Interface:K modal logic

Here are some theorems designed to illustrate modal logic. We build on classical propositional logic. The theorems here hold in the weakest commonly studied modal logic, called K.

We add two concepts to propositional logic, 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.  param (PROPOSITIONAL Interface:Classical_propositional_calculus ) term (formula (□ formula)) term (formula (◊ formula))

var (formula p q) 

Necessity can be collected across a disjunction and collected or distributed across a conjunction  stmt (NecessityDisjunctionCollection  (((□ p) ∨ (□ q)) → (□ (p ∨ q)))) stmt (NecessityConjunction  ((□ (p ∧ q)) ↔ ((□ p) ∧ (□ q)))) 

The converse of  does not hold but there is result which is a bit like its converse.  stmt (NecessityDisjunctionDistribution  ((□ (p ∨ q)) → ((□ p) ∨ (◊ q)))) 

Possibility can be distributed across a conjunction.  stmt (PossibilityConjunction  ((◊ (p ∧ q)) → ((◊ p) ∧ (◊ q)))) 

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

Here is a rule form of and the corresponding rule for the biconditional.  stmt (addNecessity ((p → q)) ((□ p) → (□ q))) stmt (buildNecessity ((p ↔ q)) ((□ p) ↔ (□ q))) 

Similar rules hold for possiblity.  stmt (addPossibility ((p → q)) ((◊ p) → (◊ q))) stmt (buildPossibility ((p ↔ q)) ((◊ p) ↔ (◊ q))) 

Possibility of an implication is equivalent to an implication involving one necessity and one possibility.  stmt (PossibilityNecessityImplication  ((◊ (p → q)) ↔ ((□ p) → (◊ q)))) 

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

Necessity and possibility are related by negations  stmt (NecessityPossibility  ((□ p) ↔ (¬ (◊ (¬ p))))) stmt (PossibilityNecessity  ((◊ p) ↔ (¬ (□ (¬ p)))))

stmt (NegationNecessity  ((¬ (□ p)) ↔ (◊ (¬ p)))) stmt (NegationPossibility  ((¬ (◊ p)) ↔ (□ (¬ p)))) </jh>

Possibility distributes or collects across a disjunction.  stmt (PossibilityDisjunction  ((◊ (p ∨ q)) ↔ ((◊ p) ∨ (◊ q)))) </jh>

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.