Philosophical Applications
One of the applications of 'possible worlds' semantics and the 'accessibility relation' is 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) is nomologically necessary means that is true at all possible worlds that are nomologically accessible from the actual world. In other words, 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. (Notice however that if the objection raised earlier is true, each different theory of the nature of physical laws would be 'possible' and 'necessary,' since the euclidean concept depends on the idea about 'possibility' and 'necessity'). 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 can be necessarily without being necessarily necessarily . 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 in the world is interpreted as "it is possible to transform into a world in which 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 is epistemically accessible from for an individual in if and only if does not know something which would rule out the hypothesis that . We can ask whether the relation is transitive. If knows nothing that rules out the possibility that and knows nothing that rules the possibility that, it does not follow that knows nothing which rules out the hypothesis that . 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.
Read more about this topic: Accessibility Relation