Published in P F Strawson and A Chakrabarti (eds) Concepts, Properties and Qualities, Ashgate, 2006, pp. 205-223. How not to trivialise the Identity of Indiscernibles Gonzalo Rodriguez-Pereyra 1. The Principle of Identity of Indiscernibles (PII, hereafter) says that no two things differ solo numero. That means that when two things differ numerically there is also a further difference between them. This further difference I shall call qualitative difference. This qualitative difference can be an internal difference between things or a difference as to how things are related to things. These differences can also be explained in terms of properties: in the former case qualitative difference consists in a difference with respect to intrinsic properties, in the latter case it consists in a difference with respect to relational properties. Relational properties may depend on the identity of the relatum (or on the identity of relata of the relatum), or they may be purely qualitative. Properties that depend on the identity of a relatum, like being two miles from the Eiffel Tower, are often called impure properties.1 Those that do not depend on the identity of a relatum, like being two miles from a tall tower, are called pure properties. Since intrinsic properties do not depend on the identity of any relatum they are also classified as pure. But given my understanding of 'qualitative difference', both pure and impure properties can make a qualitative difference.2 1 Often, but not always. In Individuals Strawson calls them universals-cum-particulars (1959: 137). 2 No doubt my understanding of the phrase 'qualitative difference' is idiosyncratic, since normally only pure properties would be taken to make a qualitative difference. But I have found no better phrase to express what I want to express, namely that difference which is not merely numerical difference, i.e. which is not a solo numero difference. As I said, what I mean by qualitative difference is any difference that is not merely a numerical difference. But differing with respect to some impure properties, for instance differing with respect to the impure property of being two miles from the Eiffel Tower, is more than differing merely numerically. In what follows, the reader should bear in mind that in this paper qualitative difference is neither synonymous nor coextensive with difference with respect to pure properties. Given that the difference can be captured in terms of properties, PII is normally taken to assert either that there are no two things that share all their properties or, in its necessitated version, that there can be no two such things. Here is a formal statement of the nonnecessitated version: PII (x)(y)[(F)(Fx ≡ Fy) ⊃ x = y)]3 Both in the necessitated and the non-necessitated version the domain of properties over which one quantifies is crucial for the truth of PII. Indeed the more one restricts the domain of properties quantified over the more likely for PII to come out false. And if one puts no restriction at all in the domain of properties quantified, then PII comes out true but, as it is often pointed out, trivially true. The triviality of this version of PII depends on certain properties being included in the domain of property quantification. I shall call those properties that render PII trivial trivialising properties. Since PII, far from being a trivial principle, is one of the most substantive and controversial ideas in metaphysics, it is important to determine which properties must be excluded from the domain of quantification to get a metaphysically serious version of PII. If one excludes from the domain of quantification all and only trivialising properties then one ensures a non-trivial and to that extent a metaphysically serious version of PII. That version will be the weakest non-trivial version of PII. For every other non-trivial version of PII will entail the truth of the version that excludes all and only trivialising properties. But that it is the weakest does not make it unworthy of metaphysical discussion. That weakest version will make a non-trivial claim and establishing it may be as difficult as establishing other versions of PII. 3 If one takes second order variables to range over sets then the principle in the text is merely a settheoretical analogue of PII, rather than PII itself. One can also express PII as a first order principle, e.g. (x)(y)(z)[(x has z ≡ y has z) ⊃ (x = y)], where 'x has z' is true if and only if z is a property of x. In this paper I shall stick to the canonical second order formulation in the text. But whether the principle must be formulated as a first or second order principle is not relevant for the purposes of the present paper. 2 I shall not discuss whether the weakest non-trivial version of PII is true. But it is significant that, as it will have become clear in §7, the weakest non-trivial version of PII will quantify over some impure properties, like being the father of a or being one meter apart from a. Thus I disagree with Peter Strawson, who said that the only version of PII which is worth discussing is one according to which there exists, for every thing, some description in purely universal or general terms such that only that thing answers to that description (Strawson 1959: 120).4 Indeed normally the following three versions of the principle are distinguished (here I present only the non-necessitated variant of each version): PII1: No two things share all their intrinsic properties PII2: No two things share all their pure properties PII3: No two things share all their properties PII1 is the strongest version, PII3 the weakest. Philosophers typically claim or suggest that these are the only three versions of PII and that PII3 is a trivial version of PII (to cite only a few: Adams 1979: 11; Forrest 2002, §1; van Cleve 2002: 389-90). While I agree that PII3 is trivial, I disagree that the other two are the only non-trivial versions of the principle. Indeed part of the significance of the discussion to follow is that shows the existence and importance of the following version of PII, weaker than PP2 but stronger than PII3: PII2.5: No two things share all their non-trivialising properties I take PII2.5 to be the weakest non-trivial