Wolfgang Schwarz

The Principle of Recombination for Properties

As a principle of plentitude, Recombination for Individuals is far too weak. If there happens to be nothing that is both red and dodecagonal, the recombination principle for individuals gives us no world where anything is. Likewise, if it happens that no red thing is on top of a blue thing, the principle gives us no world where this is different. But combinatorial reasoning seems to give us such worlds.

The relevant recombinations are recombinations of properties (red, dodecagonal) and relations (on top of). What we want, then, is a principle of recombination that tells us that for any distribution pattern of properties and relations, there is a corresponding possible world.

The properties and relations here must be restricted. Red and not-red can't be freely recombined, nor is there a world where a lonely object is an uncle. The principle should be restricted to intrinsic properties, and to some class of properties that are non-redundant in the sense that their distribution pattern doesn't supervene on the distribution pattern of one of their proper subsets. Lewis's 'fundamental properties' seem to fit that description.

Suppose we've identified some such class of properties and relations. (I'm not sure this can be done in any non-circular way. But suppose.) Let's call them 'fundamental'. The principle now says:

PRP: For any distribution pattern of fundamental properties and relations, there is a world that realizes the pattern.

What does this mean? First, what is a distribution pattern? One might think of it as a big Ramsey sentence: (Exy...)(Fx & ~Gx & ... & ~Fy & Gy & ... & Rxy & ... & (Az)(z=x v z=y v ...)) with F,G,R,... naming all fundamental properties. For a world to realize a pattern would then mean that the pattern is true at the world.

We could shorten these sentences to something like (Exy)(Fx & Gy & Rxy) by stipulating that if the sentence doesn't say that x has F, then it represents x as not having F, and if it doesn't mention any objects besides x and y, then it represents there being no such objects.

I prefer to think of a pattern not as a sentence, but as a more abstract mathematical structure, something like a partial mapping from sequences of ordinals into the power set of fundamental properties. The ordinals represent the individual inhabitants of a world. The mapping {((0),{F,G}), ((1),{F}), ((0,1),{R})} would represent a world with two individuals (represented by 0 and 1), of which one is both F and G, the other is only F, and the first stands in relation R to the second. In general, a world w realizes a pattern p iff there is a bijection b from the inhabitants of w to the ordinals in the sequences in the domain of p such that individuals x_1,...,x_n from w have a fundamental n-place property F iff F is a member of p(b(x_1),...,b(x_n)). (The mappings should be restricted so as to respect the arity of the fundamental properties.)

For a Lewisian Modal Realist, the current principle is too strong: it leaves out the privileged role of spacetime relations by allowing for worlds with spatiotemporally isolated parts. It seems to me that disallowing such worlds is a serious defect in Lewis's Modal Realism, so I wouldn't blame the principle. On the other hand, it wouldn't be too hard to build the restriction into the principle. For now, let's not. (If we have patterns that correspond 1-1 with the worlds, why not take the patterns to be the worlds?)

A bigger problem again arises from mereology. What are the individuals in the domain of the existential quantifiers, or of the bijections b: are they mereological atoms, or arbitrary objects? They shouldn't be atoms only, as there is no guarantee that fundamental properties can only be instantiated by atoms. (Maybe large things can have charge -1, which is fundamental. At any rate, there could be emergent fundamental properties, or non-atomic gunk.)

But nor should they be arbitrary objects. Otherwise the principle gets both too weak and too strong. It gets too weak because it doesn't tell us that there are three worlds satisfying (Exyz)(Fx & Gy & Hz): one in which some F-thing is a fusion of a G-thing and an H-thing, another in which some G-thing is a fusion of an F-thing and an H-thing, and a third one in which some H-thing is a fusion of an F-think and a G-thing. It gets too strong because it tells us that there is a world satisfying (Exy)(Fx & Gy): a world with two inhabitants and no fusion, contradicting mereological universalism.

I here assume that parthood is not among the fundamental relations. What if we add it into the mix? We still get the impossible pattern (Exy)(Fx & Gy), and we get more, e.g. (Exy)(Fx & Gy & xPy). The problem is that parthood doesn't fit the constraints on fundamental relations: it isn't freely recombinable. So we shouldn't add it.

Instead, we should add mereological structure to our patterns. In the sentential framework, we could introduce a fusion operator '+' so that a worldbook can say (Exy)(Fx & Gy & H(x+y)). Then we might interpret the quantifiers to range only over distinct objects. If the worldbook says nothing about a certain fusion, it represents the fusion to have no fundamental properties.

In the more mathematical construction, we could replace the ordinals by sets of ordinals, representing fusions of the things represented by the individual ordinals. Hence a pattern is now a partial mapping from sequences of sets of ordinals into the power set of fundamental properties, and a world w realizes a pattern p iff its inhabitants divide exhaustively into distinct parts X such that there is a bijection b from the X to the ordinals in the sequences in the domain of p such that individuals y_1,...,y_n from w have a fundamental n-place property F iff each y_i is the fusion of the members of some set x_i of things from X such that F is a member of p(b(x_1),...,b(x_n)).

This has a similar structure to my proposal for the principle of recombination for individuals, and that's no accident: Since the distribution of duplicates of individuals in a world is determined by the distribution of properties and relations in that world, the recombination principle for properties should entail the recombination principle for individuals.

One more problem: I've assumed that fundamental properties are freely recombinable over non-distinct objects. But that seems wrong. An object consisting of 1000 distinct parts all of which weigh 1 gram can't itself weigh just 1 gram (or so it seems). I'm not sure how to deal with this problem. We get a similar problem if we take each individual mass or distance as a single fundamental property (as Lewis suggests somewhere); recombination of properties would then allow for things that weigh both 1 and 2 grams, and for things that are both 1 and 2 meters apart. But that seems to be the wrong treatment of magnitudes anyway. I don't know how best to treat magnitudes. So my plan is to add the present problem as a further constraint on a satisfactory account of magnitudes. It follows that any satisfactory account of magnitudes will solve the problem!