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Intuitionism vs. classicism. A mathematical attack on classical logic. (English) Zbl 1358.03004
Studies in Theoretical Philosophy 2. Frankfurt am Main: Vittorio Klostermann (ISBN 978-3-465-03906-8/pbk). xvi, 269 p. (2015).
This book carefully examines the conflict between classical logic (stemming from Frege) and intuitionist logic (stemming from Brouwer). The arguments fall into four classes. (i) McCarty’s uniformity principle, de Swart’s continuity principle, and the like. Against these is the lack of any reason to accept the underlying existential claims. (ii) Prawitz’s theory of grounds, which can be modified to permit the provability of disjunctions that instantiate the law of excluded middle. (iii) Dummett’s manifestation argument, which needs to take into account that meaning and understanding are compositional. It is then shown that compositional manifestation of understanding is consistent with classical truth-conditional semantics. (iv) Proof-theoretic arguments, undermined by the presentation of harmonious collections of rules of inference for classical calculi. The findings suggest that Dummett’s hope for a meaning-theoretic resolution to the conflict is ill-founded, and that it may be that there can be no non-question-begging resolution.

03-03 History of mathematical logic and foundations
01A60 History of mathematics in the 20th century
03B20 Subsystems of classical logic (including intuitionistic logic)
03B05 Classical propositional logic
03B10 Classical first-order logic