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→Syntax for modal operators: Correct a false statement. [(A is possible) and (B is possible)] does not entail that [(A&B) is possible], so the two are not equivalent. Tags: Mobile edit Mobile web edit |
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{{Short description|Logical operator in modal logic}}
A '''modal connective''' (or '''modal operator''') is a [[logical connective]] for [[modal logic]]. It is an [[binary function|operator]] which forms [[proposition]]s from propositions. In general, a modal operator has the "formal" property of being non-[[truth function|truth-functional]] in the following sense: The truth-value of composite formulae sometimes depend on factors other than the actual truth-value of their components. In the case of alethic modal logic, a modal operator can be said to be truth-functional in another sense, namely, that of being sensitive only to the distribution of truth-values across possible worlds, actual or not. Finally, a modal operator is "intuitively" characterized by expressing a modal attitude (such as [[Logical truth|necessity]], [[Logical possibility|possibility]], [[belief]], or [[knowledge]]) about the proposition to which the operator is applied.<ref name="garson">{{cite book |last1=Garson |first1=James |title=The Stanford Encyclopedia of Philosophy |date=2021 |publisher=Metaphysics Research Lab, Stanford University |edition=Summer 2021 |url=https://plato.stanford.edu/archives/sum2021/entries/logic-modal/ |access-date=5 February 2024 |chapter=Modal Logic}}</ref>
== Syntax for modal operators ==
{{unreferenced|section|date=February 2024}}
The syntax rules for modal operators <math>\Box</math> and <math>\Diamond</math> are very similar to those for universal and existential [[Quantifier (logic)|quantifiers]]; In fact, any formula with modal operators <math>\Box</math> and <math>\Diamond</math>, and the usual [[Logical connective|logical connectives]] in [[propositional calculus]] (<math> \land,\lor,\neg,\rightarrow,\leftrightarrow </math>) can be [[Rewriting#Logic|rewritten]] to a [[De dicto and de re|''de dicto'']] normal form, similar to [[prenex normal form]]. One major caveat: Whereas the universal and existential quantifiers only binds to the [[Propositional variable|propositional variables]] or the [[Predicate variable|predicate variables]] following the quantifiers, since the modal operators <math>\Box</math> and <math>\Diamond</math> quantifies over [[Accessibility relation|accessible]] [[Possible world|possible worlds]], they will bind to any formula in their [[Scope (logic)|scope]]. For example, <math>(\exists x (x^2 = 1)) \land (0 = y)</math> is logically equivalent to <math>\exists x (x^2 = 1\land 0 = y)</math>, but <math>(\Diamond (x^2 = 1)) \land (0 = y)</math> is not logically equivalent to <math>\Diamond (x^2 = 1\land 0 = y)</math>; Instead, <math>\Diamond (x^2 = 1\land 0 = y)</math>
When there are both modal operators and quantifiers in a formula, different order of an adjacent pair of modal operator and quantifier can lead to [[De dicto and de re#Representing de dicto and de re in modal logic|different semantic meanings]]; Also, when [[multimodal logic]] is involved, different order of an adjacent pair of modal operators can also lead to different semantic meanings.
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