Editing Modal logic (section)

{{Short description|Type of formal logic}}
{{Use dmy dates|date=May 2024}}
'''Modal logic''' is a kind of [[logic]] used to represent statements about [[Modality (natural language)|necessity and possibility]]. In [[philosophy]] and related fields
it is used as a tool for understanding concepts such as [[knowledge]], [[obligation]], and [[causality|causation]]. For instance, in [[epistemic modal logic]], the [[well-formed_formula|formula]] <math>\Box P</math> can be used to represent the statement that <math>P</math> is known. In [[deontic modal logic]], that same formula can represent that <math>P</math> is a moral obligation. Modal logic considers the inferences that modal statements give rise to. For instance, most epistemic modal logics treat the formula <math>\Box P \rightarrow P</math> as a [[Tautology_(logic)|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 system]]s that include [[unary operation|unary]] operators such as <math>\Diamond</math> and <math>\Box</math>, representing possibility and necessity respectively. For instance the modal formula <math>\Diamond P</math> can be read as "possibly <math>P</math>" while <math>\Box P</math> can be read as "necessarily <math>P</math>". In the standard [[Kripke semantics|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 ''[[Accessibility relation|accessible]]'' [[possible worlds]]. In particular, <math>\Diamond P</math> is true at a world if <math>P</math> is true at ''some'' accessible possible world, while <math>\Box P</math> is true at a world if <math>P</math> is true at ''every'' accessible possible world. A variety of proof systems exist which are sound and complete with respect to the semantics one gets by restricting the accessibility relation. For instance, the deontic modal logic '''D''' is sound and complete if one requires the accessibility relation to be [[serial relation|serial]].

While the intuition behind modal logic dates back to antiquity, the first modal [[axiomatic system]]s were developed by [[C. I. Lewis]] in 1912. The now-standard relational semantics emerged in the mid twentieth century from work by [[Arthur Prior]], [[Jaakko Hintikka]], and [[Saul Kripke]]. Recent developments include alternative [[topology|topological]] semantics such as [[neighborhood semantics]] as well as applications of the relational semantics beyond its original philosophical motivation.<ref name="bluebible">{{cite book |last1= Blackburn |first1= Patrick |last2= de Rijke |first2= Maarten |last3= Venema|first3= Yde |date=2001 |title= Modal Logic |url=https://books.google.com/books?id=pbb_Asgoq0oC&dq=rijlke%20blackburn%20venema%20modal%20logic&pg=PP1 |series=Cambridge Tracts in Theoretical Computer Science |publisher=Cambridge University Press|isbn= 9780521527149 }}</ref> Such applications include [[game theory]],<ref name="openminds">{{cite book |last= van Benthem |first= Johan |date=2010 |title= Modal Logic for Open Minds |url= https://pdfs.semanticscholar.org/9bea/866c143326aeb700c20165a933f583b16a46.pdf |archive-url= https://web.archive.org/web/20200219165057/https://pdfs.semanticscholar.org/9bea/866c143326aeb700c20165a933f583b16a46.pdf |url-status= dead |archive-date= 2020-02-19 |publisher= CSLI|s2cid= 62162288 }}</ref> [[Moral theory|moral]] and [[legal theory]],<ref name="openminds" /> [[web design]],<ref name="openminds" /> [[Multiverse (set theory)|multiverse-based set theory]],<ref>{{cite journal |last1= Hamkins |first1= Joel|date=2012 |title=The set-theoretic multiverse |journal= The Review of Symbolic Logic |volume=5 |issue=3|pages= 416–449|doi=10.1017/S1755020311000359|arxiv= 1108.4223|s2cid= 33807508}}</ref> and [[social epistemology]].<ref>{{cite journal |last1=Baltag |first1=Alexandru |last2=Christoff |first2=Zoe |last3=Rendsvig |first3=Rasmus |last4=Smets |first4=Sonja |date=2019 |title= Dynamic Epistemic Logics of Diffusion and Prediction in Social Networks. |journal=Studia Logica |volume=107 |issue=3 |pages=489–531 |doi=10.1007/s11225-018-9804-x |s2cid=13968166 |doi-access=free }}</ref>

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