Lewis on Set Theory David Lewis in the short monograph

§3. Protestations of Innocence

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§3. Protestations of Innocence

Lewis's mereological assumptions amount to what mathematicians call the theory of a complete Boolean algebra, or ignoring gunk, a complete atomic Boolean algebra, consisting of a number of atoms — just how many will turn out to be the great issue — and arbitrary fusions thereof. For what mathematicians call completeness amounts to what mereologists call unrestricted composition, permitting arbitrary fusions. There is just one departure from the usual mathematical approach, the dropping of the assumption of a null item in the algebra, which is one instance where Lewis does object to introducing an "ideal element" to round out a system. The minor complications this course involves him in will be ignored in the exposition below.

By completeness or unrestricted composition, absolutely abitrary unions or joins are possible, as are absolutely arbitrary intersections or meets except those that would turn out to be null, and a complement exists for anything except the universal item or fusion of all things, whose complement would be null. Lewis, by the way, calls that universal fusion "Reality," and idiosyncratically calls two things whose intersection would be null — in other words, two things that are nonoverlapping or disjoint — "distinct," a word more normally used to mean nonidentical.

Despite upholding unrestricted composition, yielding fusions of scattered, heterogeneous parts — fusions critics have considered monstrous and mythical, or in a word, chimeras — Lewis wishes to claim a kind of ontological innocence, comparable to that of first-order logic, for mereology. The claim of ontological innocence is largely based on a variant of Donald Baxter's thesis of "composition as identity," the claim that when one thing is the fusion of many things, "They are it and it is them."

Lewis takes this identity thesis somewhat less than literally, claiming that the relation of things to the fusion thereof is, though not strictly speaking identity, at any rate "analogous" to identity. It is difficult, however, to see how anything less than literal identity could suffice for ontological innocence. It is difficult, also, to see how Lewis can be acquitted of question-begging when he argues that one respect in which there is analogy is in ontological innocence.

For Lewis, the plural includes the singular: He does not object to the kind of counting in of limiting or degenerate cases involved in reading "there are some things" as "there are one or more things" (though he would object to reading it as "there are zero or more things"). The relation he finds analogous to identity amounts to the relation some things, the xs, bear to other things, the ys, just in case (i) the fusion of the xs is identical with the fusion of the ys, and (ii) either there is just one single x or there is just one single y or both. This includes identity of the usual kind, between one single x and one single y, as a special case.

Apart from symmetry, however, the relation in question lacks the usual formal properties of identity. It is not reflexive, since some two or more things, the xs, never stand in this relation to themselves, or to any other two or more things, the ys. And though it is transitive in the sense that when a single x bears this relation to some ys and those ys bear the same relation to a single thing z, then x is identical with z, it is intransitive in the sense that even when some two or more things, the xs, bear this relation to as single thing y and this y bears the same relation to some two or more things, the zs, the xs still do not bear this relation to the zs.

Nor need the xs then be identical with the zs as plural identity is usual understood. For the usual understanding requires that each single thing among the xs be also among the zs, whereas the eight ranks of a chessboard bear the Lewis relation to that chessboard, and the chessboard bears the Lewis relation to its eight files, while the ranks are not the files. Above all, as Lewis acknowledges, the indiscernibility of identicals fails utterly for plural things and their single fusions, since they are many while it is one. In our example, the ranks are horizontal while the files are vertical; and though there are eight ranks and eight files, there are sixty-four squares, whose fusion is again the same old chessboard.

It may be that in some sense the fusion is nothing over and above the things it is the fusion of, as Lewis asserts; but the things seem to be something over and above their fusion, consisting of that fusion plus a particular mode of division. Needless to say, the "plus" here is not mereological summation or fusion. In subsequent discussion on "composition as identity" — see Sutton (2008) for an overview — critics have outnumbered defenders. Yet even on this, his weakest point, Lewis's discussion, in his inimitable style, remains well worth reading.

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