The axioms of S5

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.