version of PII. And since not all impure properties are trivialising properties, PII2.5 quantifies over some impure properties. The main aim of the paper is then to specify the class of trivialising properties. But I want to produce a philosophically illuminating specification of such a class. That is, I want to be able to say 4 There is another dimension in which I would disagree with Strawson, since in that passage he also makes the necessitated version of PII the only one worth discussing (Strawson 1959: 120). But I cannot discuss this issue here. 3 what makes a certain property trivialising. This is why I shall specify the class of trivialising properties intensionally rather than merely extensionally. In the next section I shall introduce the paradigmatic kind of trivialising properties. In §3 I shall discuss other trivialising properties and I shall argue against giving a characterisation of trivialising properties in terms of the notion of entailment between properties. In §4 I shall discuss, and eventually find unsatisfactory, another way of characterising trivialising properties, in terms of the notion of property containment. In §§5-6 I shall present and discuss my own characterisation of trivialising properties. §7 is a brief conclusion. 2. When one quantifies over all properties of things, what PII asserts is that no two things share all their properties. This version of PII is true but trivially true, as it has been widely recognised. To see why this version of PII is true consider property (1) below, which is an instance of what I call properties of identity:5 (1) being identical to a. If one quantifies over all properties, PII is true because if any things a and b share all their properties, including (1), then they are identical and so they are not two things. We can deploy the structure of the argument in the following way: (i) a and b share all their properties. (ii) a has the property of being identical to a. 5 It is important to be clear what properties of identity are. Having recourse to the property abstraction λ-operator makes that clear. The λ-operator binds a variable from a first order open sentence to designate the property expressed by that open sentence. Thus properties of identity are those that in their λ-expression the open sentence from which the λ-operator binds a variable consists only of an identity sign flanked by an individual variable and an individual constant. So properties like being identical to a and being identical to b are properties of identity because their λ-expressions are, respectively, '(λx)(x = a)' and '(λx)(x = b)'. But the properties of being identical to something or being self-identical are not properties of identity. Their λ-expressions, '(λx)(x = y)' and '(λx)(x = x)', do not satisfy our characterization. 4 (iii) b has the property of being identical to a. (iv) Therefore, a = b. Since (1) asserts the indiscernibility of a and b, the argument (i)-(iv) derives identity from indiscernibility by using a property of identity. (i) is the assumption of indiscernibility between a and b needed to get the argument started. (ii) follows, assuming properties of identity, from the general law that everything is self-identical, i.e. (x)(x = x). (i) and (ii) entail (iii), which states that b has the property of being identical to a, but if b has the property of being identical to a, it follows that b is identical to a, i.e. (iv).6 So, if a and b are indiscernible, they are identical, i.e. PII is true.7 This argument for PII is simple and clear and it turns the denial of PII into a contradiction, since denying it would amount to saying that there are two non-identical particulars that share all their properties, including their properties of identity, and therefore they are identical. But note that this argument supports PII only because it is a case of a general argument that can be applied to every two particular things that are supposed to be indiscernible. Taken in itself the argument only proves that there is nothing indiscernible from a, not that there is no pair of indiscernibles. But, since everything is self-identical, this argument can be generalized. Other instances of the argument will use other properties of identity, like being identical to b, being identical to c, or any others. So properties of identity make PII true. But they make it trivially true. No doubt the proof of this version of PII has an undeniable air of triviality, but what matters here is not the triviality of the proof but the triviality of what is proved.8 For it is trivial to claim that no numerically distinct things share all their properties, including their properties of identity. Properties of identity are trivialising properties, since they do not make a qualitative 6 That b is identical to a follows only on the assumption that if a thing has property F then it is F. This is unexceptionable, and should not be confused with the more controversial principle that if a thing is F then it has property F. 7 Informal versions of (i)-(iv) appear in Brody 1980: 9, Katz 1983: 37, Legenhausen 1989: 626, and Whitehead and Russell 1925: 57. 8 The trivialising nature of properties of identity is recognised in, among others, Adams 1979: 11, Ayer 1954: 29, Black 1952: 155, Katz 1983: 37-8, Legenhausen 1989: 626, O'Connor 1954: 103-4. 