A Brief Introduction to the Intuitionistic Propositional Calculus

For a classical mathematician, mathematics consists of the discovery of preexisting mathematical truth. This understanding of mathematics is captured in Paul Erdös’s notion of “God’s Book of Mathematics,” which contains the best mathematical definitions, theorems, and proofs, and from which fortunate mathematicians are occasionally permitted read a page. Intuitionism takes the position that mathematical objects are mental constructions. Intuitionistic epistemology centers on proof, rather than truth. Thus, intuitionists analyze propositional combinations of mathematical statements in terms of what it takes to prove them, and a proof of φ∧ψ consists of a proof of φ together with a proof of ψ, a proof of φ ∨ ψ consists of a proof of φ or a proof of ψ, while a proof of φ ⇒ ψ consists of an algorithm that converts proofs of φ into proofs of ψ. For an intuitionist, a propositional formula is a tautology if it can be proven, e.g., α ⇒ α is an intuitionistic tautology because to convert a proof p of α into a proof of α we simply return p. The classical mathematician believes in the soundness of mathematical reasoning, i.e., that everything provable is true, and therefore all intuitionistic tautologies are also classical tautologies. The intuitionistic mathematician has no reason, however, to believe that everything that is true is provable (since he does not possess in his mind a construction that enables him to pass from an arbitrary but true mathematical statement φ to an intuitionistic proof of φ). Indeed, it is not difficult to give propositional formulae that are classical, but not intuitionistic, tautologies. The standard example of a classical tautology that is not an intuitionistic tautology is the law of the excluded middle, α ∨ ¬α. The law of the excluded middle is true classically because α must be either true or false,