On the ontological commitment of mereology

In Parts of Classes [1991] David Lewis argues that, like logic, but unlike set theory, mereology is “ontologically innocent”. Prima facie, Lewis’ innocence thesis seems to be ambiguous. On one side, he seems to argue that, given certain objects Xs, referring to their sum is ontologically innocent because there is not a new entity as referent of the expression “the sum of the Xs”. So, talking of the sum of the Xs would simply be a different way of talking of the Xs, looking at them as a whole. However, on the other side, Lewis’ innocence is not understood as a mere linguistic use, where sums are not reified. He himself claims that the innocence of mereology is different from that of plural reference, where the reference to some objects does not require the existence of a single entity picking up them in a whole. In the case of plural quantification “we have many things, in no way do we mention one thing that is the many taken together”. Instead, in the mereological case: “we have many things, we do mention one thing that is the many taken together, but this one thing is nothing different from the many” ([1], 87). But, due to the fact that Lewis explicitly uses sums as outright objects, we think that Lewis’ innocence thesis cannot be understood but in the sense that, even if the sum of the Xs is a well determined object, distinct from the Xs, the existence of such an object is to be necessarily accepted from whom which has already accepted the existence of the Xs. In other words, committing oneself to the existence of the Xs would be an implicit commitment to some other entities and – among them – the sum of the Xs. On the other hand, the existence of the set of the Xs would not be implicitly guaranteed by the existence of the Xs. The aim of the paper is to argue that – for a certain use of mereology, weaker than Lewis’ one – an innocence thesis similar to that of plural reference is defendable. In order to give a definite account of plural reference, we use the idea of a plural choice. Then, we propose a virtual theory of mereology, where the role of individuals is played by plural choices of atoms. A choice is not an authentic object, its existence is merely potential and it consists in the act of performing it. Accordingly, in order to interpret a formal first order mereological language, as Goodman calculus of individuals (CG), we introduce a potential semantic of plural choices. We argue that our development of virtual mereology, grounded on the notion of plural choice, is ontologically innocent in a way completely analogous to that of plural reference: our claim is that mereological sums – unlike atoms – are not real objects. Referring to a sum of atoms is nothing but a way of referring to certain atoms. Our approach is adequate to interpret a first order mereological language. It is inadequate for Lewis’ mereology, because his plural quantification on all objects is incompatible with our notion of plural choice, where just atoms are capable of being chosen.