Modal Operator
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A modal connective (or modal operator) is a
logical connective In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. They can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary co ...
for
modal logic Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend other ...
. It is an operator which forms
proposition In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...
s from propositions. In general, a modal operator has the "formal" property of being non-
truth-functional In logic, a truth function is a function that accepts truth values as input and produces a unique truth value as output. In other words: The input and output of a truth function are all truth values; a truth function will always output exactly one ...
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 Necessary or necessity may refer to: * Need ** An action somebody may feel they must do ** An important task or essential thing to do at a particular time or by a particular moment * Necessary and sufficient condition, in logic, something that i ...
,
possibility Possibility is the condition or fact of being possible. Latin origins of the word hint at ability. Possibility may refer to: * Probability, the measure of the likelihood that an event will occur * Epistemic possibility, a topic in philosophy ...
,
belief A belief is an attitude that something is the case, or that some proposition is true. In epistemology, philosophers use the term "belief" to refer to attitudes about the world which can be either true or false. To believe something is to take i ...
, or
knowledge Knowledge can be defined as awareness of facts or as practical skills, and may also refer to familiarity with objects or situations. Knowledge of facts, also called propositional knowledge, is often defined as true belief that is distinc ...
) about the proposition to which the operator is applied. See also Garson, James, "Modal Logic", The Stanford Encyclopedia of Philosophy (Summer 2021 Edition), Edward N. Zalta (ed.), URL =


Syntax for modal operators

The syntax rules for modal operators \Box and \Diamond are very similar to those for universal and existential quantifiers; In fact, any formula with modal operators \Box and \Diamond, and the usual
logical connectives In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. They can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary c ...
in
propositional calculus Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...
( \land,\lor,\neg,\rightarrow,\leftrightarrow ) can be rewritten to a ''de dicto'' normal form, similar to
prenex normal form A formula of the predicate calculus is in prenex normal form (PNF) if it is written as a string of quantifiers and bound variables, called the prefix, followed by a quantifier-free part, called the matrix. Together with the normal forms in prop ...
. 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 \Box and \Diamond quantifies over
accessible Accessibility is the design of products, devices, services, vehicles, or environments so as to be usable by people with disabilities. The concept of accessible design and practice of accessible development ensures both "direct access" (i. ...
possible worlds Possible Worlds may refer to: * Possible worlds, concept in philosophy * ''Possible Worlds'' (play), 1990 play by John Mighton ** ''Possible Worlds'' (film), 2000 film by Robert Lepage, based on the play * Possible Worlds (studio) * ''Possible Wo ...
, they will bind to any formula in their
scope Scope or scopes may refer to: People with the surname * Jamie Scope (born 1986), English footballer * John T. Scopes (1900–1970), central figure in the Scopes Trial regarding the teaching of evolution Arts, media, and entertainment * Cinem ...
. For example, (\exists x (x^2 = 1)) \land (0 = y) is logically equivalent to \exists x (x^2 = 1\land 0 = y), but (\Diamond (x^2 = 1)) \land (0 = y) is not logically equivalent to \Diamond (x^2 = 1\land 0 = y); Instead, \Diamond (x^2 = 1\land 0 = y) is logically equivalent to (\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 A multimodal logic is a modal logic that has more than one primitive modal operator. They find substantial applications in theoretical computer science. Overview A modal logic with ''n'' primitive unary modal operators \Box_i, i\in \ is called an ...
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 Interpreting is a translational activity in which one produces a first and final target-language output on the basis of a one-time exposure to an expression in a source language. The most common two modes of interpreting are simultaneous interp ...
modal operators in modal logic, including at least: alethic,
deontic In moral philosophy, deontological ethics or deontology (from Greek language, Greek: + ) is the normative ethics, normative ethical theory that the morality of an action should be based on whether that action itself is right or wrong under a s ...
,
axiological Axiology (from Greek , ''axia'': "value, worth"; and , ''-logia'': "study of") is the philosophical study of value. It includes questions about the nature and classification of values and about what kinds of things have value. It is intimately c ...
, epistemic, and
doxastic Doxastic logic is a type of logic concerned with reasoning about beliefs. The term ' derives from the Ancient Greek (''doxa'', "opinion, belief"), from which the English term ''doxa'' ("popular opinion or belief") is also borrowed. Typically, a ...
.


Alethic

Alethic modal operators (M-operators) determine the fundamental conditions of
possible worlds Possible Worlds may refer to: * Possible worlds, concept in philosophy * ''Possible Worlds'' (play), 1990 play by John Mighton ** ''Possible Worlds'' (film), 2000 film by Robert Lepage, based on the play * Possible Worlds (studio) * ''Possible Wo ...
, especially
causality Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the cau ...
, time-space parameters, and the action capacity of persons. They indicate the
possibility Possibility is the condition or fact of being possible. Latin origins of the word hint at ability. Possibility may refer to: * Probability, the measure of the likelihood that an event will occur * Epistemic possibility, a topic in philosophy ...
,
impossibility In contract law, impossibility is an excuse for the nonperformance of duties under a contract, based on a change in circumstances (or the discovery of preexisting circumstances), the nonoccurrence of which was an underlying assumption of the ...
and
necessity Necessary or necessity may refer to: * Need ** An action somebody may feel they must do ** An important task or essential thing to do at a particular time or by a particular moment * Necessary and sufficient condition, in logic, something that i ...
of actions, states of affairs, events, people, and qualities in the possible worlds.


Deontic

Deontic In moral philosophy, deontological ethics or deontology (from Greek language, Greek: + ) is the normative ethics, normative ethical theory that the morality of an action should be based on whether that action itself is right or wrong under a s ...
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 Axiology (from Greek , ''axia'': "value, worth"; and , ''-logia'': "study of") is the philosophical study of value. It includes questions about the nature and classification of values and about what kinds of things have value. It is intimately c ...
modal operators (G-operators) transform the world's
entities An entity is something that exists as itself, as a subject or as an object, actually or potentially, concretely or abstractly, physically or not. It need not be of material existence. In particular, abstractions and legal fictions are usually ...
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 Doxastic logic is a type of logic concerned with reasoning about beliefs. The term ' derives from the Ancient Greek (''doxa'', "opinion, belief"), from which the English term ''doxa'' ("popular opinion or belief") is also borrowed. Typically, a ...
modal operators express belief in statements.


Boulomaic

Boulomaic modal operators express desire. {{logic Modal logic Logic symbols Logical connectives