||It has been suggested that this article be merged with modal logic and S5 (modal logic). (Discuss) Proposed since November 2012.|
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 and its dual possibly . To deal with the new formulas of the form and , the following rules complement the inference apparatus of K:
- the distribution axiom
- necessitation rule
The logic M is K plus the axiom:
In S5 formulas of the form can be simplified to where is formed by any (finite) number of either or operators or both. The same stands for formulas of the form which can be simplified to .
- 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