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 it's 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:
- (M)
which restricts the accessibility relation of the Kripke frame to be reflexive.
The modal logic S5 is obtained by adding the axiom:
- (5)
The (5) axiom restricts the accessibility relation, of the Kripke frame to be euclidean, i.e. .
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 .
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