5 difference. They must be excluded from the domain of quantification to get a metaphysically serious version of PII. 3. If properties of identity were the only trivialising properties, our problem would be trivial. But although they are paradigmatic trivialising properties, there are other trivialising properties. For if properties of identity trivialise PII, then so do conjunctive properties like (2): (2) being identical to a and being green. Suppose a certain thing a is green; one can then show that there is nothing indiscernible from it in the following way: (v) a and b share all their properties. (vi) a has the property of being identical to a and being green. (vii) b has the property of being identical to a and being green. (viii) Therefore, a = b. This argument is generalisable in the relevant way, for even if not every thing is identical to a and green, everything has some conjunctive property one of whose conjuncts is a property of identity and the other is some other property. So conjunctive properties like (2) establish PII. But what they establish is trivial, since all the work in the proof is done by (1). The reason why in the argument above the real work is done by the property of being identical to a is that everything having the property of being identical to a and being green must have the property of being identical to a. So a and b cannot share (2) without being identical. In other words, property (2) entails property (1) in the sense that nothing having the former can lack 6 the latter.9 Since every property entails itself, if all and only trivialising properties entail properties of identity, then we have a solution to our problem: D1 F is a trivialising property =def. F entails a property of identity. But D1 is wrong, if only because there are trivialising properties that do not entail any properties of identity.10 Consider property (3): (3) being numerically distinct from a. Property (3) is the complement of property (1), the property of being identical to a. I shall call any complement of a property of identity a property of difference. Property (3) does not entail property (1) and, in general, properties of difference do not entail properties of identity. Yet property (3) can be used to show there is nothing indiscernible from a. For since indiscernibles are those that share exactly the same properties, indiscernibles are those that lack exactly the same properties, and so we can run the following argument: (ix) a and b lack exactly the same properties. (x) a lacks the property of being numerically distinct from a. (xi) b lacks the property of being numerically distinct from a. (xii) Therefore, a = b.11 9 This is the sense in which I shall conceive of property entailment in this essay and this is the usual way of conceiving property entailment. See, for instance, Carnap 1988: 17, Katz 1983: 44 , and Lewis 1983: 199. 10 It may be thought that the problem with D1 is that, given that it is necessary that everything is selfidentical and that if any thing is self-identical then that thing has a property of identity, all properties entail properties of identity and so, according to D1, all properties trivialise PII. This would show D1 to be wrong, since some properties, like being green, do not trivialize PII. But this is not a problem for D1. For even if it is necessary that everything is self-identical and that if any thing is self-identical then that thing has a property of identity, it does not follow that all properties entail properties of identity. For all that follows from this is that every property F is such that it is necessary that if any thing has F, then that thing has some property of identity. But it does not follow from it that for every property F there is a property of identity F* such that it is necessary that if any thing has F, then that thing has F*. And only in this latter case does every property F entail a property of identity. 7 Again, although this argument refers to a and b only, the argument is generalisable to apply to any two things supposed indiscernible, and thus properties of difference establish PII. But the thesis established is trivial, for the work in such arguments is again done by properties of identity, since lacking a property of difference is equivalent to having a property of identity. So properties of difference must also be excluded from the domain of quantification of any metaphysically serious of PII. Properties of difference are not the only trivialising properties that do not entail properties of identity. Consider property (4): (4) being numerically distinct from a or not being green. Property (4) is the complement of property (2). Since no two things can lack property (4), it can be used to deploy an argument similar to (ix)-(xii). Since that argument is generalisable in the relevant way, properties like (4) prove PII. But it should be clear that the thesis thereby established is trivial since all the work in such arguments is done by the property of being identical to a. For lacking the property of being numerically distinct from a or not being green is equivalent to having the property of being identical to a and being green, and no two things can share this one because they would share the non-shareable property of being identical to a. 11 I have found no version of (ix)-(xii) in the literature, but something similar to it is in Katz 1983: 40. There are also arguments that derive discernibility from numerical difference, like the following two, which I shall deploy using the λ-operator: (i2) a ≠ b (ix3) a ≠ b (ii2) (λx)(x = a)(a) (x3) ¬(λx)(x ≠ a)(a) (iii2) ¬(λx)(x = a)(b) (xi3) (λx)(x ≠ a)(b) (iv2) (∃F)(Fa & ¬Fb) (xii3) (∃F)(¬Fa & Fb) These arguments are contrapositive versions of (i)-(iv) and (ix)-(xii). Informal versions of (i2)-(iv2) and (ix2)-(xii2), or of mixtures of them, appear in Adams 1979: 11, Ayer 1954: 29, Bergmann 1953: 77, Black 1952: XX, Broad 1933: 172-3, Greenlee 1968: 760, McTaggart 1921: 96, O'Connor 1954: 1034, Odegard 1964: 204 and Russell 1959: 115. The arguments (i)-(iv), (i2)-(iv2), (ix)-(xii) and (ix2)-(xii2) are clearly related to each other, but they have never been clearly differentiated and, sometimes, they are thought of as a single argument (e.g. Adams 1979: 11, footnote 11, seems to confound several of them). 