nLab
intuitionistic logic

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.

Definition

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

Properties

Unlike classical logic, intuitionistic logic has the disjunction? and existence? properties: any proof of $⊢A\vee B$ must contain a proof of either $⊢A$ or $⊢B$, and similarly any proof of $⊢\exists x.F(x)$ must construct a term $t$ and a proof of $⊢F(t)$. 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 ${\Pi }_1^0$ formulas; that is, formulas of the form $\forall x:N.\exists y:N.F(x,y)$. 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.

Related concepts

• BHK interpretation

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

• Robert Harper, Practical Foundations for Programming Languages