15.1T.1A Structural Abstractionism. Possible worlds represent what they do in virtue of their internal structure.
What sorts of structures might do the trick? An initial strategy is to identify common objects that represent the world somehow, and discern how it is that those objects do their representing. One then builds worlds on analogy with those common representational objects. Two sorts of things spring to mind: linguistic items like sentences and stories, and pictures of various sorts like paintings and photographs. Linguistic and Pictorial Abstractionism are respectively attempts to model the way worlds represent on analogies with linguistic items and pictures.
15.1T.1A.1 Linguistic Abstractionism. Structural Abstractionism is true, and worlds represent in the way linguistic things do, namely, by having components which represent things and which are arranged in according to a kind of “grammar”.
15.1T.1A.2 Pictorial Abstractionism. Structural Abstractionism is true, and worlds represent in the way pictures do, namely, by having components that literally share features with what they represent.
We examine these views in turn, after a comment about whether Structural Abstractionism can supply a reductive account of modality.
We have seen that Magical Abstractionism is committed to the irreducibility of modal facts, though there are ways of being a Magical Abstractionist while holding that actuality in particular is reducible. The question that will occupy us here is whether Structural Abstractionists might embrace the reducibility of all modal facts. There is some reason to think that it might. Representation was modal according to Magical Abstractionism precisely because no account of it was on offer. Maybe a structural account of representation can supply the resources to reduce all the other recalcitrant modality as well. The key, it seems, is to find a reductive account of the notion of a maximal possible proposition. Recall that proposition p is maximal if and only if for any proposition p′, either p entails p′ or p entails not-p′. There isn't anything obviously modal about that. Further, we might try to give the following reductive account of what it is for a proposition to be possible: proposition p is possible if and only if p does not entail a contradiction (that is, if and only if p does not entail some proposition and its negation). Again, nothing obviously modal there.
However, entailment is modal. Proposition p entails proposition q if and only if necessarily, if p is true then q is true. One might try to get around this by saying that p entails q if and only if p and not-q are inconsistent. But again, inconsistency is an implicitly modal notion. Two propositions are inconsistent if and only if it is possible to derive a contradiction from them, or if and only if it is not possible for the two to be true together.
It seems, therefore, that Structural Abstractionism, like Magical Abstractionism, is best understood as embracing fundamental modal truths.
15.2.1 Linguistic Abstractionism
Linguistic Abstractionism is the view that worlds represent in the way linguistic things do. That is, worlds are something like sentences or very long and detailed stories. But how do linguistic items represent? Consider a sentence like ‘Elsie is a dog.’ This sentence represents Elsie's being a dog, or represents Elsie's exemplifying the property of being a dog. It does this by having a word, ‘Elsie’, which represents Elsie, and a string of words, ‘is a dog’, which represents the property of being a dog, and arranges those parts according to a grammar which represents that the thing represented by the one thing exemplifies the thing represented by the other thing. Sentences, then, contain parts (words and phrases) that represent certain things in the world, and are arranged according to a grammar that represents a certain arrangement of those things represented by those parts. Worlds, then, according to Linguistic Abstractionism, are built up out of parts that stand for things like words do, and which are put in certain quasi-grammatical arrangements, a propositional “grammar”, that represent the arrangements of those represented things. That is how worlds represent.
Importantly, one needn't subscribe to the view sketched above about how sentences represent in order to embrace Linguistic Abstractionism. Absolutely everyone has to say something about how words and phrases and grammars contribute to the meanings of sentences, and Linguistic Abstractionists simply want to say that the way propositional elements contribute to the representational features of propositions, and therefore of worlds, need be no different from one's favored view of linguistic representation.
What sorts of parts one opts for needn't detain us much at this point. But David Lewis recommends that Linguistic Abstractionists opt for a “Lagadonian” language (an allusion to Swift's Gulliver's Travels). In a Lagadonian language, the name for each thing is that thing itself. So, the proposition that Obama is a Democrat contains both Barack Obama himself (as the proposition's name for Obama) and the property of being a Democrat (as the proposition's predicate for having that property). Importantly, Linguistic Abstractionists have traditionally been Actualists, and so will use only actually existing things to build their worlds.
15.2.1.1 Combinatorialism.
A very simple variety of Linguistic Abstractionism takes possible worlds to be mathematical combinations of basic individuals and natural properties. (To see why this can plausibly be considered a version of Linguistic Abstractionism, recall the Lagadonian language noted above.) This sort of view corresponds to an account of propositions developed by Ludwig Wittgenstein in the Tractatus Logico-Philosophicus (1922/1961). Consequently, this account of possibility is often called ‘Tractarian’.
Def D15.3 Tractarian Propositions. A Tractarian proposition is an ordered n+1-tuple consisting of an n-ary natural (fundamental) property and n basic individuals.
