However, there may be a further problem once Concretism is added to Neo-Humeism. It's one thing to say that it is reasonable to believe that the one and only, metaphysically unique actual world is simple and regular, and quite another to believe that I just happen to be located in one of the few simple and regular concrete worlds. The first has some plausibility as a fundamental principle of rationality, even if it is somewhat ad hoc. The second seems hopelessly optimistic beyond all reason (see Pruss 2011: 117–119 for a formalization of this argument).
4. The problem of non-indexical uses of ‘actual.’ There are linguistic data that Modal Indexicalists cannot account for. Consider (15):
(15) The actual world might not have been actual.
The first use of ‘actual’ in (15) does seem to be indexical, but the second use cannot be, since (15) seems obviously true. If world Alpha is the actual world, Modal Indexicalists must read (15) as equivalent to (16):
(16) Alpha might not have been Alpha.
But (16) is obviously false. However, (17) seems both true and equivalent to (15):
(17) Alpha might not have been actual.
Contrast, for example:
(18) The equator is here, but it might not have been here.
(18) hardly makes any sense. It seems to be saying that the equator might not have been the equator. (17), in contrast, is asserting that there is some property, actuality, which Alpha has but which it might have lacked. The trouble is that Indexicalists don't have the option of reading (15) as (17).
Modal Indexicalists might respond that (17) can be understood as true, since ‘might have been’ is to be interpreted as ‘is so in some other world’. This reading is suggested by (19):
(19) There is a possible world in which Alpha is not actual.
The problem is that ‘actual’ in (19) is not functioning indexically. If it were, (19) would be asserting the same thing as (20):
(20) There is a possible world in which Alpha is not Alpha.
But (20) is obviously false. How then, can we understand (19) as true? Modal Anti-Indexicalists will argue that (19) should be taken as a somewhat non-standard way of saying what is expressed by (21):
(21) There is a possible world w such that, were w actual, Alpha would not be actual.
According to Anti-Indexicalists, by talking about what is ‘actual’, we are talking about a special property of being actual that Alpha has and other worlds lack but could have had. We are not merely using an indexical to refer to our local world.
The best that that Concretists can do to make sense of (19) is to suppose that it is really making use of ‘actuality’ in ‘scare quotes’:
(19′) The actual world might not have been ‘actual’.
(19′) could be understood as pointing out that, when people in other possible worlds use the word ‘actual’, they are not referring to Alpha:
(22) There are other possible worlds in which the word ‘actual’ does not refer to Alpha.
(22) is certainly true, but Anti-Indexicalists can plausibly claim that the true meaning of (19) is given by the non-indexical (21), and not by (22).
5. The problem of isolation. If all of Concretism's possible worlds really do exist, then it seems that there is just One Really Big World, containing myriad parallel universes, rather than many possible worlds, as Concretism insists. We might put the point in the form of a question. What makes something a world, rather than part of a world? What, that is, isolates worlds from one another? Put another way, what makes something part of the same world as something else? What, that is, makes two things worldmates?
There are three plausible answers to these questions. First, one might say that the worldmate relation is fundamental, that it is just a primitive of the Concretists' theory. On this view, there is no informative answer to the question of what makes two things worldmates. This undermines the Concretists' reduction strategy, since the worldmate relation appears to be primitively modal if it's primitive at all. Why? Because one now has fundamental facts about which possible world something is a part of. Given that Concretism becomes quite implausible if it doesn't at least offer a reductive account of modality, this situation would be rather unhappy for Concretism.
Second, one might think that two things are worldmates if and only if they are causally interrelated. On this view, worldmate relations are causal relations. This view couples poorly with Neo-Humeism, since it appears to require fundamental causal relations. That is not, of course, a principled objection to the view, but it is worth noting given the standard combination of Concretism with Neo-Humeism.
Third, one might think that two things are worldmates if and only if they are spatiotemporally interrelated. On this view, worldmate relations are spatiotemporal relations. Both the causal and spatiotemporal reductions of the worldmate relation face the following problem. It seems that there might have been several parallel universes which are spatially, temporally and causally isolated from each other. If the worldmate relation is reduced either to causal relations, spatiotemporal relations or a combination of these, this is just not possible. Such parallel universes would count as different worlds, and so could not be universe-size parts of the same world. To avoid this problem, Concretists must give up the claim that a possible world is a maximal aggregation of spatiotemporally and/or causally connected things or deny that parallel universes are possible. Both of these moves are problematic. The first strips Concretists of any reductive account of the worldmate relation. The second suggests that Concretists cannot have as many worlds as there are possibilities. Both are seriously problematic for Concretism's reduction of modal facts to facts about possible worlds.
14.4 Conclusion
Concretism provides a simple and reductive account of possibility according to which proposition is possibly true just in case it is true in or with exclusive reference to some concrete, spatiotemporally connected universe. A proposition is actually true (for us) when it is true in our own local universe. Although attractively simple, Concretism faces some serious challenges, some of which we have canvassed above.
