User:Marc Schroeder/sandbox4

Constructive logic, refers to systems of symbolic logic that differ from classical logic by more closely mirroring the notion of constructive proof. Constructive logics do not assume the law of the excluded middle and double negation elimination, which are fundamental inference rules in classical logic.

There exists a variety of constructive logics, the most prominent of which is intuitionistic logic, based on L.E.J. Brouwer's intuitionism, formalized by Arend Heyting.

Other logics have also been said to be constructive, like, for instance,
 * the minimal logic of Ingebrigt Johansson,
 * the logics of David Nelsons.

Paraconsistent logic
Intuitionistic logic is related by duality to a paraconsistent logic known as Brazilian, anti-intuitionistic or dual-intuitionistic logic. The subsystem of intuitionistic logic with the FALSE axiom removed is known as minimal logic and some differences have been elaborated on above.

Intermediate logics
Any finite Heyting algebra that is not equivalent to a Boolean algebra defines (semantically) an intermediate logic. On the other hand, validity of formulae in pure intuitionistic logic is not tied to any individual Heyting algebra but relates to any and all Heyting algebras at the same time.

For a schema not involving negations, consider the classically valid $$(A\to B)\lor(B\to A)$$. Adopting this over intuitionistic logic gives the intermediate logic called Gödel-Dummett logic.

Many-valued logic
Kurt Gödel's work involving many-valued logic showed in 1932 that intuitionistic logic is not a finite-valued logic. (See the section titled Heyting algebra semantics above for an infinite-valued logic interpretation of intuitionistic logic.)