---
tag: ECYRK9C2ZT
kind: statement
role: proposition
name: The axioms of S5
sources:
  - wikipedia: en:12708106@r1375088158
    locator: S5 (modal logic), section The axioms of S5
    note: adopted
    mention: wikipedia://en/12708106#MPF6
    sections: [Statement]
inventory: [7RRQE7VXRA]
---

## Statement

The following makes use of the modal operators $\Box$ ("necessarily") and $\Diamond$ ("possibly").

S5 is characterized by the axioms:

- **K**: $\Box(A\to B)\to(\Box A\to\Box B)$;
- **T**: $\Box A \to A$,

and either:

- **5**: $\Diamond A\to \Box\Diamond A$;
- or both of the following:
  - **4**: $\Box A\to\Box\Box A$, and
  - **B**: $A\to\Box\Diamond A$.

The (5) axiom restricts the accessibility relation $R$ of the Kripke frame to be Euclidean, i.e. $(wRv \land wRu) \implies vRu$, thereby conflating necessity with possibility under idempotence.