Note
1. Strictly speaking, this is only true if the number of possible worlds mirrors the number of possibilities, something we will assume here. One could reject this, but we won't get into the reasons for doing so.
15
Abstractionism: Worlds as Representations
Given the troubles with Concretism and its immense implausibility even on top of those problems, many philosophers have sought to give an account of possible worlds that doesn't commit one to a plenitude of universes. These philosophers tend to think that possible worlds, rather than being concrete, are abstract. That is, they are more like properties, propositions, and sets than they are like rocks, donkeys, and angels. This view about possible worlds is Abstractionism:
15.1T Abstractionism. Possible worlds are maximal possible abstract objects. (=14.1T.1A)
Some Abstractionists think that worlds are maximal possible states of affairs. States of affairs are much like ways, though understood as abstracta that can obtain or fail to obtain in virtue of the way things are. So, for example, the state of affairs of grass's being green obtains because grass is green. One might think of the state of affairs of grass's being green as obtaining because of the fact that grass is green. For Concretists, ways are facts. For Abstractionists, ways are states of affairs. Other states of affairs include THP's being taller than RCK, the United States having 50 states, grass's being purple, a square's having five sides, and so on. A state of affairs is possible if and only if it is possible for it to obtain. A square's having five sides is not possible: it simply could not obtain. But grass's being purple is possible, despite that it does not obtain in fact. A state of affairs s is maximal if and only if for any state of affairs s′, s either includes or precludes s′. State of affairs s includes state of affairs s′ if and only if necessarily, if s obtains then s′ obtains; s precludes s′ if and only if, necessarily, if s obtains then s�
�� does not obtain. States of affairs are maximal possible states of affairs if they are, in a sense, complete descriptions of the way the world could be, such that no more information is needed to fully characterize the world. Such states of affairs are, on this view, possible worlds. Some states of affairs are too “small” to be worlds because they aren't complete descriptions of the way things might be.
Other Abstractionists think that worlds are maximal possible propositions. This view is structurally similar to the view that possible worlds are maximal possible states of affairs. Obtaining and failing to obtain are replaced with truth and falsity, and maximality is characterized using entailment rather than inclusion and preclusion: proposition p is maximal if and only if for any proposition p′, p entails either p′ or not- p′. Still other Abstractionsists think that worlds are maximal possible classes of propositions. On this view, maximality is understood in terms of membership. Class of propositions S is maximal if and only if for any proposition p, either p is a member of S or not-p is a member of S.
A third group of Abstractionists take worlds to be maximal structural universals (see Section 10.3) that can only be properties of reality as a whole. The actual world is the one world that has an instance; all other worlds are uninstantiated maximal structural universals. A possible world is a property that has the second-order property of possibly having an instance. A world-universal u is maximal in the sense that, for every other structural universal u*, u's being instantiated either entails that u* is also instantiated or else entails that u* is not instantiated.
The distinctions among these types of Abstractionism won't matter for our purposes, since the most useful taxonomy of Abstractionist views categorizes them using the ways they make sense of how possible worlds represent possibilities, rather than in terms of the categorial nature of worlds. So far, we have not addressed this question. In what follows, we consider three ways that Abstractionists might account for how possible worlds represent that such-and-such. There is Magical Abstractionism (Section 15.1), according to which that question has no informative answer. Then there are two views that answer the question by appealing to structural features of worlds. There is Linguistic Abstractionism (Section 15.2.1), according to which possible worlds represent in the way that languages do. And there is Pictorial Abstractionism (Section 15.2.2), according to which possible worlds represent in the way that pictures do.
15.1 Magical Abstractionism
Magical Abstractionism is the view that possible worlds represent in a primitive way. On this view, all we can say about the way worlds represent is that a world w represents that p if and only if it is necessarily true that if w were actual, then p:
15.1T.1T Magical Abstractionism. A world w represents that p if and only if it is necessarily true that if w were actual, then p, and there is no substantive account to give about why possible worlds represent what they do.
Alvin Plantinga (1974) is most plausibly read as a Magical Abstractionist. He says nothing about why worlds represent as they do, and offers the right-hand side of the first conjunct of the above definition as his account of representation. Importantly, according to Magical Abstractionism, worlds do not represent in virtue of having some type of internal structure. Any parts, and any arrangements of parts, that worlds might have are simply irrelevant to what they represent. Worlds just have representational features, and that is that.
It is easy to see that Magical Abstractionism embraces the existence of fundamental modal truths. If Magical Abstractionism is true, there are metaphysically fundamental modal truths even in the characterizations of possible worlds suggested above. For ease, focus on worlds as maximal possible propositions. (Similar remarks apply to the other views.) The notion of a possible proposition is explicitly modal: p is possible if and only if it is possible for p to be true, or p could be true. Entailment is modal as well, and so maximality is modal. Magical Abstractionism has no resources to reduce these appeals to modality, given that the view cannot offer a reductive account of representation. Matters will become complicated when we consider Structural Abstractionism below. Structural Abstractionists attempt to supply a substantive account.
