In the modal system S5 the accessibility relation, R, which allows for the ability to determine truth-values in the system, is such that accessibility is transitive, reflexive, and symmetrical. That is, given a semantics of possible worlds for the syntactical system of S5, we are able to quantify over possible worlds in such a way that we can refer to entities in the actual world (w*), in merely possibly worlds, and from merely possible worlds back to the actual world. This assumes that all worlds are mutually accessible.[i] This robust system breaks down, however, under a combinatorialist picture of possible worlds ontology, wherein possible worlds are merely re-combinations of actually existing entities.[ii]In other words, all that exists for the combinatorialist are actual particulars and instantiated universals; any ontology beyond this is alien. From this it will follow that, even given an infinitely robust actuality, wherein all universals are instantiated, the system S5 will be rejected upon a combinatorialist view.

S5 is derived from the conjunction of less robust systems, namely T and S4.[iii] S5 is characterised by the sentence A→□◊A ; or, equivalently, given system T, ◊A→□◊A. This is highly intuitive in that S4 is saying, roughly, that if a proposition P is necessary then its necessity is necessary (i.e. it’s modal status *as that proposition* is necessary). S5 is saying something analogous for possible propositions, namely that for any proposition P, if P has the modal status of being possible then it has that modal status necessarily.[iv] This seems to best capture our intuitions that a necessary truth is one that is true *in all possible worlds*. S5 tells us that the accessibility relation between worlds is symmetric. That is, all worlds, relative to each other, are accessible to each other. If there is some proposition P in a world w, such that ◊A, then, relative to the set of all possible worlds, it is necessary that àP is true *because, no matter which world we view P from it is possible*.[v] This satisfies our intuition that, broadly speaking, “things could not have been such that it would be impossible for things to have been as they in fact are.”[vi]

David Armstrong’s combinatorialism is apt for examination for my purposes.[vii] His is motivated by independent ontological restrictions prior to modal considerations; that is, he is convinced that naturalism[viii] is the correct ontological worldview. He is also convinced that modality is not primitive, worlds are not “out there, viewable by a strong telescope” as Kripke might think a Lewisian might hold, nor are possible worlds anything other than *merely possible*. All that exists is actual and, hence, only the actual exists. Therefore, if we are to speak of “possible worlds” it had better be reducible to talk of actual entities. Armstrong reduces possible worlds to fictional entities[ix] but this need not be the only option. As he puts it, “Each possible world is a different fiction about the way the world is. Logical space is the great fiction of a book of all these fictions: the book of worlds.”[x] In more detail, possible worlds are conjunctions of “atomic states of affairs”[xi] such that some worlds could have less (i.e. contracted worlds), there is never a propertyless individual (i.e. individuals always come with properties and properties always are properties of some individual(s)), and, further, all simple properties and relations in a world must be compossible (i.e. the world must make up the totality of states of affairs, not allowing contradictions).[xii]

The combinatorialist accepts what David Lewis called the ‘principle of re-combination’.[xiii] This is not to say that Lewis is a combinatorialist, but rather that, as Lewis wanted, “patching together parts of different possible worlds yields another possible world.”[xiv] Melia is apt in marking the point that accepting this principle does not automatically make one a combinatorialist. Lewis is recombining *duplicates* into further possible worlds; the combinatorialist is recombining particulars and universals.[xv] Thus, worlds are not arrangements of counterparts or even duplicates but genuinely new combinations of actually existing entities.

