nLab
intuitionistic logic

Contents

Idea

Intuitionistic logic was introduced by Arend Heyting as a logic for Brouwer's intuitionistic mathematics. It applies more generally to constructive mathematics and so may also be called constructive logic.

Beware the terminological ambiguity: Some people insist that “intuitionistic logic” refers to Brouwerian intuitionism, which includes axioms that contradict classical logic; but other people use “intuitionistic” to mean the same as what in other contexts is called “constructive”, i.e. mathematics without the principle of excluded middle or the axiom of choice but nothing added that contradicts them. Some people (particularly material set theorists) use “constructive” to mean predicative constructive and “intuitionistic” to mean impredicative constructive.

Definition

Intuitionistic logic is most easily described as classical logic without the principle of excluded middle (A¬A\vdash A \vee \neg{A}) or the double-negation rule (¬¬AA\neg\neg{A} \vdash A). It may also be defined by starting with Gentzen's sequent calculus for classical logic (with ¬\neg but not \bot) and restricting to sequents ΓΔ\Gamma \vdash \Delta where Δ\Delta may contain at most one formula, or by starting with sequent calculus with \bot and restricting to such sequents where Δ\Delta must contain exactly one formula.

Properties

Double negation

The double negation translation says that a proposition PP is provable in classical logic precisely if its double negation ¬¬P\not \not P is provable in constructive logic

Disjunction property

Unlike classical logic, intuitionistic logic has the disjunction? and existence? properties: any proof of AB\vdash A \vee B must contain a proof of either A\vdash A or B\vdash B, and similarly any proof of x.F(x)\vdash \exists x.\,F(x) must construct a term tt and a proof of F(t)\vdash F(t). These properties are what justify our calling intuitionistic logic ‘constructive’.

On the other hand, (classical) Peano arithmetic is conservative over (intuitionistic) Heyting arithmetic when restricted to Π 1 0\Pi^0_1 formulas; that is, formulas of the form x:N.y:N.F(x,y)\forall x\colon N.\, \exists y\colon N.\, F(x,y). Roughly speaking, classical logic can be just as ‘constructive’ as intuitionistic logic as far as proving the totality of functions \mathbb{N} \to \mathbb{N} is concerned.

References

The observation that the poset of open subsets of a topological space (the internal logic of the sheaf topos) serves as a model for intuitionistic logic is apparently originally due to

  • Alfred Tarski, Der Aussagenkalkül und die Topologie, FundamentaMathemeticae 31 (1938), pp. 103–134.

A textbook account in the context of programming languages is in section 30 of

Revised on May 9, 2017 05:05:18 by Urs Schreiber (131.220.184.222)