Axiom S5

The modal axiom 5 is a sentence in the language of propositional modal logic, which states that if possibly p, then necessarily possibly p. In the standard notation:

$\Diamond p \to \Box\Diamond p$

When added to the distribution axiom K (i.e. $\Box(p \to q) \to (\Box p \to \Box q)$ ) and the reflexivity axiom T (i.e. $\Box p \to p$; sometimes called M), in the presence of the Necessitation Rule (if $\vdash p$, then $\vdash\Box p$), it yields the widely used modal logic S5.

Among other desirable features, S5 allows the reduction of strings of modal operators to the final element of the string (i.e. $\vdash \triangle_{1}\triangle_{2}...\triangle_{n}p \leftrightarrow \triangle_{n}p$, where each $\triangle_{i}$ stands for either a $\Diamond$ or a $\Box$). For a detailed discussion of the properties of S5, consult any standard introductory text in modal logic, e.g. Chellas (1980) or Hughes & Creswell (1996).

Note also that there exists a number of alternative axiomatizations of S5 which do not employ the axiom 5 (of course, 5 remains a theorem of S5 independently of a given axiomatization). For example, in the axiomatization described above, 5 can be replaced with its dual, $\Diamond\Box p \to \Box p$.

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

External links

• http://home.utah.edu/~nahaj/logic/structures/axioms/CMpLMp.html
• http://plato.stanford.edu/entries/logic-modal/