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. A formula's truth value at one possible world can depend on the truth values of other formulas at other accessible possible worlds. In particular, ◊ P {\displaystyle \Diamond P} is true at a world if P {\displaystyle P} is true at some accessible possible world, while ◻ P {\displaystyle \Box P} is true at a world if P {\displaystyle P} is true at every accessible possible world. A variety of proof systems exist which are sound and complete with respect to the...