Interface:Axioms of K modal logic

Modal logic builds on propositional logic and adds two concepts, necessity and possibility. The formula  means that   is necessarily true and   means that   is possibly true. They behave a little bit like the quantifiers ∀ and ∃ of predicate logic, although they don't quantify a particular variable.

Here we present axioms for the weakest commonly studied modal logic, called K.

 param (PROPOSITIONAL Interface:Classical_propositional_calculus )

term (formula (□ formula))

var (formula p q) 

The rule of necessitation is analogous to the axiom of generalization of predicate logic.  stmt (necessitate (p) (□ p)) 

We can distribute necessity across an implication.  stmt (NecessityImplication  ((□ (p → q)) → ((□ p) → (□ q)))) 

We define possibility in terms of necessity.  def ((◊ p) (¬ (□ (¬ p)))) 

Referenced works

 * George Edward Hughes, M. J. Cresswell (1996), A new introduction to modal logic, ISBN 978-0415126007.
 * Garson, James (first published Tue Feb 29, 2000; substantive revision Fri Oct 2, 2009) "Modal Logic", Stanford Encyclopedia of Philosophy, Edward N. Zalta (ed.). Fall 2009 version.