8 This suggests that trivialising properties are those having or lacking which entails having a property of identity. Since lacking a property is having its complement, we may attempt to define trivialising properties as follows: D2 F is a trivialising property =def. (1) F entails a property of identity or (2) the complement of F entails a property of identity. But D2 is wrong, for not all trivialising properties satisfy it. Consider properties (5) and (6): (5) being identical to a or being green (6) being identical to a or not being green. Neither (5) nor its complement entail a property of identity. The same is true of (6) and its complement. Many things could and do have them and many things could and do lack them. Nevertheless they make PII true. For nothing can have both of them. But they make it trivially true, because having both of them entails having property (1). It is only thanks to this entailment that together they make PII true. The same is true of the complements of properties (5) and (6), properties (7) and (8) respectively: (7) being numerically distinct from a and not being green (8) being numerically distinct from a and being green For although neither having nor lacking either (7) or (8) entails having a property of identity, lacking both of them does entail having (1), the property of being identical to a. In general, properties like (7) and (8) are such that lacking both of them entails having a property of identity. So such properties make PII trivially true. 9 One might decide to exclude from the property domain of PII only one property from such pairs like (5) and (6). But this would be arbitrary. There is no reason to count either of them as trivialising that does not apply to the other. Thus we should count both of them as trivialising properties. Ditto for (7) and (8). This suggests we replace D2 by D3 below: D3. F is a trivialising property =def. (a) F or its complement entails a property of identity, or (b) F or its complement is the conjunct of a conjunctive property which entails a property of identity and the other conjunct(s) neither individually nor jointly entail a property of identity. D3 rightly makes (5) and (6) trivialising properties, since they are the conjuncts of (5)&(6), and so they satisfy condition (b) in D3. Ditto for (7) and (8). But D3 has a crucial defect: it counts as trivialising some properties that are not. Consider properties (9) and (10) below: (9) being green. (10) being (identical to a or not green) and being green.12 The defect of D3 is that it makes (9) a trivialising property, for (9) satisfies the second disjunct in its definiens. In effect, (9) is a conjunct of (10), which entails a property of identity, but the other conjunct of (10), namely (6), entails no property of identity. But (9) is the property of being green. And the property of being green is a paradigmatic nontrivialising property. I do not see how to solve this difficulty in terms of the notion of entailment. But even if there is such a satisfactory solution, trying to define trivialising properties as those that somehow or other entail properties of identity is marred from the beginning. For any definition that counts properties that entail properties of identity as trivialising properties assumes that no pure properties entail properties of identity. But suppose things had pure 12 Let us resort to λ-formulations to make clear what property (10) is. Where 'Gx' stands for 'x is green', (10) is the following property: (λx)((x = a ∨ ¬Gx) & Gx). 10 individual essences. Imagine, for the sake of example, that being the greatest philosopher was the individual essence of Plato. In that case the property of being the greatest philosopher would entail a property of identity, namely the property of being identical to Plato. If all things have pure individual essences then PII is true, and it is true thanks to these pure individual essences. But a property like being the greatest philosopher does not trivialise PII. If what one proves is that numerically different things must have different pure individual essences then one has established that every numerical difference goes accompanied by a qualitative difference – and this is no triviality.13 The point can perhaps be better appreciated by considering Leibniz's position. For Leibniz all things (individual substances) have qualitative essences, expressed by their complete concepts (and therefore more complex than anything like the property of being the greatest philosopher). These essences entail properties of identity and so they guarantee PII. But no doubt Leibniz's was not a trivial version of PII. Whether or not things have pure individual essences is not the question here. The point is simply that, whether or not things have such essences, D3 is inadequate as a characterisation of trivialising properties. First, if things have such essences, D3 is extensionally wrong. Second, even if things do not have such essences, a formulation of a non-trivial version of PII should not presuppose that things do not have such essences. Third, even if things do not have such essences, and all trivialising properties satisfy D3, the mere conceptual possibility of things having pure individual essences shows that D3 does not tell us why trivialising properties trivialise PII. 4. We need a different kind of definition of trivialising properties. Since it seems clear that trivialising properties are those related in some special way to properties of identity, the question is: how are trivialising properties related to properties of identity, if not by entailment? 13 To make the point of this paragraph I do not need to invoke pure individual essences. Invoking the possibility of impure individual essences that are not trivialising would have been good enough. But the point is more forcefully made by invoking the possibility of pure individual essences. 11 The intuitive answer is that they contain properties of identity. And this marks the difference between a trivialising property like (6), which contains the property of identity (1), and a non-trivialising property like (9), which does not contain any property of identity. This approach looks promising. Indeed Katz proposes a solution in terms of a notion of containment. But what Katz does is to specify the class of trivialising predicates, which is not the same as specifying the class of trivialising properties if only because presumably, as Katz acknowledges, there are properties which no predicate expresses. What is Katz's definition of trivialising predicates? He first introduces what he calls basic identity properties (BIPs) as follows: F is a BIP if and only if (1) it is possible that (∃x)(Fx) and (2) it is necessary that (x)(y)(Fx & Fy ⊃ x = y) Let us call predicates expressing BIPs BIP-predicates. Katz says that a BIP-predicate is a trivialising predicate and that a predicate 'P' contains a BIP-predicate essentially provided 'P' contains a BIP-predicate but is not logically equivalent to a predicate that does not: 'x is numerically distinct from a' contains a BIP-predicate essentially, but 'x is green and (identical to a or numerically distinct from a)' does not. Then Katz says that a predicate 'P' expresses a trivialising property if and only if 'P' contains a BIP-predicate essentially or 'P' may be defined in terms of some predicate that does. This, of course, makes such properties like (1)−(8) above, and also (10) and others, trivialising properties. Needless to say, this does not make a property like (9) a trivialising property.14 14 I have altered Katz's terminology. He calls BIP-predicates identity-predicates and he calls trivialising properties identity-properties. So what he actually says is that 'a predicate, P, expresses an identityproperty if and only if P contains an identity predicate essentially or may be defined in terms of some predicate that does' (Katz 1983: 41). I changed Katz's terminology because of its potential for confusion. 