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.

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 ($\vdash A \vee \neg{A}$) or the double-negation rule ($\neg\neg{A} \vdash A$). It may also be defined by starting with Gentzen's sequent calculus for classical logic (with $\neg$ but not $\bot$) and restricting to sequents $\Gamma \vdash \Delta$ where $\Delta$ may contain at most one formula, or by starting with sequent calculus with $\bot$ and restricting to such sequents where $\Delta$ must contain exactly one formula.

Properties

Double negation

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

Disjunction property

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

Classicality principles

The principle of excluded middle is not provable in intuitionistic logic, and if we assume it then the logic becomes classical logic. But there are other principles that are provable classically but not intuitionistically, but which are weaker than full PEM, such as

• various omniscience principles such as LPO and LLPO
• the double-negation shift
• de Morgan's law, i.e. weak excluded middle
• the world's simplest axiom of choice?

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, Fundamenta Mathematicae 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