Intuitionistic Logic

Intuitionistic logic is a system of logic in which proofs are required to be constructive. Its propositional and predicate forms modify the proof theories of classical propositional calculus and first-order logic, respectively.

Intuitionistic propositional logic can be described as classical propositional calculus in which the axiom schema

(1)

is replaced by

(2)

Similarly, intuitionistic predicate logic extends intuitionistic propositional logic to formulas containing quantifiers.

Every formula provable in intuitionistic logic is also provable in classical logic, but some basic theorems of classical logic do not hold in intuitionistic logic. For example, the law of the excluded middle

(3)

does not hold in intuitionistic propositional logic.

Here are some examples of propositional formulas that are not provable in intuitionistic propositional logic:

(4)

(5)

Assuming nonempty domains and that is not free in , examples of first-order formulas that are not provable in intuitionistic predicate logic are

(6)

(7)

Truth tables for propositional connectives define the interpretation of classical propositional calculus over the domain of two elements: true and false. This interpretation is a model of classical propositional calculus, that is, tautologies and only tautologies are formal theorems. In contrast, no single finite algebra characterizes intuitionistic propositional logic, although each unprovable formula fails in some finite Heyting algebra.

Heyting algebras give algebraic semantics for intuitionistic propositional logic. Their meet, join, and implication interpret conjunction, disjunction, and implication, while 0 and 1 interpret false and true, respectively.

Classical proof by contradiction does not in general establish in intuitionistic logic. A contradiction derived from establishes only . Intuitionistic proofs are constructive, which is reflected in the following properties. Intuitionistic propositional logic has the disjunction property: If is provable in intuitionistic propositional calculus, then either or is provable in intuitionistic propositional calculus. Intuitionistic predicate logic has the existence property: In a language with at least one closed term, if is a formula without free variables and is provable in intuitionistic predicate logic, then there is a closed term such that is provable in intuitionistic predicate logic.

The deduction theorem holds in intuitionistic propositional and predicate logics. Glivenko's theorem (Glivenko 1929) captures the relation between intuitionistic and classical logics: If is provable in classical propositional calculus, then is provable in intuitionistic propositional calculus. This theorem does not extend to intuitionistic predicate logic.