Concept
Accessibility relation
Overview
An accessibility relation is a relation which plays a key role in assigning truth values to sentences in the relational semantics for modal logic. In relational semantics, a modal formula's truth value at a possible world
w
{\displaystyle w}
can depend on what is true at another possible world
v
{\displaystyle v}
, but only if the accessibility relation
R
{\displaystyle R}
relates
w
{\displaystyle w}
to
v
{\displaystyle v}
. For instance, if
P
{\displaystyle P}
holds at some world
v
{\displaystyle v}
such that
w
R
v
{\displaystyle wRv}
, the formula
◊
P
{\displaystyle \Diamond P}
will be true at
w
{\displaystyle w}
. The fact
w
R
v
{\displaystyle wRv}
is crucial. If
R
{\displaystyle R}
did not relate
w
{\displaystyle w}
to
v
{\displaystyle v}
, then
◊
P
{\displaystyle \Diamond P}
would be false at
w
{\displaystyle w}
unless
P
{\displaystyle P}
also held at some other world
u
{\displaystyle u}
such that
w
R
u
{\displaystyle wRu}
.
Accessibility relations are motivated conceptually by the fact that natural language 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 is 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
w
R
v
{\displaystyle wRv}
if
v
{\displaystyle v}
is compatible with the information that is available to the speaker in
w
{\displaystyle w}
.
This idea can be extended to various applications of modal logic. In epistemic logic, one can use an epistemic notion of accessibility where
w
R
v
{\displaystyle wRv}
for an individual
I
{\displaystyle I}
if
I
{\displaystyle I}
does not know something which would rule out the hypothesis that
w
′
=
v
{\displaystyle w'=v}
. In deontic logic, one can say that
w
R
v
{\displaystyle wRv}
if
v
{\displaystyle v}
is a morally ideal world given the moral standards of
w
{\displaystyle w}
. In the application of modal logic to computer science, possible worlds can be understood as representing possible states of a system and the accessibility relation can be understood as representing state transitions (see Kripke structure (model checking)). Then
w
R
v
{\displaystyle wRv}
if the system can transition from state
w
{\displaystyle w}
to state
v
{\displaystyle v}
.
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.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
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Any kind of dissimilarity is also of this kind: if the colour of A is unlike the colour of B, then the colour of B is unlike the colour of A. Relations of this sort are called _symmetrical_. Thus a relation is symmetrical if, whenever it holds between A and B, it also holds between B and A. All relations that are not symmetrical are called _non-symmetrical_. Thus "brother" is non-symmetrical, because, if A is a…
Bertrand Russell, Our Knowledge of the External World as a Field for Scientific Method in Philosophy
States what Accessibility relation is.
=132.= The next step, in defining a form of externality, is obtained from the idea of _dimensions_. Positions, we have seen, are defined solely by their relations to other positions. But in order that such definition may be possible, a finite number of relations must suffice, since infinite numbers are philosophically inadmissible. A position must be definable, therefore, if knowledge of our form is to be possible…
Bertrand Russell, An essay on the foundations of geometry
States what Accessibility relation is.
There is a tendency for certain attributes to maintain their concreteness (qualitativeness) in things, and for others to suggest the connection of things with other things, and so to emphasize a more abstract aspect of experience. Thus then arises a temporary and practical distinction that tends to be taken as opposition between qualities and relations.
John Dewey, Creative Intelligence
States what Accessibility relation is.
Philosophers who discussed Accessibility relation
Related ideas
Place in the tradition
Part of Deontic logic.