15.1.1 Modal Anti-Indexicalism
It is clear that Magical Abstractionists require fundamental truths about possibility, but what about actuality? Magical Abstractionists, and indeed all Abstractionists, are agreed in rejecting one type of reductive account of actuality, namely Modal Indexicalism (14.2T). If the term ‘actual’ functions as an indexical, then every possible world is a concrete entity or sum of concrete entities, like a place or a time. This is obviously incompatible with Abstractionism. Abstractionism thus requires Modal Anti-Indexicalism:
15.2T Modal Anti-Indexicalism. Some attributions of actuality are not indexical. (=14.2A)
There are several varieties of Anti-Indexicalism. The most straightforward is Simple Anti-Indexicalism, according to which the property of being actual is a simple, fundamental property.
15.2T.1 Simple Anti-Indexicalism. Actuality is a simple, fundamental property of possible worlds.
There are a number of problems with Simple Anti-Indexicalism. First, the following inference should be logically valid:
(1) Lions (actually) exist.
Therefore,
(2) The actual world contains lions.
Indeed, the two statements seem to be logically equivalent. But how can (1) and (2) entail one another if actuality is a simple property of possible worlds? (1) makes no reference to worlds at all. But to say that a world contains lions is to say that the existence of lions is part of its content. That is, it is the claim that if that world were actual, then it would be true that lions exist. To make the logical connection between (1) and (2) work, we need the following principle:
Uniqueness of the Actual World. Necessarily, one and only one world (namely, the actual world) is actual.
If the Uniqueness of the Actual World were necessarily true, then (1) and (2) would entail one another. If lions exist, then the actual world must contain the fact that they do because all worlds are maximal. The actual world, and every world for that matter, must either contain the existence or non-existence of lions, and given that there are lions, no world can be actual and contain their non-existence. However, if actuality is a fundamental property of worlds, Uniqueness of the Actual World would have to be a brute necessity.1 Given that actuality is a fundamental property, there is no immediate reason that it might not characterize more than one world. Clearly, actuality can be exemplified by many things, since many propositions are actual. Since worlds are just propositions, without Uniqueness of the Actual World, there is no reason multiple worlds might not be actual. Therefore, we cannot explain Uniqueness of the Actual World in terms of the impossibility of two worlds being actual, and its truth must be a brute necessity. This is a cost of Simple Anti-Indexicalism because we should minimize the postulation of brute necessities, according to Ockham's Razor and Structuralism:
PMeth 1.6 Sixth Corollary of Ockham's Razor: Other things being equal, prefer the theory that posits the smallest class of metaphysical possibilities.
PMeth 3 Structuralism. Other things being equal, adopt the theory that explains metaphysical impossibilities in terms of the essential structure of things.
It is worth looking for an alternative to Simple Anti-Indexicalism.
If Simple Anti-Indexicalism is false, what could actuality be? Suppose that we assume two primitive necessities: (1) there is exactly one possible world corresponding to each way the world could be, and (2) things correspond to a way the world could be if and only if things would be that way if that world were actual. Things are in fact some way or other, and necessarily so. Consequently, there must be exactly one possible world that corresponds to the way things in fact are. We could identify a total way things could be with a set of propositions, namely, the propositions that would be true if things (as a whole) were that way.
Consequently, we could assume the following three the
ses:
15.2T.2 Actual-Truth-Defined Anti-Indexicalism. A possible world is actual if and only if it corresponds to the class of (actually) true propositions.
Completeness of Actual Truth. Necessarily, there is a unique, maximal class that contains exactly the propositions that are (actually) true.
Existence and Uniqueness of Worlds. Every maximal class of propositions corresponds to exactly one world (either possible or impossible).
These three theses jointly entail the necessity of the existence of a unique actual world. Completeness of Actual Truth requires that there necessarily is a class containing the true propositions, and Existence and Uniqueness of Worlds implies that exactly one world corresponds to that class. Since Actual-Truth-Defined Anti-Indexicalism defines truth in terms of correspondence with truth, it follows that there is necessarily one and only one actual world. One consequence of this: we won't be able to make use of the Many Worlds account of ontological vagueness, discussed in Section 12.2.2.
This account of actuality involves two primitive notions, namely, the actual truth of propositions and the correspondence relation between worlds and classes of propositions. What are we to do with this notion of ‘actual truth’? We could take it to be a special case of the actual possession of a property by a thing. A proposition is actually true if the proposition is in actual possession of the property of truth. So far, this is a theory on which actuality is a fundamental property of worlds, since we are treating the actual possession of properties as a primitive, metaphysically fundamental fact.
The Atlas of Reality Page 54
