basic constructions:
strong axioms
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.
Intuitionistic logic is most easily described as classical logic without the principle of the 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.
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.
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
A textbook account in the context of programming languages is in section 30 of