'Identity predicate' suggests a predicate that expresses a property of identity, but as we shall see below not all BIPs are properties of identity. 'Identity property' suggests a property of identity, but as we have seen not all trivialising properties are properties of identity. Katz was of course aware that not all trivialising properties are properties of identity, and although he did not realise that all BIPs are properties of identity, nothing here should be taken to imply that Katz used his terminology confusingly or confusedly. He used his terminology clearly and consistently, but nevertheless his terminology has potential for confusion. 12 What are we to say about Katz's definition of trivialising predicates? Three comments. First, that it wrongly makes trivialising properties those expressed by superlative predicates, i.e. predicates like 'being the tallest man', 'being the widest river' etc. Such superlative properties are BIPs. But superlative properties in general do not trivialise PII. Superlative properties, being BIPs, cannot be shared, e.g. no two things could be the tallest man. But they do not serve to prove PII, since not everything must have one of them. The most one can do with them is to assert that if something has a superlative property then that thing has no indiscernibles, but this, of course, is far from asserting that nothing has indiscernibles, which is what PII requires. If they do not make PII true, superlative properties cannot make it trivially true. But perhaps everything has a superlative property relative to certain reference class? I am not sure. In a world like that imagined by Max Black (1952: 156), consisting of only two indiscernible iron spheres, there seems to be no superlative property that either sphere has relative to any reference class. In any case, even if everything has a superlative property relative to some reference class, the problem with Katz's proposal is that it makes those properties that are superlative relative to no class (or relative to the most inclusive class) trivialising, which they are not. In any case the difficulty with superlative properties can be met by just letting BIPs be properties of identity – in that case superlative properties will not counts as trivialising properties for they will not be expressed by trivialising predicates. Second, Katz does not explain what it is for a predicate to contain another. So it is not clear which predicates contain BIP-predicates and which do not, and therefore it is not clear which predicates express trivialising properties and which do not. For although it may be intuitively clear that the predicate 'is green' does not contain any BIP-predicate, intuition suggests that the predicate 'thinks about a' and 'is one meter apart from a' contain the BIPpredicate 'is identical to a'. But these predicates do not express trivialising properties, since properties like thinking about a and being one meter apart from a can be shared and they do not make PII true unless conjoined with properties like being identical to a or not thinking 13 about a or being identical to a or not being one meter apart from a. But it is these latter properties that trivialise PII. Third, it is possible to define trivialising properties, rather than predicates, since all we need is a precise notion of property containment. For instance one may introduce a notion of property containment via some stipulations like the following: Every property contains itself. Every property that is a function of other properties contains those properties (i.e. a conjunctive property contains its conjuncts; a disjunctive property contains its disjuncts; a negative property contains its negated property). The relation of property containment is transitive. Then, following Katz, we say that a property F contains a property of identity essentially provided F contains a property of identity but is not logically equivalent to a property that does not. Then, to avoid the problem of superlative properties, we define trivialising properties in terms of their containment of properties of identity, rather than BIPs: D4 F is a trivialising property =def. F contains a property of identity essentially. D4 rightly counts properties (1)-(8) and (10) as trivialising properties. Furthermore, thanks to the precise specification of the containment relation, it rightly excludes properties like thinking about a and being one meter apart from a from the class of trivialising properties. But D4 has several problems. First, it does not seem to count as trivialising the property of being a member of {a}. For being a member of {a} does not seem to be a property of identity, nor the complement of a property of identity, nor a conjunctive or disjunctive property having a property of identity as one of its conjuncts or disjuncts. But being a member of {a} is a trivialising property. For one could argue for PII thus: If a and b have all their properties in common, then since a has the property of being a member of {a}, b has this 14 property as well; but since whatever is a member of {a} is identical to a, a = b. But clearly what is doing the work here is the property of being identical to a, which must be had by whatever is a member of {a}. Being a member of {a} seems to contain the property of being identical to a in the sense that it is a relational property whose relatum ({a}) is specified