Def D15.4 Tractarian Worlds. A Tractarian world is a set of Tractarian propositions (intuitively, precisely those Tractarian propositions that would be true if the corresponding possible world were actual).
15.3T Combinatorialism. Every Tractarian world corresponds to a possible world.
15.3A Anti-Combinatorialism. Some Tractarian worlds correspond to no possible world.
Combinatorialists believe that every Tractarian world represents a real possibility. That is, every mathematically possible assignment of basic n-ary relations to n particular things could correspond to a way things could be. If I'm a particular thing and being an electron is a basic property, then there is a possible world in which I'm an electron. If being red and being green are basic properties, then it's possible for something to be both red and green.
Combinatorialists owe us an account of what to do with Tractarian worlds that contain no proposition involving some actual individual or some actual property. Suppose there is a Tractarian world that includes no proposition involving Socrates. Should we understand the Tractarian world as corresponding to the possibility in which Socrates does not exist or in which Socrates exists but has no natural properties? If the Tractarian world contains no proposition involving some natural property (like being negatively charged), should we understand it as corresponding to a world in which negative charge does not exist, or one in which it exists but has no instances? Can individuals exist without natural properties? Can natural properties exist without instances? The most natural way of taking the Tractarian picture would be the one Wittgenstein himself endorsed, according to which each basic individual and each natural property exists in all possible worlds. Every basic thing is a necessary being (see Skyrms 1981).
Presently, we sketch very briefly arguments for and against Combinatorialism.
arguments for combinatorialism
Combinatorialism offer a simple, reductive account of possibility. Whether or not something is possible is just a mathematical question: does a structure of the right kind exist?
Combinatorialism minimizes the class of metaphysical necessities. There are no necessary connections between distinct individuals or natural properties. This argument appeals, of course, to Ockham's Razor (PMeth 1.4).
arguments against combinatorialism
It's implausible to suppose that everything exists necessarily. We can imagine many things not existing at all. For example, we can imagine a Big Bang that results in far fewer fundamental particles than the actual Big Bang produced. Combinatorialism is committed to counting this as absolutely impossible.
Furthermore, it certainly seems possible that there might have existed things and properties that don't actually exist. The Big Bang could have produced more particles and even more basic kinds of particles than it actually did, but Combinatorialists must deny this.
The motivation for Combinatorialism, as illustrated by David Armstrong, involves a strong aversion to brute necessities of any kind. All necessities are supposed to be purely logical or conceptual in character. However, the attempt to avoid all brute necessities seems to be quixotic. Even the Tractarian Combinatorialists admit some inexplicable necessities. For example, why do two-place relations, like greater-than or part-of, require two relata in order to make a complete possible fact? Why couldn't there be just one relatum, or any number of relata? Why does every possible fact require both one natural property and one or more basic individuals? Why couldn't a complete fact contain just properties or just individuals? Once we admit that there are some brute necessities, it seems reasonable to be open to further necessities, so long as these are needed to fit the theory to the rest of the data.
Like other versions of Linguistic Abstractionism, Combinatorialism faces the problem of alien possibilities, to which we turn presently.
15.2.1.2 The problem of alien possibilities.
Linguistic Abstractionists face the problem of alien possibilities. They have a problem providing the sentences needed to represent certain possibilities involving non-actual items. Consider a Max Black world, consisting of two indiscernible steel balls, neither of which actually exists. The sentences representing this world cannot be in the pure Lagadonian language, since there are in fact no steel balls to act as names of themselves. We'll have to introduce some non-Lagadonian names, by pressing some abstract objects into service as names. However, the connection between any such name and its referent will be merely conventional or arbitrary. How then can we make each name refer to a different ball, when there are in fact no balls to refer to? How can we ensure, in other words, that if the sentence were true, that it would then designate one determinate possibility rather than another? In technical terms, Linguistic Abstractionists conflate what seem to be distinct possibilities into a single class of sentences, since there is no way for such sentences to distinguish between things that don't actually exist but would have been distinct, had they existed.
This conflation problem arises in other cases as well. Consider a second thought-experiment involving a Nietzschean world in which the same history endlessly repeats itself, but with different entities playing the historical roles in each epoch. In such a world, there would be a different person in the Nietzsche-role on each repetition. We can use an infinite class of sentences to represent such a possibility, but we cannot distinguish certain permutations of such a world. For example, suppose the Nietzsche-role occupier from the first epoch changed places with the Nietzsche-role occupier from the second epoch. This seems to be a different possibility from the one in which they are in their original epochs, but Linguistic Abstractionists cannot distinguish the possibilities. Since neither Nietzsche-role occupier actually exists, a Lagadonian language is inadequate, and there is no way to ensure that the “name” in a non-Lagadonian language which is putatively about the first Nietzsche-role occupier uniquely picks out that thing rather than the second, and vice versa.
