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{{Short description|Logical operator in modal logic}}

A '''modal operator''' is a [[logical connective]], in the language of a [[modal logic]], which forms propositions from propositions. In general, a modal operator is ''formally'' characterised by being non-[[truth function|truth-functional]], and ''intuitively'' characterised by expressing a modal attitude (such as necessity, possibility, belief, or knowledge) towards the proposition which it is applied to.

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 ==

In literary and fiction theory, the concept of '''modal operators''' has been explored by Lubomir [[Dolezel]] in Heterocosmica (1998), a book that articulates a complete theory of literary fiction based on the idea of [[possible worlds]]. Dolezel works with the concept of modalities that play the crucial role in ''formative operation'', i.e. in shaping narrative worlds into orders that have the potential to produce stories. Based on the theories of [[modal logic]], Dolezel introduces a set of modal systems that are appropriated for fictional [[semantics]], expanding on the table used by [[Georg Henrik von Wright]] (1968). There are four kinds of '''modal operators''' that function in the modal systems: [[alethic]], [[deontic]], [[axiological]] and [[epistemic]].

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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> logically entails <math>(\Diamond (x^2 = 1)) \land \Diamond(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.

== Examples ==

* In the [[alethic moods|alethic]] [[modal logic]] of [[Clarence Irving Lewis|C.I. Lewis]], the modal operator <math>\Box</math> expresses necessity: if the proposition ''A'' is read as "it is true that ''A'' holds", the proposition <math>\Box</math>''A'' is read as "it is ''necessarily'' true that ''A'' holds".

* In the [[tense logic]] (more commonly now called [[temporal logic]]) of [[Arthur Prior|A.N. Prior]], the proposition ''A'' is read as "''A'' is true at the present time"; '''F''' ''A'', as "''A'' will be true at some time in the future"; and '''G''' ''A'', as "''A'' is true now and will always be true".

* The previous two examples are of [[unary]] or [[Monad (category theory)|monadic]] modal operators. As an example of a [[dyadic]] modal operator -- which produces a new proposition from ''two'' old propositions -- is the operator '''P''' in the dyadic [[deontic logic]] of [[Georg Henrik von Wright|G.H. von Wright]]. '''P'''(''A'',''B'') expresses that "''A'' is obligatory under the circumstances ''B''".

== Modality interpreted ==

{{Mathlogic-stub}}

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There are several ways to [[interpretation (logic)|interpret]] modal operators in modal logic, including at least:

[[alethic modality|alethic]], [[deontic logic|deontic]], [[axiology|axiological]], [[epistemic modal logic|epistemic]], and [[doxastic logic|doxastic]].

===Alethic===

[[Category:Modal logic]]

[[Alethic modality|Alethic]] modal operators (M-operators) determine the fundamental conditions of [[possible worlds]], especially [[causality]], time-space parameters, and the action capacity of persons. They indicate the [[logical possibility|possibility]], [[Subjunctive possibility|impossibility]] and [[Logical truth|necessity]] of actions, states of affairs, events, people, and qualities in the possible worlds.

=== Deontic ===

[[pl:Operator modalny]]

[[Deontic logic|Deontic]] modal operators (P-operators) influence the construction of possible worlds as proscriptive or prescriptive norms, i.e. they indicate what is prohibited, obligatory, or permitted.

=== Axiological ===

[[Axiology|Axiological]] modal operators (G-operators) transform the world's [[wikt:entity|entities]] into values and disvalues as seen by a social group, a culture, or a historical period. Axiological modalities are highly subjective categories: what is good for one person may be considered as bad by another one.{{clarification needed|date=April 2022}}

=== Epistemic ===

[[Epistemic logic|Epistemic]] modal operators (K-operators) reflect the level of knowledge, ignorance and belief in the possible world.

=== Doxastic ===

[[Doxastic logic|Doxastic]] modal operators express belief in statements.

=== Boulomaic ===

Boulomaic modal operators express desire.

== References ==

{{reflist}}

{{logic}}

[[Category:Modal logic|Operator]]

[[Category:Logic symbols]]

[[Category:Logical connectives]]