in terms that depend on the identity of a and so, in that sense, on the property of being identical to a. But if we redefine containment so as to make the property of being a member of {a} contain the property of being identical to a, then we should make sure we avoid making thinking about a or, even more to the point, being the only lover of a, contain a property of identity. But even if this can be done, a definition of trivialising properties in terms of a notion of property containment will still be lacking, even if extensionally correct. The problem with such a definition is that it does not explain why trivialising properties are trivialising properties. Why should properties containing properties of identity trivialise PII? It is not evident why this should be the case. Furthermore it is clear that merely containing a property of identity is not what makes a property trivialising, since there are properties, like being green and (being identical to a or being numerically distinct from a), which contain properties of identity but do not trivialise PII. This is why D4 does not define trivialising properties purely in terms of containment, since the explanation of what it is for a property to contain a property of identity essentially makes reference not only to the properties it contains but also to the properties it is logically equivalent to. But the relation of equivalence is not a containment relation.15 It may be claimed that this lack of purity is not a symptom of explanatory deficiency. Why should it matter that the definition defines trivialising properties purely in terms of property containment? Even if it does not define them purely in terms of property containment, that does not show that D4 fails to provide an explanation of why trivialising 15 Katz's definition of trivialising predicates has the same feature of not being purely in terms of containment. 15 properties are trivialising. Perhaps what explains why they are trivialising properties is that they contain a property of identity and are equivalent only to properties that do? But why should such properties be trivialising properties? It is not clear why this should be the case. Furthermore, there are reasons to doubt that this is what explains why trivialising properties are trivialising. Consider property (5). It contains a property of identity and is not logically equivalent to a property that does not. But this does not show it is a trivialising property – after all, property (5) can be shared and so it does not suffice to establish PII. The same applies to (6). Someone may say that even if (5) and (6) can be individually shared, they are such that in virtue of what they contain the pair of them cannot be shared (i.e. no two things can have both of them). But this does not explain why they are trivialising properties, since (6) and (9) are also such that in virtue of what they contain the pair of them cannot be shared. But (9) is not a trivialising property. It may be claimed that the relevant difference between (5) and (9) is that (5) contains a property of identity while (9) does not. But we saw four paragraphs back that there are properties that contain properties of identity but do not trivialise PII. Furthermore, saying that it is in virtue of containing a property of identity that (5) is prevented from being shared with (6) does not work. For it is no less by virtue of containing the property of being green than containing a property of identity that (5) is prevented from being shared with (6). But it is by virtue of containing the property of being green that (9) is prevented from being shared together with (6). 5. This second objection to D4 applies only if we are interested in more than mere extensional correctness. But extensional correctness cannot be the goal of our inquiry. For extensional correctness per se does not provide an explanation of why trivialising properties trivialise PII. So even if we hit an extensionally correct definition of trivialising properties, we may still have serious difficulties in recognizing it as a correct definition, for there may be properties such that it is not intuitively clear whether they trivialise PII. Furthermore, even if 16 we knew that a certain definition is extensionally correct, it may not be philosophically illuminating. For we might know that without having answered the question of why trivialising properties trivialise PII. But knowing what features make trivialising properties trivialise PII puts us in a position to define trivialising properties: trivialising properties are those that have the features in question. So the question I shall answer in this section is What makes trivialising properties trivialise PII? Trivialising properties are those that can be used to establish a trivial version of PII. So in order to find out what makes trivialising properties trivialise PII, we first need to understand why the trivial version of PII is trivial. Once we know this it should be easy to see what makes trivialising properties trivialise PII, namely that they have those features that enable them to be used to establish a trivial version of PII. The trivial version of PII is the version established by arguments like the ones considered in §§2-3. I shall focus on argument (i)-(iv), which by using properties of identity is a paradigmatic trivialising argument. So why do arguments like (i)-(iv) establish a trivial thesis? PII is meant to be a thesis about the connection between qualitative identity and numerical identity, namely that qualitative identity entails numerical identity: there cannot be qualitative identity without numerical identity. Equivalently, there cannot be solo numero difference: things that differ numerically must also differ qualitatively. But (i)-(iv) establishes that if a and b share all their properties, and therefore are qualitatively identical, they are numerically identical because they share a property of identity. But sharing a property of identity is being numerically identical. So what the argument shows is that qualitatively identical things that are numerically identical are numerically identical. This is trivial. In other words, the argument establishes only that any numerically different things