Wolfgang Schwarz

The Principle of Recombination for Individuals

Many versions of the recombination principle are floating around in the literature. Most of them are principles for individuals, saying, roughly, that you get a possible world by patching together (copies of) arbitrary parts of other possible worlds. (I will have more on principles for properties later.)

It is surprisingly difficult to make this precise. All attempts I know of fail in one way or another. To illustrate some of the pitfalls, let's begin with this classic version from Daniel Nolan's "Recombination Unbound".

PRI-1: For any objects in any worlds, there is a world that contains any number of duplicates of all of those objects.

A minor problem here concerns the quantifiers. PRI-1 is not meant to be read as:

PRI-1a: For any objects X in any worlds, there is a world w such that for any number n, w contains n duplicates of each X.

This is too weak. It tells us only that patching together infinitely many copies of any things from any worlds results in a new world: w contains n copies for any number n. Swapping the quantifiers helps a little:

PRI-1b: For any objects X in any worlds and any number n, there is a world w such that w contains n duplicates of each X.

But that still doesn't give us a world with exactly three copies of Berlin and two of New York, as the same number n is used for all X. We need several numbers, one for each of the X. Plural quantification over number-thing pairings gets it right:

PRI-1c. For any pairs X of a number and an object from any world, there is a world w such that, for each (n,e) among the X, w contains n duplicates of e.

All this is only a minor problem as due to some magical behaviour of the word "any" in English, PRI-1 at least seems to have one reading on which it expresses the same as PRI-1c.

The bigger problem is how to interpret "containing n duplicates": does that mean i) containing exactly n duplicates, or ii) containing at least n duplicates?

If "exactly n", the principle becomes too strong. It entails that there are impossible worlds containing, say, exactly three duplicates of Berlin and two of Kreuzberg. But Kreuzberg is part of Berlin, hence every copy of Berlin contains a copy of Kreuzberg; so there is no such world.

That's why Daniel opts for (ii): "at least n". However, on this reading, the principle becomes too weak again. It is compatible with everything having a copy in every possible world. Recombination should allow us not only to multiply, but also to subtract parts of worlds. On reading (i), we could exclude an object from a world by pairing it with zero. But on reading (ii), this only means the world contains zero or more copies of the object.

I once thought the principle could be fixed by adding "and nothing else":

PRI-1d: For any objects in any worlds, there is a world that contains any number of duplicates of all of those objects, and nothing else, i.e. nothing distinct from them all.

This (or anyway the more complicated version paralleling PRI-1c) correctly entails that nothing has a copy in every world. But, like PRI-1c, it doesn't tell us that there is a world with exactly three copies of Berlin. All worlds might have either zero or infinitely many Berlins.

As a first step towards a solution, consider this rather different approach.

PRI-2: For any objects e1 and e2 from any worlds, there is a world that contains a copy of e1 and, distinct from that, a copy of e2, and nothing else.

To get a world consisting of three Berlins, two New Yorks and one additional Kreuzberg, you apply the principle several times. First select Berlin as both e1 and e2 to get a world with just two Berlins; then select the fusion of those two as e1, and Berlin again as e2, etc.

Unfortunately, PRI-2 does not give us a world with infinitely many copies of Berlin, and it doesn't provide for worlds patched together from copies of things out of infinitely many worlds. And it doesn't give us worlds with only a single atom, let alone empty worlds. We might try:

PRI-2a: For any objects in any worlds, there is a world containing as distinct parts any number of duplicates of each of them, and nothing else.

The "any numbers" here are exact numbers rather than at-least numbers, and the "distinct parts" clause is meant to generalize the "distinct from that" in PRI-2. The idea is to let the numbers count only duplicates 'added' to the resulting world for the selected object. Sadly, I doubt that the grammar of English allows PRI-2a to have this intended meaning. Here is a more precise version.

PRI-2b: For any pairs X of a number and an object from any world, there is a world containing, for each (n,e) of the X, n duplicates of e which are distinct from any duplicates of e that are a proper part of an object duplicating an object in another pair among the X.

For instance, beginning with the pairs (3, Berlin), (2, New York), (1, Kreuzberg), PRI-2b gives us a world consisting of three Berlins, two New Yorks and one additional Kreuzberg, making it 4 Kreuzbergs in total.

The problem with PRI-2b is that it is pretty much incomprehensible. <update>And that it doesn't work; see comments below.</update> Here's yet another, slightly simpler approach -- and my current favourite.

PRI-3. For any pairs X of a number and an object from any world, there is a world dividing exhaustively into distinct things Y that correspond 1-1 with the X in such a way that each of the Y is mapped to one of the X whose non-number component it duplicates.

This time, the numbers are merely indices to allow for repetition in the plural quantification. Thus we get our example world by starting out with, for instance, the pairs (1,Berlin), (2,Berlin), (3,Berlin), (1,New York), (2,New York), (1, Kreuzberg). By PRI-3, there is a world consisting entirely of 6 distinct parts that correspond to those six pairs in such a way that three of them duplicate Berlin, two New York and one Kreuzberg -- the non-number components of the pairs they correspond to.

Throughout, all "numbers" should be understood as cardinals, not merely naturals, to allow for infinite multiplication. (To get worlds with only one or zero inhabitants, we should read "for any pairs" as "for any (zero or more) pairs". For worlds with proper-class many inhabitants, the mapping in PRI-3 will need to be a proper class; without proper classes, we're in trouble, because the quantification over mappings probably can't be pluralized away.)

PRI-3 is tolerable, I believe. But it's still odd that the intuitive idea of the recombination principle gets so complicated when properly spelled out. Have I missed a simpler alternative?