Modal operator: Difference between revisions

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

{{expert-subject|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>

{{not verified|date=February 2008}}

In [[modal logic]], a '''modal operator''' is an [[operator]] which forms [[proposition]]s from propositions. In general, a modal operator has the "formal" property of being non-[[truth function|truth-functional]], and is "intuitively" characterised by expressing a modal attitude (such as [[necessity]], [[possibility]], [[belief]], or [[knowledge]]) about the proposition to which the operator is applied. The concrete examples in this entry relate modality to [[literary theory]].

== Syntax for modal operators ==

==Literary theory==

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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).

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.

==Modality interpreted==

There are four established interpretations of the modal operator of modal [[modal logic]]:

== Modality interpreted ==

[[alethic]], [[deontic]], [[axiological]] and [[epistemic]].

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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===

[[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.

[[Alethic]] modal operators (M-operators) determine the fundamental conditions of [[fictional worlds]], especially [[causality]], time-space parameters, and the action capacity of persons. They indicate the possibility, impossibility and necessity of actions, states of affairs, events, people, and qualities in the [[fictional worlds]]. Alethic modal operators play an important role in distinguishing a natural [[fictional world]] from the supernatural and intermediate ones. “The natural world generates stories of human condition” and such stories tend to be tragic from the very beginning, for example [[J.W. Goethe]]’s ''[[The Sorrows of Young Werther]]''. The structure of the supernatural worlds is usually revealed by the alethic modal operators (a) when they show the presence of physically impossible beings in the [[fictional world]] (gods, spirits, monsters); (b) when selected natural-world persons are granted properties and action capacities that are not available to ordinary persons of that world: becoming invisible, flying on a carpet, etc.; (c) when inanimate objects are anthropomorphized (for example the legend of Don Juan in [[Alexander Pushkin]]’s ''The Stone Guest''.) The intermediate worlds are usually represented in dreams within a [[fictional world]].

=== Deontic ===

[[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.

===Deontic Doxastic ===

[[Deontic]] modal operators (P-operators) influence the construction of [[fictional worlds]] as proscriptive or prescriptive norms, i.e. they indicate what is prohibited, obligatory, or permitted in the fictional world. The deontic marking of actions is the richest source of narrativity; it generates the famous triad of the fall (violations of a norm – punishment), the test (obligation fulfilled – reward), and the predicament (conflict of obligations) stories. An example in literature of a story of the fall would be Mme. De Renal, in [[Stendhal]]’s ''[[The Red and the Black|Le Rouge et le Noir]]'' (1830). An example of a deontic alien in literature, a person who “exempts himself from the world’s codex and follows his own principles” is Raskolnikov, the protagonist of F.M. [[Dostoevsky]]’s ''[[Crime and Punishment]]'' (1866).

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

===Axiological===

[[Axiological]] modal operators (G-operators) transform the world’s [[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. “Subjective abnegation of the world’s axiological order generates the story of the axiological alien” of which there are two kinds – the [[nihilist]] and the axiological rebel. A nihilist negates the axiological order of the world and replaces it with a subjective axiology with a single operator: indifference. An example of the nihilist axiological alien in literature is [[Pechorin]], the protagonist of [[Mikhail Lermontov]]’s ''[[A Hero of Our Time]]'' (1839).

===Epistemic Boulomaic ===

Boulomaic modal operators express desire.

[[Epistemic]] modal operators (K-operators) reflect the level of knowledge, ignorance and belief in the [[fictional world]]. The epistemic imbalance in the fictional world of a story produces a “mystery story,” which is usually the basic model of [[detective fiction]]. Epistemic code can also be perceived at the core of the [[Bildungsroman]], where the protagonist undergoes a transformation from ignorance (of self) to knowledge (of self). An example of such transformation in German literature would be [[J.W. Goethe]]’s ''[[Wilhelm Meister's Apprenticeship]]'' (1795).

== ExamplesReferences ==

{{reflist}}

* 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''".

[[Category:Modal {{logic]]}}

[[Category:Literary theory]]

[[Category:Modal logic|Operator]]

[[pl:Operator modalny]]

[[Category:LiteraryLogic theorysymbols]]

[[Category:Logical connectives]]