Accessibility relation

An accessibility relation is a binary relation $R\,\!$ between possible worlds which has very powerful uses in both the formal/theoretical aspects of modal logic as well as in its applications to things like epistemology, metaphysics, value theory, and computer science. In informal terms, it is the notion that modal statements (it is possible that x, it is necessary that x) may not take the same truth value in all possible worlds, or in other words that the set of possible worlds varies depending on which possible world you are in, i.e. from any given possible world, some other possible worlds may be accessible, and others may not be.

Basic review of (propositional) modal logic

In order to really understand what an accessibility relation is a bit of background explanation of the basics of modal logic is required (but see modal logic for a more in-depth discussion). For purposes of simplicity, we will here stick to propositional modal logic. Propositional modal logic is just traditional propositional logic with the addition of two key unary operators: $\Box$ which means "it is necessary that..." and $\Diamond$ which stands for "it is possible that...". These operators can be attached to a single sentence to form a new compound sentence. For any sentence $A\,\!$ (simple or compound), we can therefore form the compound sentences $\Box A$ and $\Diamond A$.

Now, using $p\,\!$, $q \,\!$, etc. to stand for statements of our language, $x\,\!$, $y\,\!$, etc. to stand for objects, and $P\,\!$, $Q\,\!$, etc. to stand for predicates, we can write out the six basic axioms of almost all modal logics:

• $\Box p \leftrightarrow \lnot \Diamond \lnot p$
• $\Diamond p \leftrightarrow \lnot \Box \lnot p$
• $\Box (p \land q) \leftrightarrow (\Box p \land \Box q)$
• $(\Box p \lor \Box q) \rightarrow \Box (p \lor q)$
• $\Box (p \to q) \to (\Box p \to \Box q)$

Most of the other axioms concerning the modal operators are controversial and not widely agreed upon. Here are the most commonly used and discussed of these:

(T) $\Box p \rightarrow p$

(4) $\Box p \rightarrow \Box \Box p$

(5) $\Diamond p \rightarrow \Box \Diamond p$

(B) $p \rightarrow \Box \Diamond p$

Here, "(T)","(4)","(5)", and "(B)" represent the traditional names of these axioms (or principles).

According to the traditional possible worlds semantics of modal logic, the compound sentences that are formed out of the modal operators are to be interpreted in terms of quantification over possible worlds, subject to the relation of accessibility. The relation of accessibility can now be defined as an (uninterpreted) relation $R(w_1,w_2)\,\!$ that holds between possible worlds $w_1\,\!$ and $w_2\,\!$ only when $w_2\,\!$ is accessible from $w_1\,\!$. Using a procedure called standard translation, a formula in modal logic can be translated into a formula of first-order logic which makes the meaning of the boxes and diamonds in modal logic explicit.

Importance of AR in formal semantics

Letting $w^*\,\!$ denote the actual world, we have the following two fundamental translational schema for possible world semantics:

• (TS) Necessarily $p\,\!$ means that $p\,\!$ is true at every possible world $w\,\!$ such that $R(w^*,w)\,\!$. Possibly $p\,\!$ means that $p\,\!$ is true at some possible world $w\,\!$ such that $R(w^*,w)\,\!$.

To see the power and usefulness of the accessibility relation on a technical/formal level, note that the following relationships hold:

• Axiom (T) holds if the accessibility relation $R\,\!$ is reflexive. If every world is accessible to itself, then any world in which $A\,\!$ is true will be a world from which there is an accessible world in which $A\,\!$ is true.

• Axiom (4) holds if $R\,\!$ is transitive. $\Box A$ is true at a world $w\,\!$ only when $A\,\!$ is true at every world $w'\,\!$ accessible from $w\,\!$. Hence, $\Box \Box A$ is true at a world $w\,\!$ only when $A\,\!$ is true at every world accessible from every world accessible from $w\,\!$.

• Axiom (5) holds if $R\,\!$ is euclidean. $\Diamond A$ is true at a world $w\,\!$ if and only if $A\,\!$ is true at some world accessible from $w\,\!$. $\Box \Diamond A$ is true at a world $w\,\!$ if and only if, for every world $w'\,\!$ accessible from $w\,\!$, there is a world $w''\,\!$ accessible from $w'\,\!$ at which $A\,\!$ is true. The euclidean property guarantees the truth of this. If $A\,\!$ is true at a world accessible from $w\,\!$, then if that world is accessible from every other world accessible from $w\,\!$, it will be true that for every world accessible from $w\,\!$ there is an accessible world in which $A\,\!$ is true.

