Editing Modal logic (section)

== Syntax of modal operators ==

Modal logic differs from other kinds of logic in that it uses modal [[operator (mathematics)|operator]]s such as <math>\Box</math> and <math>\Diamond</math>. The former is conventionally read aloud as "necessarily", and can be used to represent notions such as moral or legal [[obligation]], [[knowledge]], [[determinism|historical inevitability]], among others. The latter is typically read as "possibly" and can be used to represent notions including [[permission (philosophy)|permission]], [[ability]], compatibility with [[evidence]]. While [[well-formed formula]]s of modal logic include non-modal formulas such as <math>P \land Q</math>, it also contains modal ones such as <math>\Box(P \land Q)</math>, <math> P \land \Box Q</math>, <math>\Box(\Diamond P \land \Diamond Q)</math>, and so on. 

Thus, the [[language (logic)|language]] <math>\mathcal{L}</math> of basic [[propositional logic]] can be [[recursive definition|defined recursively]] as follows.

#If <math>\phi</math> is an atomic formula, then <math>\phi</math> is a formula of <math>\mathcal{L}</math>.
#If <math>\phi</math> is a formula of <math>\mathcal{L}</math>, then <math>\neg \phi</math> is too.
#If <math>\phi</math> and <math>\psi</math> are formulas of <math>\mathcal{L}</math>, then <math>\phi \land \psi</math> is too.
#If <math>\phi</math> is a formula of <math>\mathcal{L}</math>, then <math>\Diamond \phi</math> is too.
#If <math>\phi</math> is a formula of <math>\mathcal{L}</math>, then <math>\Box \phi</math> is too.

Modal operators can be added to other kinds of logic by introducing rules analogous to #4 and #5 above. Modal [[predicate logic]] is one widely used variant which includes formulas such as <math>\forall x \Diamond P(x) </math>. In systems of modal logic where <math>\Box</math> and <math>\Diamond</math> are [[dual (mathematics)|duals]], <math>\Box \phi</math> can be taken as an abbreviation for <math>\neg \Diamond \neg \phi</math>, thus eliminating the need for a separate syntactic rule to introduce it. However, separate syntactic rules are necessary in systems where the two operators are not interdefinable.

Common notational variants include symbols such as <math>[K]</math> and <math>\langle K \rangle</math> in systems of modal logic used to represent knowledge and <math>[B]</math> and <math>\langle B \rangle</math> in those used to represent belief. These notations are particularly common in systems which use multiple modal operators simultaneously. For instance, a combined epistemic-deontic logic could use the formula <math>[K]\langle D \rangle P</math> read as "I know P is permitted". Systems of modal logic can include infinitely many modal operators distinguished by indices, i.e. <math>\Box_1</math>, <math>\Box_2</math>, <math>\Box_3</math>, and so on.