Modal logic

Modal logic is a kind of logic used to represent statements about necessity and possibility. In philosophy and related fields

it is used as a tool for understanding concepts such as knowledge, obligation, and causation. For instance, in epistemic modal logic, the formula

◻

P

{\displaystyle \Box P}

can be used to represent the statement that

P

{\displaystyle P}

is known. In deontic modal logic, that same formula can represent that

P

{\displaystyle P}

is a moral obligation. Modal logic considers the inferences that modal statements give rise to. For instance, most epistemic modal logics treat the formula

◻

P

→

P

{\displaystyle \Box P\rightarrow P}

as a tautology, representing the principle that only true statements can count as knowledge. However, this formula is not a tautology in deontic modal logic, since what ought to be true can be false.

Modal logics are formal systems that include unary operators such as

◊

{\displaystyle \Diamond }

and

◻

{\displaystyle \Box }

, representing possibility and necessity respectively. For instance the modal formula

◊

P

{\displaystyle \Diamond P}

can be read as "possibly

P

{\displaystyle P}

" while

◻

P

{\displaystyle \Box P}

can be read as "necessarily

P

{\displaystyle P}

". In the standard relational semantics for modal logic, formulas are assigned truth values relative to a possible world.