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Intermediate logics are intermediate between

intuitionistic logic and classical logic in the sense

that they contain theorems that are not provable in intuitionistic logic, without giving rise to the whole of classical logic. Such logics are also called superintuitionistic or subclassical.

There are several different intermediate logics, often obtained by adding one or more axioms to intuitionistic logic. Examples of such axioms are:

the weak excluded middle: ¬¬ P ∨ ¬ P.
(P → Q) ∨ (Q → P), giving Gödel- Dummett logic, also called LC.
(¬ P → (Q ∨ R)) → (( ¬ P → Q) ∨ ( ¬ P → R))

The list is not complete.

The tools for studying intermediate logics are similar to those used for intuitionistic logic, such as Kripke semantics.

References

Toshio Umezawa. On logics intermediate between intuitionistic and classical predicate logic. Journal of Symbolic Logic, 24(2):141–153, June 1959.

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