An accessibility relation is a
relation which plays a key role in assigning truth values to sentences in the
relational semantics
Kripke semantics (also known as relational semantics or frame semantics, and often confused with possible world semantics) is a formal semantics for non-classical logic systems created in the late 1950s and early 1960s by Saul Kripke and André Jo ...
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 ot ...
. In relational semantics, a modal formula's truth value at a ''
possible world''
can depend on what's true at another possible world
, but only if the accessibility relation
relates
to
. For instance, if
holds at some world
such that
, the formula
will be true at
. The fact
is crucial. If
did not relate
to
, then
would be false at
unless
also held at some other world
such that
.
Accessibility relations are motivated conceptually by the fact that
natural language
In neuropsychology, linguistics, and philosophy of language, a natural language or ordinary language is any language that has evolved naturally in humans through use and repetition without conscious planning or premeditation. Natural languages ...
modal statements depend on some, but not all alternative scenarios. For instance, the sentence "It might be raining" is not generally judged true simply because one can imagine a scenario where it was raining. Rather, its truth depends on whether such a scenario is ruled out by available information. This fact can be formalized in modal logic by choosing an accessibility relation such that
iff
is compatible with the information that's available to the speaker in
.
This idea can be extended to different applications of modal logic. In epistemology, one can use an epistemic notion of accessibility where
for an individual
iff
does not know something which would rule out the hypothesis that
. In
deontic modal logic, one can say that
iff
is a morally ideal world given the moral standards of
. In application of modal logic to computer science, the so-called possible worlds can be understood as representing possible states and the accessibility relation can be understood as a program. Then
iff running the program can transition the computer from state
to state
.
Different applications of modal logic can suggest different restrictions on admissible accessibility relations, which can in turn lead to different validities. The mathematical study of how validities are tied to conditions on accessibility relations is known as ''modal correspondence theory''.
See also
*
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 ot ...
*
Possible worlds
*
Propositional attitude
A propositional attitude is a mental state held by an agent toward a proposition.
Linguistically, propositional attitudes are denoted by a verb (e.g. "believed") governing an embedded "that" clause, for example, 'Sally believed that she had won ...
*
Modal depth
In modal logic, the modal depth of a formula is the deepest nesting of modal operators (commonly \Box and \Diamond). Modal formulas without modal operators have a modal depth of zero.
Definition
Modal depth can be defined as follows. Let MD(\p ...
References
* Gerla, G.; ''Transformational semantics for first order logic''
Logique et Analyse No. 117–118, pp. 69–79, 1987.
* Fitelson, Brandon; ''Notes on "Accessibility" and Modality'', 2003.
* Brown, Curtis; ''Propositional Modal Logic: A Few First Steps'', 2002.
* Kripke, Saul; ''Naming and Necessity'', Oxford, 1980.
*
*
List of most of the more popular modal logics.
{{DEFAULTSORT:Accessibility Relation
Modal logic
Binary relations