nLab axiom K (modal logic)

Disambiguation: For axiom K as a principle of type theory, see axiom K (type theory)

Idea

In modal logic, axiom K, named after Saul Kripke, is a basic principle which almost all versions of propositional modal logic satisfy

$$K \colon \Box(p \to q) \to \Box p \to \Box q.$$

Developments

It is possible to consider a variant in dependent type theory (Spitters)

$$Dependent\; K \colon \Box \Pi_{y:A} B(y) \to \Pi_{x: \Box A} \Box B [open\; x/y].$$

References

• Bas Spitters et al., Modal Dependent Type Theory and Dependent Right Adjoints, (slides)