---
tag: FDFQFWPM4X
kind: statement
role: proposition
name: Common modal axiom schemata
sources:
  - wikipedia: en:911960@r1374082003
    locator: Kripke semantics, section Semantics of modal logic > Correspondence and completeness > Common modal axiom schemata
    note: adopted
    mention: wikipedia://en/911960#WA4R
    sections: [Statement]
inventory: [XJNAXC8N0J]
---

## Statement

The following table lists common modal axioms together with their corresponding classes. The naming of the axioms often varies; Here, axiom **K** is named after Saul Kripke; axiom **T** is named after the truth axiom in epistemic logic; axiom **D** is named after deontic logic; axiom **B** is named after L. E. J. Brouwer; and axioms **4** and **5** are named based on C. I. Lewis's numbering of symbolic logic systems.

Axiom **K** can also be rewritten as $\Box [(A\to B)\land A]\to \Box B$, which logically establishes [[modus ponens|0Y0MQYST0M]] as a rule of inference in every possible world.

Note that for axiom **D**, $\Diamond A$ implicitly implies $\Diamond\top$, which means that for every possible world in the model, there is always at least one possible world accessible from it (which could be itself). This implicit implication $\Diamond A \rightarrow \Diamond\top$ is similar to the implicit implication by [[existential quantifier on the range of quantification|0FYQMKFE77]].