differ in their properties of identity, without requiring that they differ in any other property. But difference with respect to a property of identity is numerical difference, not qualitative 17 difference. So all the argument establishes is that numerically distinct things are numerically distinct. This is trivial. So (i)-(iv) establishes a trivial version of PII because it only establishes the numerical difference of numerically distinct things. What the argument establishes is only the letter of the principle when formulated using unrestricted property quantifiers – that no two things can share all their properties – but it does not establish the spirit of the principle – that there cannot be qualitative identity without numerical identity. What features of properties of identity account for this? That differing with respect to them is differing numerically. For since differing qualitatively is more than differing numerically, simply establishing a difference with respect to properties of identity establishes only a numerical difference, not a qualitative difference. I am not saying that properties of identity are trivialising because differing with respect to them entails no more than a numerical difference. I am saying that they are trivialising because differing with respect to them is differing numerically. This is so even if differing with respect to properties of identity entails a qualitative difference. In that case one still cannot establish a qualitative difference by simply establishing a difference with respect to properties of identity: one needs to invoke the fact that a difference with respect to properties of identity entails a qualitative difference. This also applies to properties of difference. They are the complements of properties of identity. So having a property of difference is lacking a property of identity and lacking a property of difference is having a property of identity. So differing with respect to properties of difference is differing with respect to a property of identity and so differing with respect to them is differing numerically. Thus establishing a difference with respect to a property of difference only establishes a numerical difference, not a qualitative difference. So properties of difference are trivialising properties. There are two things to distinguish. One is what the trivialising character of properties of identity and difference consists in; the other is what makes them have that 18 character. The trivialising character of properties of identity and difference consists in that merely establishing a difference with respect to them only establishes a numerical difference between the things in question. What makes properties of identity and difference have that character is that being numerically different is differing with respect to those properties. The trivialising character is common to all and only trivialising properties. Every property such that merely establishing a difference with respect to it only establishes that the things in question are numerically different is a trivialising property. Such a property can be used to establish a trivial version of PII and so it is a trivialising property. And every property such that merely establishing a difference with respect to it establishes more than a numerical difference is such that establishing a difference with respect to it establishes a qualitative difference. So, since establishing a difference with respect to it cannot be used to establish a trivial version of PII, such a property is not trivialising. But although the trivialising character is common to all and only trivialising properties, only in the case of properties of identity and difference what accounts for their trivialising character is that differing with respect to them is differing numerically.16 So how can a property F be trivialising without being a property of identity or difference? Even if differing with respect to F is not differing numerically, differing with respect to F may consist in differing numerically. If differing with respect to F may consist in differing numerically, merely establishing a difference with respect to F only establishes a numerical difference. So if differing with respect to F may consist in differing numerically, F is a trivialising property. Differing with respect to F may consist in differing numerically if and only if differing with respect to F may consist in differing with respect to a property of identity or property of difference. So properties such that differing with respect to them may consist in differing with respect to a property of identity or difference are trivialising properties. 16 Note that the trivialising character of properties of identity and difference consists in that merely establishing a difference with respect to them only establishes that the things in question are numerically different – not that the things in question differ only with respect to properties of identity and difference. The latter is not true. For instance, if and b differ with respect to properties of identity and difference then they differ with respect to conjunctive properties having their properties of identity and difference as conjuncts. 19 How can a property be such that differing with respect to it may consist in differing with respect to a property of identity or difference? Consider conjunctive properties. A conjunctive property is such that having or lacking it is having or lacking other properties. So differing with respect to F&G is simply differing with respect to F or G or both. So differing with respect to F&G may consist in differing with respect to F. So some conjunctive properties containing properties of identity are such that a difference with respect to them may simply consist in a difference with respect to a property of identity. Consider the property of being identical to a and being green. Differing with respect to it consists in differing with respect to either of its conjuncts. So differing with respect to it may consist in differing with respect to the property of being identical to a. And so differing with respect to the property of being identical to a and being green may simply be differing numerically. The same applies to