Abstract

Brouwer's criticism of mathematical proofs making essential use of the tertium non datur had a surprisingly late response in logical circles. Among the diverse reactions in the mid 1920s and early 1930s, it is possible to delimit a coherent body of opinions on these questions: (1) whether Brouwer's denial of the tertium non datur meant only the abandonment of this classical law or, beyond that, the affirmation of its negation; (2) whether one or both of these alternatives were logically inconsistent; and (3) whether Brouwer's line of argument was forced to take resort to the very law it was designed to refute. The controversy centred around a series of articles by Marcel Barzin and Alfred Errera who fought against the intuitionistic critique, missed their victory because of conceptual confusions and fallacious reasoning, but emerged unconvinced from the debate in the late 1930s. The controversy is of interest to the historiography of formal logic since it stimulated the clarification not only of the concepts of formal validity, decidability, many-valued systems of logic and non-classical systems generally, but also of the distinction between object and meta-level, and between a formal system and its semantics. Most important, the debate, by putting pressure on the intuitionistic camp to make their ideas more precise, seems to have given the decisive motivation towards Heyting's answer to this demand by his axiomatization of intuitionistic logic in 1930