Modal operator


Modal operator

In modal logic, a modal operator is an operator which forms propositions from propositions. In general, a modal operator has the "formal" property of being non-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.

Contents

Literary theory

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).

Modality interpreted

There are four established interpretations of the modal operator of modal modal logic: alethic, deontic, axiological and epistemic.

Alethic

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 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 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).

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

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).

Examples

  • In the alethic modal logic of C.I. Lewis, the modal operator \Box expresses necessity: if the proposition A is read as "it is true that A holds", the proposition \BoxA is read as "it is necessarily true that A holds".
  • In the tense logic (more commonly now called temporal logic) of 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 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 G.H. von Wright. P(A,B) expresses that "A is obligatory under the circumstances B".

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