Editing Rule of inference (section)

=== Modal logics ===
{{main|Modal logic}}
Modal logics are formal systems that extend propositional logic and first-order logic with additional logical operators. Alethic modal logic introduces the operator <math>\Diamond</math> to express that something is possible and the operator <math>\Box</math> to express that something is necessary. For example, if the <math>P</math> means that "Parvati works", then <math>\Diamond P</math> means that "It is possible that Parvati works" while <math>\Box P</math> means that "It is necessary that Parvati works". These two operators are related by a rule of replacement stating that <math>\Box P</math> is equivalent to <math>\lnot \Diamond \lnot P</math>. In other words: if something is necessarily true then it is not possible that it is not true. Further rules of inference include the necessitation rule, which asserts that a statement is necessarily true if it is provable in a formal system without any additional premises, and the distribution axiom, which allows one to derive <math>\Diamond P \to \Diamond Q </math> from <math>\Diamond (P \to Q)</math>. These rules of inference belong to system K, a weak form of modal logic with only the most basic rules of inference. Many formal systems of alethic modal logic include additional rules of inference, such as system T, which allows one to deduce <math>P</math> from <math>\Box P</math>.<ref>{{multiref | {{harvnb|Garson|2024|loc=Lead section, § 2. Modal Logics}} | {{harvnb|Sider|2010|pp=[https://books.google.com/books?id=-KkPEAAAQBAJ&pg=PA171 171–176, 286–287]}} }}</ref>

Non-alethic systems of modal logic introduce operators that behave like <math>\Diamond</math> and <math>\Box</math> in alethic modal logic, following similar rules of inference but with different meanings. [[Deontic logic]] is one type of non-alethic logic. It uses the operator <math>P</math> to express that an action is permitted and the operator <math>O</math> to express that an action is required, where <math>P</math> behaves similarly to <math>\Diamond</math> and <math>O</math> behaves similarly to <math>\Box</math>. For instance, the rule of replacement in alethic modal logic asserting that <math>\Box Q</math> is equivalent to <math>\lnot \Diamond \lnot Q</math> also applies to deontic logic. As a result, one can deduce from <math>O Q</math> (e.g. Quinn has an obligation to help) that <math>\lnot P \lnot Q</math> (e.g. Quinn is not permitted not to help).<ref>{{harvnb|Garson|2024|loc=§ 3. Deontic Logics}}</ref> Other systems of modal logic include [[Temporal logic|temporal modal logic]], which has operators for what is always or sometimes the case, as well as [[Doxastic logic|doxastic]] and [[epistemic modal logic]]s, which have operators for what people believe and know.<ref>{{multiref | {{harvnb|Garson|2024|loc=§ 1. What is Modal Logic?, § 4. Temporal Logics}} | {{harvnb|Sider|2010|pp=[https://books.google.com/books?id=-KkPEAAAQBAJ&pg=PA234 234–242]}} }}</ref>