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'BUNDLE' INDIVIDUALITY
'Principles' of individuality that involve sets or 'bundles' of properties or attributes must face the problem of multiple instantiability which can be expressed in the further question: what is it that guarantees that some other entity could not possess the same set or subset of properties? Failure to come up with such a guarantee would be disastrous for this approach. One response is to invoke some set or subset of properties, together with some further principle which ensures that no other 18218t1911s entity could possess that set or subset. Thus, one might take the relevant properties to be or include spatio-temporal ones and invoke the Impenetrability Assumption (IA), to the effect that no two entities can occupy the same spatial location at the same time; that is, the points of space-time are either monogamous or virginal. 10 As we shall see, this is a common response to this issue not only in philosophy but also in (classical) physics. More generally, the requisite guarantee has been sought in Leibniz's Principle of Identity of Indiscernibles (PII), which insists that there cannot be two, or more, individuals which have all relevant properties in common.
This principle occupies a pivotal position in the debate between the adherents of positions (a) and (b) above. Not surprisingly there has been considerable discussion regarding its status. On the one hand, it has been regarded as necessary, in the sense that the very supposition that there might be two things exactly alike in all their properties is self-contradictory. Leibniz himself suggested that it followed from his 'Complete Notion of an Individual', 11 according to which the notion of an individual contains all the predicates that individual will possess. By definition, no two individuals can possess the same Complete Notion or, equivalently, the same set of properties. Such a view motivates the 'bundle' theory by effectively resolving individuality into the (complete) set of properties. 12
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Elsewhere, 13 he argues that it follows from his Principle of Sufficient Reason, since if there were two indiscernible objects, they would have to be created in different places but God would have no reason for choosing between the two alternatives that arise through considering a permutation of the particles. This suggests that the Principle is regarded as contingent and Leibniz does present the image of someone scurrying about the garden unable to find two leaves exactly alike. Of course, in response to the claim that, nevertheless, we can imagine two such indiscernibles, Leibniz retorts that such a situation does not constitute a 'genuine' possibility, suggesting that the necessary/contingent distinction is perhaps too crude for this form of metaphysics.
Leaving the historical exegesis aside, there are several well-known arguments to the effect that the Principle cannot be necessary, based on similar sorts of imaginings. 14 Under the pressure of these arguments, supporters of the Principle have staged a retreat to the less forceful claim that it is only contingently true 15 and that, even if no longer regarded as an a priori logical truth of metaphysics, it may still be a useful methodological principle. 16 It is precisely the former claim that we will be examining in the context of classical and quantum physics in later chapters.
Now, indiscernibility, or, equivalently, indistinguishability, 17 is here understood in terms of the mutual possession (however that is understood) by the individual (however that is understood!) of the relevant properties. The question as to which are the relevant properties obviously assumes a crucial importance and different forms of the Principle can be conceptually delineated depending on the set chosen. Rewriting it in terms of second order logic with equality, it can be expressed thus:
where a and b are any two individual terms and F is a variable ranging over the possible attributes of these individuals.
The question is, what sort of attributes should be included in the range of F?
If the attribute 'being identical with a', which is certainly true of a, is included, then PII becomes a theorem of second order logic. 18 Leaving self-identity aside for the moment, different forms of PII, of increasing logical strength, can be delineated, according to the kinds of attributes F is taken to range over. Thus the weakest form, which we shall label PII(1), states that it is not possible for there to exist two individuals possessing all their properties (relational and non-relational) in common, whereas what we shall call PII(2) excludes those which can be described as spatio-temporal from this set. An even stronger form, PII(3), results if the set is taken to include only monadic, non-relational properties. 19
It has been argued that of the three, PII(1) must be necessary, since no two individuals can possess exactly the same spatio-temporal properties or enter into exactly the same spatio-temporal relations. 20 However, this obviously involves the Impenetrability Argument (IA) as an implicit premise and, as we shall see, the status of the latter in the context of modern physics is contentious. 21 Furthermore, both PII(1) and allow relations into the bundle and therefore carry the implication that these may confer individuality. Many philosophers vigorously dispute this on the grounds that, to cite one form of argument, since relations presuppose numerical diversity, they cannot account for it. 22 If such arguments are accepted then PIIs and must both be abandoned and the strongest form alone admitted, with the variable F covering strictly monadic properties only. Of course, this exclusion of relations from the bundle only applies to those that are irreducibly relational. It is commonly agreed that Leibniz himself believed that all relations-particularly spatial
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ones-could be reduced to monadic properties, thereby collapsing all forms of the Principle into PII(3). 23 Again, it is not our intention to delve into the history here; we mention this issue simply because of the possibility that physics presents us with examples of such irreducible relations.
The standings of PIIs and do not depend on IA and so it is possible to drop either or both without having to abandon the latter assumption. Examples of individuals which are indistinguishable in the sense of possessing a common set of non-spatio-temporal properties would obviously be enough to show that PII(2) is contingently false. If this set includes monadic properties then PII(3) will obviously be false as well. Equally obviously, however, if the individuals concerned possess only relational but non-spatio-temporal properties then PII(3) is saved. As we shall see, this touches on the further issue of how we regard properties in physics.
As we have said, the 'bundle' view of individuality needs some form of PII to guarantee individuation. Shifting to the claim that the Principle is contingently true leaves this view in a precarious situation and invites consideration of the physical nature of the world in which it is supposed to hold. We shall examine the impact of these considerations in what follows.
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