deficiencies of ordinary language, which encouraged ambiguities and the smuggling in of implicit premisses. In the Begriffsschrift, therefore, he proposed a new, formal language, free of ambiguities, and with as few as possible rules of inference. The new language was a great advance on previous formalizations of logic: the propositional connectives (including material implication) were introduced, and the interdefinability relations between these connectives were recognized; quantifiers were invented-a crucial innovation, for the Aristotelian subject-predicate logic was, as Frege saw, too weak to express arithmetical concepts; an axiomatization (subsequently shown to be complete) of propositional and predicate logic was provided; and logical definitions were given of 'hereditary property' and 'ancestor'-a vital preliminary to the definition of arithmetical concepts in logical terms. Part of the explanation for the poor reception of this pioneering work lies, no doubt, in Frege's somewhat forbidding two-dimensional symbolism. He expressly intended to exploit the two-dimensionality of the writing