If one takes the combinatorial picture sketched here to heart, then the outlook is further outlined as the following. The combinatorialist is an “actual-world chauvinist”[xvi] in that only the actual world is a genuine world. Consequently, our pool of possibilities is limited to the re-combinations possible from the pool of entities in the actual world. This has the result that, strictly speaking, there cannot be what Lewis called “alien properties.” These are properties such that, from the point of view of a world w1, in some world w2 there are properties that are not possible at w1. That is, they are totally cut off from w1. This seems to actually be a criticism that can be leveled at the combinatorialist in that, given the possibility of contracted worlds, if w1 is a contracted world, relative to w2, then it seems there are universals in w2 that are alien to w1. This point is more salient when w1 is really the actual world, w*. It seems that there is a world w2 such that, relative to w*, it has more universals. If w2 does have more universals than w* then those universals are alien (that is, they are not instantiated in or by any actually existing entity; they are *merely possible* yet somehow existing). On the strong actualism of combinatorialism this is unnacteptable: if the actual world is the only genuine world then the actual universals (universals instantiated in/by actual particulars) set the limit for possible universals (and, similarly, for individuals). The only way, the combinatorialist will argue, for the possibility of alien universals to arise is if one takes possible worlds to be of the same ontological kind as the actual world. But, *pace* Lewis, this is not so; possible worlds add no extra ontological commitments for the combinatorialist. The actual world could not *itself* be less than it is[xvii] and thus any world w relative to w* cannot outstrip it (allowing that re-combination does not genuinely outstrip the actual). Therefore, there are no alien entities.[xviii]

This line of reasoning has a subtle and not straightforward consequence, but one which it would seem any combinatorialist of the sort described must, and does, accept. It follows from allowing contracted worlds relative to the actual world.[xix] Suppose that there is a world w1 such that it is not the actual world and such that it is constituted solely by the individual *a* and properties *S* and *G* such that *Sa* and *Ga* are true at w1. Further, suppose that accessible to w1 is w2, constituted solely by *Sa*. W2 is lacking the property *G*. Now, w2 is accessible relative to w1, given that all entities in w2 are also entities, recombined or otherwise, of w1. W1, however, is not accessible to w2, and this is the upshot W1 contains an alien property relative to w2; from the standpoint of w2 there is something in w1 that should not exist (*if w2 were the actual world!*). And so, reflexivity and transitivity in modality would still hold but symmetry breaks down. If symmetry were to hold among w1 and w2 (much less also to w*) then, relative to an un-countable number of contracted possible worlds, there would *actually* exist alien properties (and this cannot happen!). The result is simple: S5 is lost, S4 is the most we can have as a modal system. We cannot “get to” the actual world from a contracted world; the actual world is not a possible world relative to a world with fewer entities.[xx]

Perhaps, it could be intoned, that S5 can be saved if we stipulate that *every possibility* is realised in the actual world, making it an infinitely rich world. That is, if we start out with a *maximally* rich actual world, can we then have enough entities to bring back symmetry and with it S5? To ask this question is to entirely miss the combinatorialist’s point concerning the actual world and alien universals. No matter what is in the actual world, no matter how rich it is, there will be some world w such that, relative to w*, w has fewer properties; from the point of view of w, then, w* has alien properties (they don’t exist!). Hence, they would not be accessible from w and thus symmetry does not obtain and we are left with S4.

Notes

[i] See Pruss, Alexander. *Actuality, Possibility, and Worlds*. New York: The Continuum International Publishing Group, 2011 (14).

[iii] System T says that A→◊A and also that □A→A; S4 says that □A→□□A.

[iv] Konyndyk, Kenneth. *Introductory Modal Logic*. Notre Dame: University of Notre Dame Press, 1986, (51).

[v] To elaborate further, if one were able to stand at world w1 and “see” world w2 where ◊A is true, then it would be the case that, for *any* world w*n*, w2 would be accessible to w*n* and ◊A would be true. Therefore, for any proposition P, ◊A→□◊A.

[viii] Naturalism should be taken, as it seems Armstrong intends it to be taken, to mean roughly that actuality is limited to the universe as modern science tells us it happens to be. That is, there is only one spatiotemporally bound universe and all that exists is in it.

[ix] Armstrong, David. *A Combinatorial Theory of Possibility*. Cambridge: Cambridge University Press, 1989, (49).

