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| {{Merge|modal logic|S5 (modal logic)|date=November 2012}}
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| == Axiom (5) ==
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| '''Axiom (5)''' extends the [[modal logic]] ''M'', to form the [[modal logic]] '''[[S5 (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 [[tautology (logic)|tautologies]], and inference apparatus with [[substitution (logic)|substitution]] and [[modus ponens]], but extending the syntax with the modal operator ''necessarily'' <math>\Box</math> and its dual ''possibly'' <math>\Diamond</math>.
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| To deal with the new formulas of the form <math>\Box \varphi</math> and <math>\Diamond \varphi</math>, the following rules complement the inference apparatus of '''K''':
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| : the distribution axiom <math>\Box(\varphi \implies \psi) \implies (\Box \varphi \implies \Box \psi)</math>
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| : necessitation rule <math>\frac{\varphi}{\ \Box \varphi\ }</math>
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| The logic ''M'' is '''K''' plus the axiom:
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| : (M) <math> \Box \varphi \implies \varphi </math>
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| which restricts the [[accessibility relation]] of the [[Kripke frame]] to be reflexive.
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| The [[modal logic]] '''[[S5 (modal logic)|S5]]''' is obtained by adding the [[axiom]]:
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| : (5) <math>\Diamond \varphi\implies\Box\Diamond \varphi</math>
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| The (5) axiom restricts the [[accessibility relation]] <math>R</math>, of the [[Kripke frame]] to be euclidean, i.e. <math>(wRv \land wRu) \implies vRu </math>.
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| In '''[[S5 (modal logic)|S5]]''' formulas of the form <math>OOO\ldots\Box\varphi</math> can be simplified to <math>\Box\varphi</math> where <math>OOO\ldots</math> is formed by any (finite) number of either <math>\Box</math> or <math>\Diamond</math> operators or both. The same stands for formulas of the form <math>OOO\ldots\Diamond\varphi</math> which can be simplified to <math>\Diamond\varphi</math>.
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| ==References==
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| *Chellas, B. F. (1980) ''Modal Logic: An Introduction''. Cambridge University Press. ISBN 0-521-22476-4
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| *Hughes, G. E., and Cresswell, M. J. (1996) ''A New Introduction to Modal Logic''. Routledge. ISBN 0-415-12599-5
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| ==External links==
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| * http://plato.stanford.edu/entries/logic-modal/
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| [[Category:Modal logic]]
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| [[Category:Axioms of modal logic]]
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| [[Category:Mathematical axioms]]
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