Modal Logic: a question concerning accessibility

Question

I’m reading a lot about modal logic lately, right now Lewis “On the Plurality of Worlds” and Priests “Introduction to Non-classical Logics”.

It is postulated that the different worlds have nothing to do with each other. Everything that belongs to this world does not belong to another world, nothing here can cause something there (whereever there is, this question is not aiming at any ontological questions I think. For it is irrelevant how they exist).

If there is no relation between the worlds, what is the notion of accessibility? And how could it be that there are two worlds that don’t access each other?

EDIT:

I think, after reading the answers, what I am really asking is this:

Suppose there are two worlds, w1 and w2, and I want to know wether w1Rw2, how will I know?

What does it mean for w1 and w2 to be accessible?

What are examples of two possible worlds that don’t share an accessibility relation (What is it that makes them unaccessible)?

Answer 1

You can do the semantics for QML in one of two ways, either variable domain or constant domain. To keep worlds isolated (i.e., to prohibit overlap) as Lewis does in his modal realism requires a variable domain approach (since the domain of each world's quantifier is disjoint from any other world). You can read about these two approaches in the second half of Priest's book, chs 14 and 15 of the second edition.

It is worth emphasizing that this prohibition against overlapping worlds is a metaphysical postulate that is part of Lewis's metaphysical view modal realism. It is in no means forced upon you by modal logic itself.

If there is no relation between the worlds, what is the notion of accessibility? And how could it be that there are two worlds that don´t access each other?

Ah, but there are relations that hold between the worlds. There are similarity relations for Lewis and accessibility relations in the model theory of modal logic. The relations between worlds that Lewis forbids are spatiotemporal and causal relations. His thesis is that worlds are spatiotemporally and causally isolated from one another--- not that no relations exist between worlds at all.

If you have an S5 modal logic (as Lewis assumes in OPW) then you can, as Eric's answer points out, ignore accessibility since every world is accessible from every world (since accessibility is an equivalence relation in S5). If, however, you start off with two isolated worlds w1 and w2 then you will have two disjoint equivalence classes of worlds (where each class is as large as the totality of Lewis's "logical space"). This isn't generally a problem in the metaphysical applications of modal logic (like Lewis's modal realism), though, since usually we start from the actual world and I don't know of anyone who has argued that two worlds can both be actual (at the same time).

What does it mean for w1 and w2 to be accessible?

What it means is simply that the propositions true at those worlds can be relevant for assessing the truth value of modal propositions at the world w1 and w2 are accessible from. For instance, if w0Rw1 and w0Rw2 then in order for []p (necessarily p) to be true at w0, p will have to be true at w1 and w2.

What are examples of two possible worlds that dont share an accesibility relation (What is it that makes them unaccessible)?

In weaker systems than S5 (or in the bizarre version of S5 I described above) you will often have worlds that are inaccessible to one another. For instance, in K there are no restrictions placed on accessibility and so you cannot be sure that any worlds access any others. As a weird side effect, you can have a world w1 in which both []p and ~p are true. Since accessibility isn't reflexive in K you don't need p to be true at w1 in order for []p to be true and so there is no contradiction.

What it is for a world w1 to be "inaccessible" is simply for the propositions true at w1 to make no difference in the evaluate of modal statements at worlds that don't access w1.

To answer a question you didn't ask: but why does Lewis assume S5?

Well, there seems to be wide agreement that S5 captures the notion of logical necessity. This is why Lewis calls the space of possible worlds logical space.

But often you want to consider other sorts of necessity/modality. For example, in provability logic you are concerned with provability rather than possibility. Provability logic is generally thought to be intermediate in strength between S4 and S5 (I've seen it claimed to be S4.2 or S4.3 most frequently).

Answer 2

Shortest answer: If you have defined a proper model, you know that R(w1, w2) if in your model the pair (w1, w2) is in the extension of R. If you do not have a model you may be able to infer this from the presumed properties of R, for example from R(w0, w1) and R(w0, w2) you can infer R(w1, w2) if R is Euclidean. You define this relation in the way that characterizes the modality in question. You need to have an idea what properties the respective modality has in the first place.

Longer answer: The accessibility relation has no meaning outside the formal system in general, because its interpretation depends on the purpose for which you use the modal logic. For example, for rational belief sometimes system KD45 is used, which has a serial, transitive, and Euclidean accessibility relation. So if R(w1, w2), then some people would interpret this as "w2 is compatible with what the agent believes in w1". However, from a formal point of view this interpretation is insignificant, because what counts are the formal properties of the relation, i.e. in this case that it is serial, transitive, and Euclidean. These properties directly correspond to axioms in the proof theory of the logic. In case of KD45, for example, they correspond to (D) consistency, and (4) positive and (5) negative introspection axioms. (Axiom (K) holds by default in all normal modal logics.)

I'm sure somebody has written something about it but personally I wouldn't read too much into accessibility relations . They are more or less a technicality, although an important one.

Original, detailed answer: First of all, Priest and Lewis have different views on the question what objects can 'inhabit' different worlds. Lewis is a modal realist, he subscribes to the view that one and the same object cannot reside within two worlds at the same time, which is why he developed counterpart theory. Priest comes from a more modern tradition not based on counterpart theory. As far as I know, he allows in his systems the 'same' object to exist in different worlds where it might have different properties. (He sometimes allows much more, of course, because many of his systems are paraconsistent. Notice that sameness cannot be Leibnizian identity across worlds in this context. Cross-world identity has been debated extensively since the 70s of last century.)

Another question to take into account is whether a constant in world w can refer to an object residing in world u. I don't know whether Priest allows this, but for example Fitting & Mendelsohn do so in their book on first-order modal logic. In case of doubt, you always have check the rules for term evaluation in the logical system.

Regarding accessibility between worlds, both Lewis and Priest need to allow a binary accessibility relation between worlds for technical reasons, as long as they want to stay within the realm of normal modal logic with Kripke frames. If you leave out accessibility, you end up with a system like S5. (I say "like S5", because strictly speaking the system you end up with is S5 with global box modality or S5 with models in which all inaccessible worlds have been removed.)

There are other systems, for example modal logic with neighbourhood semantics, which generalize normal modal logic and do not have an accessibility relation. If I'm not mistaken Lewis studied some of them, too, in his work on conditional logic. However, even in these more general systems you need to restrict the sets of worlds modal operators run over somehow, for example by a 'selection function', in order to make them behave in any interesting way.

A lot has been written about what all of this means from a metaphysical point of view and no final agreement has ever been reached, for it depends on various stances one might take. For example, it makes a difference whether you're a modal realist or believe that possible worlds are ontologically reducible and, if so, in which way, and whether possible worlds are essentially models of a non-modal base language (Hintikka) or not. Cocchiarella (1989, 2007) has worked a lot on the metaphysics of modality, and you will also find many metaphysical aspects addressed by Kit Fine. These authors are not easy to read, though, and require some good technical background.

Addition: I forgot to mention that I know of no author who claims that possible worlds could causally interact with each other, but accessibility does not imply causal interaction.

Answer 3

If two worlds were accessible from each other then they are in fact one world - in the same way if there is a door between two rooms those two rooms are in fact part of one house. Hence if there are two Worlds then then are inaccessible from each other.

Although Occams Razor says there is a world, and we're in it. One can reverse the cut and say, if there is one World, why can't there be more? Perhaps all Worlds that don't violate logic exist?