Similarly, suppose you think that there could have been alien properties, properties that don't exist and are not instantiated in the actual world, like a third charge, in addition to positive and negative. These alien properties are like the Max Black spheres in that a Lagadonian language is too limited to name them, and a non-Lagadonian language will conflate possibilities involving such alien properties.
There are four solutions to the problem of alien possibilities that have been offered by Abstractionists. First, one could postulate the existence of uninstantiated haecceities or thisnesses (Def 9.2). These are properties that would, if they were instantiated, ensure that some object is identical to a certain thing. Haecceities are meant to exist necessarily, even though the objects (if any) to which they belong do not. Plantinga (1974) defends the existence of haecceities. Second, one could assume that the very same things exist in every possible world. On this view, everything exists necessarily, and nothing could exist except what actually does exist. As we have seen, this is the position of all Combinatorialists.
Third, one could assume that the actual world is special in that every possibility involving things—every de re possibility (see Section 16.1)—involves things that exist in the actual world, but that there are possible worlds that are ontologically impoverished in the sense that they do not contain enough objects to represent all the de re possibilities. This is the view of Kit Fine (1985), Robert M. Adams (1979) and others. Plantinga has labeled this position, with tongue in cheek, as ‘Existentialism’. (No connection with Sartre or Camus.) Fourth, one could suppose that many things do not exist. That is, one could go for Anti-Actualism, whether Possibilist (12.1A.1T) or Meinongian (12.1A.1A). These non-existing things can be parts of possible worlds in which they would exist (if those worlds were actual) (see Section 12.1).
Each of these solutions has its drawbacks. Existentialists have complained that uninstantiated haecceities (the first solution) are a strange or even incomprehensible kind of property. Robert Adams believes in thisnesses, but he supposes that the property of being identical to A can exist only if A itself actually exists. So, according to Adams, there is a thisness for Pope Francis, but no thisnesses for any of his possible but non-actual grandchildren.
A second problem for the first solution is that it seems to make it impossible to have genuinely de re propositions. Every apparently de re proposition is really about, not the thing itself, but its haecceity. If one thinks that Socrates could have avoided the death penalty, then one is really thinking that Socrates' haecceity (Socrateity) could have been co-instantiated with the property of avoiding the death penalty. In effect, advocates of haecceities turn all de re thoughts into de haecceitate thoughts, thoughts about a haecceity. This has some severe consequences about what thoughts we can think. Suppose, for example, that we want to assert the following:
(3) Socrates doesn't instantiate any haecceity.
According to Plantinga, this sentence must express the same proposition as the absurd (4):
(4) Socrates's haecceity is co-instantiated with the property of not instantiating any haecceity.
Similarly, when Plantinga asserts (5), he seems to be asserting something substantive, not the trivial proposition expressed by (6):
(5) Whatever instantiates Socrates' haecceity is necessarily identical to Socrates.
(6) Whatever instantiates Socrates' haecceity necessarily instantiates Socrates' haecceity.
Thus, the first solution comes at a significant cost in terms of what genuinely de re propositions can be formulated.
As we have seen in discussing Combinatorialism, the second solution is problematic because it seems that some things exist only contingently, and that there could have been other things than those that actually exist.
The main difficulty for the third solution is that it is hard to believe that we are so lucky as to exist in one of the relatively few special worlds that contain everything that enters into every de re possibility. Christopher Menz
el (1991) has proposed a way of modeling alien possibilities that involves using abstract objects to act as the substitutes or surrogates for the missing entities that exist in other possible worlds but not in the actual world. This can certainly be done, since there are plenty of such abstract objects around. However, Abstractionists must believe that possible worlds are identical to, not merely represented by, Menzelian representations. So Menzel's theory must postulate a deep, metaphysical difference between de re possibilities involving only actually existing things and alien de re possibilities.
With respect to the fourth solution, we've already discussed the pros and cons of Anti-Actualism in Section 12.1, so we won't repeat them here.
Finally, Combinatorialists must simply deny that alien possibilities are really possible. No individual could exist other than the individuals that do exist, and no property could exist other than the properties that have actual instances.
15.2.2 Pictorial Abstractionism
Pictorial Abstractionism is the view that worlds represent in the way pictures do. That is, worlds are like photographs or maps. But how do photographs and maps represent? Consider a picture of a family of four, with a dad, a mom, a son, and a daughter. To make things easier, give them names: Tim, Jamie, Lyle, and Gretchen, respectively. How does the picture represent Tim? By having a part, the Tim-part, with features like Tim's, a red part that represents Tim's beard, a blue part that represents his shirt, and a white part that represents his shoes. It represents Tim as taller than Jamie by having the Tim-part be larger than the Jamie-part. It represents Lyle's blue eyes by having two blue parts in the head part of the Lyle-part. And it represents Gretchen's impish smile by having an impish-smile part of the head part of the Gretchen-part. Pictures, then, represent by having parts whose features match the features of the things represented, where those parts are arranged in ways that match the arrangements of the things represented.
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