# Axiom S5

Axiom (5) extends the modal logic M, to form the modal logic S5. Which in turn, consists of modal logic called K, in honour of Saul Kripke. It is the most basic modal logic, is formed with propositional calculus formulas and tautologies, and inference apparatus with substitution and modus ponens, but extending the syntax with the modal operator necessarily $\Box$ and its dual possibly $\Diamond$. To deal with the new formulas of the form $\Box \varphi$ and $\Diamond \varphi$, the following rules complement the inference apparatus of K:

the distribution axiom $\Box(\varphi \implies \psi) \implies (\Box \varphi \implies \Box \psi)$
necessitation rule $\frac{\varphi}{\ \Box \varphi\ }$

The logic M is K plus the axiom:

(M) $\Box \varphi \implies \varphi$

which restricts the accessibility relation of the Kripke frame to be reflexive.

The modal logic S5 is obtained by adding the axiom:

(5) $\Diamond \varphi\implies\Box\Diamond \varphi$

The (5) axiom restricts the accessibility relation $R$, of the Kripke frame to be euclidean, i.e. $(wRv \land wRu) \implies vRu$.

In S5 formulas of the form $OOO\ldots\Box\varphi$ can be simplified to $\Box\varphi$ where $OOO\ldots$ is formed by any (finite) number of either $\Box$ or $\Diamond$ operators or both. The same stands for formulas of the form $OOO\ldots\Diamond\varphi$ which can be simplified to $\Diamond\varphi$.

## References

• Chellas, B. F. (1980) Modal Logic: An Introduction. Cambridge University Press. ISBN 0-521-22476-4
• Hughes, G. E., and Cresswell, M. J. (1996) A New Introduction to Modal Logic. Routledge. ISBN 0-415-12599-5