One of the axioms of mereology is Unrestricted Composition. Here is David Lewis' formulation (Parts of Classes, Basil Blackwell 1991, p. 74):
Unrestricted Composition: Whenever there are some things, then there exists a fusion of those things.
A fusion is a mereological sum, so I'll use 'sum.' The axiom assures us that, for example, if there are some cats, then there exists a sum of those cats. The cats are many but the sum is one. So it is not unreasonable to think that if there are five cats that compose the sum, the sum is a sixth thing. One could argue as follows: (a) The sum is distinct from each of the cats. (b)There are five cats, each of which exists, and by UC the sum also exists. Therefore, (c) at least six things exist.
But consider this example, adapted from Donald Baxter. You proceed with six bottles of beer to the supermarket 'six items or fewer' checkout line. The attendant protests your use of the line on the ground that you have seven items: six bottles of beer plus one mereological sum. This would be an outrage, of course. The example suggests that the argument to (c) above has gone wrong.
Lewis avoids the mistake -- assuming it is one -- by pleading that "Mereology is ontologically innocent." (PC 81) That means that a commitment to a cat-sum is not a further commitment over and above the commitment to the cats that compose the sum. The cat-sum just is the cats, and they are it. This is the thesis of Composition as Identity. The xs compose the y by being identical to the y. As Lewis says,
Take them together or take them separately, the cats are the same portion of Reality either way. Commit yourself to their existence all together or one at a time, it's the same commitment either way. If you draw up an inventory of Reality according to your scheme of things, it would be double counting to list the cats and also list their fusion. In general, if you are already committed to some things, you incur no further commitment when you affirm the existence of their fusion. (PC 81-82)
I'm sorry, but this doesn't make much sense. Glance back at Unrestricted Composition. It is not a tautology. It does not say that whenever there are some things, then there are some things. It says that whenever there are some things, then there exists a fusion or sum of those things. Now if the sum of the xs is just the xs, then UC is a tautology. But if UC is not a tautology, then Composition as Identity is false. How can Unrestricted Composition and Composition as Identity both be true?
The problem is already present at the purely syntactic level. 'Y is identical to the xs' is unproblematic if the xs are identical to one another. For then the open sentence collapses into 'y is identical to x.' But if the xs are distinct from each other, then 'y is identical to the xs' is syntactically malformed. How can one thing be identical to many things? If one thing is identical to many things, then it is not one thing but many things. A contradiction ensues: the one thing is one thing and not one thing because it is many things. The gaps in the predicate '. . . is identical to ____' must either be both filled with singular terms or both filled with plural terms.
And now we come back to our main theme, eliminativist/reductivist ambiguity. Lewis wants to say that there is the sum of the xs (by Unrestricted Composition) but that the the sum of the xs is identical to the xs. So he seems to be making a reductionist claim: sums reduce to their members. But I say the thesis is unstable and topples over into eliminativism: there are no mereological sums. For if the sum is just its members, then all that exists is the members so that the sum does not exist!