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added information on modal operators as viewed by Dolezel in terms of fiction/literary theory
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.
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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.
{{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> 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}}
{{unreferenced|section|date=February 2024}}
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]]