• Axiom (B) holds if $R\,\!$ is symmetric. If $A\,\!$ is true in a world $w\,\!$, then in every world $w'\,\!$ accessible from $w\,\!$, there is a world accessible from $w'\,\!$ in which $A\,\!$ is true. Since $A\,\!$ is true in $w\,\!$, this is guaranteed to be true provided that $w\,\!$ is accessible from it, which is what symmetry says.

The result, according to David Lewis, is that "old disputes give way to new. Instead of asking the baffling question whether whatever is actual is necessarily possible, we could simply try asking: is the relation $R\,\!$ symmetric?" (David Kellogg Lewis, 1996) Which question is in no way itself baffling.

Philosophical applications

Let's take the case of applications of possible worlds semantics with the accessibility relation to physics. Instead of just talking generically about necessity (or logical necessity), the relation in physics deals with nomological necessity. The fundamental translational schema (TS) described earlier can be exemplified as follows for physics:

• (TSN) $P\,\!$ is nomologically necessary means that $P\,\!$ is true at all possible worlds that are nomologically accessible from the actual world. In other words, $P\,\!$ is true at all possible worlds that obey the physical laws of the actual world.

The interesting thing to observe is that instead of having to ask, now, "Does nomological necessity satisfy the axiom (5)?", that is, "Is something that is nomologically possible nomologically necessarily possible?", we can ask instead: "Is the nomological accessibility relation euclidean?" And different theories of the nature of physical laws will result in different answers to this question. The theory of Lewis, for example, is asymmetric. His counterpart theory also requires an intransitive relation of accessibility because it is based on the notion of similarity and similarity is generally intransitive. For example, a pile of straw with one less handful of straw may be similar to the whole pile but a pile with two (or more) less handfuls may not be. So $x\,\!$ can be necessarily $P\,\!$ without $x\,\!$ being necessarily necessarily $P\,\!$. On the other hand, Saul Kripke has an account of de re modality which is based on (metaphysical) identity across worlds and is therefore transitive.

Another interpretation of the accessibility relation with a physical meaning was given in Gerla 1987 where the claim “is possible $P\,\!$ in the world $w''\,\!$ is interpreted as "it is possible to transform $w\,\!$ into a world in which $P\,\!$ is true". So, the properties of the modal operators depend on the algebraic properties of the set of admissible transformations.

There are other applications of the accessibility relation in philosophy. In epistemology, one can, instead of talking about nomological accessibility, talk about epistemic accessibility. A world $w'\,\!$ is epistemically accessible from $w\,\!$ for an individual $I\,\!$ in $w\,\!$ if and only if $I\,\!$ does not know something which would rule out the hypothesis that $w'=w\,\!$. We can ask whether the relation is transitive. If $I\,\!$ knows nothing that rules out the possibility that $w'=w\,\!$ and knows nothing that rules the possibility that $w''=w'\,\!$, it does not follow that $I\,\!$ knows nothing which rules out the hypothesis that $w''=w\,\!$. To return to our earlier example, one may not be able to distinguish a pile of sand from the same pile with one less handful and one may not be able to distinguish the pile with one less handful from the same pile with two less handfuls of sand, but one may still be able to distinguish the original pile from the pile with two less handfuls of sand.

Yet another example of the use of the accessibility relation is in deontic logic. If we think of obligatoriness as truth in all morally perfect worlds, and permissibility as truth in some morally perfect world, then we will have to restrict out universe to include only morally perfect worlds. But, in that case, we will have left out the actual world. A better alternative would be to include all the metaphysically possible worlds but restrict the accessibility relation to morally perfect worlds. Transitivity and the euclidean property will hold, but reflexivity and symmetry will not.

Computer Science applications

In modeling a computation, a "possible world" can be a possible computer state. Given the current computer state, you might define the accessible possible worlds to be all future possible computer states, or to be all possible immediate "next" computer states (assuming a discrete computer). Either choice defines a particular accessibility relation giving rise to a particular modal logic suited specifically for theorems about the computation.

See also

• Modal logic
• Possible worlds
• Propositional attitude
• Modal depth

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.
• Lewis, David K.; Counterpart Theory and Quantified Modal Logic ^{(subscription required)}, The Journal of Philosophy, Vol. LXV, No. 5 (1968-03-07), pp. 113-126, 1968
• List of Logic Systems List of most of the more popular modal logics.

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