The Logic of Brouwer and Heyting

Intuitionistic logic consists of the principles of reasoning which were used informally by L. E. J. Brouwer, formalized by A. Heyting (also partially by V. Glivenko), interpreted by A. Kolmogorov, and studied by G. Gentzen and K. Gödel during the first third of the twentieth century. Formally, intuitionistic first-order predicate logic is a proper subsystem of classical logic, obtained by replacing the law of excluded middle A ∨ ¬A by ¬A ⊃ (A ⊃ B); it has infinitely many distinct consistent axiomatic extensions, each necessarily contained in classical logic (which is axiomatically complete). However, intuitionistic second-order logic is inconsistent with classical second-order logic. In terms of expressibility, the intuitionistic logic and language are richer than the classical; as Gödel and Gentzen showed, classical first-order arithmetic can be faithfully translated into the negative fragment of intuitionistic arithmetic, establishing proof-theoretical equivalence and clarifying the distinction between the classical and constructive consequences of mathematical axioms. The logic of Brouwer and Heyting is effective. The conclusion of an intuitionistic derivation holds with the same degree of constructivity as the premises. Any proof of a disjunction of two statements can be effectively transformed into a proof of one of the disjuncts, while any proof of an existential statement contains an effective prescription for finding a witness. The negation of a statement is interpreted as asserting that the statement is not merely false but absurd, i.e., leads to a contradiction. Brouwer objected to the general law of excluded middle as claiming a priori that every mathematical problem has a solution, and to the general law ¬¬A ⊃ A of double negation as asserting that every consistent mathematical statement holds. Brouwer called “the First Act of Intuitionism” the separation of mathematics (an exclusively mental activity) from language. For him, all mathematical objects (including proofs) were mental constructions carried out by individual mathematicians. This insight together with “the intuition of the bare two-oneness” guided the creation of intuitionistic logic and arithmetic, with full induction and a theory of the rational numbers. It was limited, by Brouwer’s early insistence on predicative definability, to the construction of entities which were finitely specifiable and thus at most denumerably infinite in number; this seemed to exclude the possibility of a good intuitionistic theory of the continuum. A later insight led Brouwer to “the Second Act of Intuitionism,” which accepted infinitely proceeding sequences of independent choices of objects already constructed (choice sequences of natural numbers, for example) as legitimate mathematical entities. He succeeded in extracting positive information about the structure of the continuum from the fact that an individual choice sequence may be known only by its finite approximations. His “Bar Theorem” (actually a new mathematical axiom) essentially accepted induction up to any denumerable ordinal, significantly extending the intuitionistically available part of classical arithmetic and analysis. His Continuity Principle, which contradicted classical logic (but not classical first-order arithmetic), made possible the development of a rich al-