Modal operator

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A modal connective (or modal operator) is a logical connective for modal logic. It is an operator which forms propositions from propositions. In general, a modal operator has the "formal" property of being non-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 necessity, possibility, belief, or knowledge) about the proposition to which the operator is applied.

Syntax for modal operators[]

The syntax rules for modal operators ${\displaystyle \Box }$ and ${\displaystyle \Diamond }$ are very similar to those for universal and existential quantifiers; In fact, any formula with modal operators ${\displaystyle \Box }$ and ${\displaystyle \Diamond }$, and the usual logical connectives in propositional calculus (${\displaystyle \land ,\lor ,\neg ,\rightarrow ,\leftrightarrow }$) can be rewritten to a de dicto normal form, similar to prenex normal form. One major caveat: Whereas the universal and existential quantifiers only binds to the propositional variables or the predicate variables following the quantifiers, since the modal operators ${\displaystyle \Box }$ and ${\displaystyle \Diamond }$ quantifies over accessible possible worlds, they will bind to any formula in their scope. For example, ${\displaystyle (\exists x(x^{2}=1))\land (0=y)}$ is logically equivalent to ${\displaystyle \exists x(x^{2}=1\land 0=y)}$, but ${\displaystyle (\Diamond (x^{2}=1))\land (0=y)}$ is not logically equivalent to ${\displaystyle \Diamond (x^{2}=1\land 0=y)}$; Instead, ${\displaystyle \Diamond (x^{2}=1\land 0=y)}$ is logically equivalent to ${\displaystyle (\Diamond (x^{2}=1))\land \Diamond (0=y)}$.

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 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 several ways to interpret modal operators in modal logic, including: alethic, deontic, axiological, epistemic, and doxastic.

Alethic[]

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 possibility, impossibility and necessity of actions, states of affairs, events, people, and qualities in the possible worlds.

Deontic[]

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[]

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.

Epistemic[]

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

Doxastic[]

Doxastic modal operators express belief in statements.

Boulomaic[]

Boulomaic modal operators express desire.