This collection of essays examines logic and its philosophy. The author investigates the nature of logic not only by describing its properties but also by showing philosophical applications of logical concepts and structures. He evaluates what logic is and analyzes among other aspects the relations of logic and language, the status of identity, bivalence, proof, truth, constructivism, and metamathematics. With examples concerning the application of logic to philosophy, he also covers semantic loops, the epistemic discourse, the normative discourse, paradoxes, properties of truth, truth-making as well as theology, being and logical determinism. The author concludes with a philosophical reflection on nothingness and its modelling.
XII. Constructivism and Metamathematics
XIIConstructivism and Metamathematics
Constructivism in the foundations of mathematics comprises a fair variety of views about mathematical theories and their properties. Constructivism is not a uniform proposal in the foundations of mathematics (see Beeson 2004, Bridges, Richman 1997, Troelstra, van Dalen 1988). It comprises – even disregarding some older views as, for instance, Kronecker’s account of mathematical reasoning or French semi-intuitionists – many special versions, in particular, intuitionism (as represented by Brouwer and his followers), Russian constructivism, Bishop’s constructivism, computable analysis, constructive analysis, predicativism, Lorenzen’s operationalism, ultrafinitism (ultra-intuitionism), or constructive type-theory. Every brand of constructivism offers a criterion C of constructivity. In my further remarks I will concentrate on intuitionism and Russian constructivism.
Assume that C refers to a given fixed criterion of constructivity. Such a criterion always generates a kind of logic considered as constructive. For example, C can postulate logic in which the law of excluded middle does not hold – that is, it can postulate the standard intuitionistic logic – or recommend the Markov principle as admissible (the symbol ¬ expresses negation, but its meaning depends on the accepted system; in the case of the Markov principle, ¬ formalizes classical negation):
(MP) ∀x(A(x) ∨ ¬ A(x)) ∧ ¬∀x¬A(x) ⇒ ∃xA(x).18
Consequently, a theory T is constructive if and only if it satisfies C. Typically, constructive systems, as formally based on a kind of constructive logic, are contrasted with theories having classical logic as their logical foundation.
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