[xv] Melia, 139. One might wonder whether it matters that the re-combinations be of particulars and universals, and, if so, if one is destined to accept haecceities. Armstrong (1989) is an example of one who wants to distinguish particulars from universals and manage to reject haecceities, given that he thinks there is no clear point where one can distinguish an entity from the properties it instantiates. Presumably Armstrong would run afoul of sorites paradoxes concerning when an entity can lose a property without ceasing to be that entity.

[xvii] Here, I think, is a genuine worry that the combinatorialist is making circular claims. That is, if the actual world is such that *it itself* could not be less than it is, then we seem to have smuggled modality back into the theory when it should have be reduced to fictional entities.

[xviii] Allowing alien entities would amount to saying that there *actually* exists entities which are possible but not *actual*, because all that exists is actual and only the actual exists. So, if there is something that exists which is not in the actual world, then it actually exists *and does not* actually exist, which is a contradiction. Q.E.D.

[xix] Contracted worlds ought, intuitively, to be allowed. Without them we would have to say that every (actual) individual and universal would be members of every possible world. But this is the definition of a necessary entity, and if such were the case then every entity would be necessary. However, most (if not all) entities are contingent! Therefore, contraction is needed. See Armstrong, 61.

Bibliography

Armstrong, David. *A Combinatorial Theory of Possibility*. Cambridge: Cambridge University Press, 1989.

Divers, John. *Possible Worlds*. London: Routledge, 2002.

Konyndyk, Kenneth. *Introductory Modal Logic*. Notre Dame: University of Notre Dame Press, 1986.

Lewis, David. *On the Plurality of World*. Oxford: Blackwell Publishing, 1986.

Loux, Michael. *The Possible and the Actual: Readings in the Metaphysics of Modality*. Ithica, New York: Cornell University Press, 1979.

Lycan, William. “The Trouble with Possible Worlds,” in: Loux, Michael. *The Possible and the Actual: Readings in the Metaphysics of Modality*. Ithica, New York: Cornell University Press, 1979.

Melia, Joseph. *Modality*. Montreal: McGill-Queen’s University Press, 2003.

Pruss, Alexander. *Actuality, Possibility, and Worlds*. New York: The Continuum International Publishing Group, 2011.

Ryan C

said:Every possibility actualized in the actual world? What would Aristotle think?!

Okay, I didn’t understand most of this post, but I still wanted to make some comment. 🙂

Kyle

said:Advanced logic, I presume? Wow.

benedico

said:Modal logic, not as advanced as it gets but a lot more fun than some things.

Chris Menzel

said:You write: “In the modal system S5 the accessibility relation, R, which allows for the ability to determine truth-values in the system, is such that accessibility is transitive, reflexive, and symmetrical. That is, given a semantics of possible worlds for the syntactical system of S5, we are able to quantify over possible worlds in such a way that we can refer to entities in the actual world (w*), in merely possibly worlds, and from merely possible worlds back to the actual world. This assumes that all worlds are mutually accessible.”

This is a common (and not terribly serious) non sequitur. From the fact R is an equivalence relation (i.e., refl, sym, and trans) it does not follow that every world is accessible from every other. Rather, in general, it follows only that worlds are grouped into isolated “galaxies” (more exactly, equivalence classes) such that only worlds within the same galaxy are mutually accessible. However, the mistake is largely innocuous because it has no effect on logical consequence: φ is an S5-logical consequence of a set of sentences Γ iff φ is true in every model of Γ in which all worlds are mutually accessible; in particular, φ is S5-valid iff it is true in every interpretation in which all worlds are mutually accessible.

As for your argument, there is no doubt this is a serious problem for the combinatorialist. A student of mine (now at Miami) took it up in his MA thesis and argued quite nicely for a form of combinatorialism that preserves S5: http://goo.gl/1igQO. The view is not without its soft spots, but it’s a creditable attempt to overcome the problem you raise.