conjunctive properties containing properties of difference, like the property of being numerically distinct from a and being green. The same is true of disjunctive properties like being numerically distinct from a or not being green and being identical to a or being green. This is why such conjunctive and disjunctive properties are trivialising properties. It should be clear now why properties like being green, being square and being hot are not trivialising properties. Differing with respect to them must consist in more than simply differing numerically: it must consist in differing with respect to colour, shape and temperature. For similar reasons impure properties like being father of a, loving b, being close to c, and being in the same place as d, are not trivialising properties. Differing with respect to these properties must be more than differing with respect to a property of identity or difference: it must be differing with respect to fathering a, loving b, being close to c, and being in the same place as d. Let e be the father of a. Even if origin is essential, and so e cannot fail to be the father of a provided a exists, there is more to being the father of a than 20 being such that a exists and being identical to e. The extra is all that is involved in fathering a. Similarly in the other cases.17 This also explains why superlative properties are not trivialising properties. Differing with respect to being the tallest man is more than differing with respect to a property of identity or difference: it is also differing with respect to height from other men. It also makes clear why there are some complex properties containing properties of identity that are not trivialising properties, namely those properties that contain properties of identity but are logically equivalent to properties that are not trivialising, like the conjunctive property of being green and (being identical to a or being numerically distinct from a). Properties like these are such that differing with respect to them must consist in differing with respect to a non-trivialising property, in this case the property of being green. Thus establishing a difference with respect to those complex properties will be establishing more than a numerical difference. Therefore such complex properties are not trivialising properties, in spite of containing properties of identity.18 6. So far, so good. But how about the property of being a member of {a}? This is a trivialising property but it does not appear to satisfy our characterisation of trivialising properties. For even if differing with respect to it will require differing with respect to being identical to a, it seems to require differing with respect to being a member of {a}, and so it seems that a difference with respect to it cannot simply consist in a difference with respect to the property of being identical to a. 17 An interesting case is the property of having all parts in common with a. This is trivialising because among the parts of a is its improper part, namely a itself. So this property leaves open the possibility that a and b differ only with respect to their improper parts, in which case they differ only numerically. Some people think that no two things can share all their proper parts. If it is true that no two things can share all their parts then there is a non-trivial version of PII that is true. But this does not make the property of having all proper parts in common with a trivialising. Differing with respect to such a property is differing more than merely numerically. Furthermore, the insight that no two things can share all their parts, if indeed it is true, is not a trivial but a substantive metaphysical insight. 18 There are other properties containing properties of identity that are not trivialising, and in this case the explanation of why they are not trivialising must be different. Consider the property of being identical to a or being numerically distinct from a. This property is not trivialising because it must be shared by everything and so, since no two things can differ with respect to it, not even a trivial version of PII can be proved by its means. The property of being identical to a and being numerically distinct from a is also such that it cannot be shared, though this time because nothing can have it. 21 Of course a is a member of {a} in virtue of being identical to a – that is, it has the property of being a member of {a} in virtue of having the property of being identical to a. But this cannot be what makes the property of being a member of {a} trivialising. For that one property is had in virtue of another only means that there is a particular relation between the two – the in virtue of relation.19 It does not mean that differing with respect to one of those properties consists in differing with respect to the other. But what if it was the case that all of the properties of a thing are had in virtue of being that thing? In that case a would have all of its properties simply in virtue of being a. Perhaps the world is like that. Perhaps things cannot share all their properties because they have their properties in virtue of being the things they are. Or perhaps things cannot share all their properties because every thing has a qualitative property that is necessarily peculiar to it in virtue of being the thing it is. In either case PII would be true but it would be non-trivially true. For even if things would be qualitative different in virtue of being numerically different, differing qualitatively would still be more than differing numerically.20 But that any thing numerically different from a must also differ from a with respect to being a member of {a} is a trivial fact. However mysterious the singleton membership relation is, it appears that differing with respect to being a member of a singleton is no more than differing numerically. How can this be? This is because, if sets exist, the identity of the members fixes what sets they belong to. And this is, in turn, because given a set S with certain things as members, there is no more to being a member of S than being one of those things. So, given {a}, there is no more to being a member of {a} than being a, i.e. being identical to a. Thus the property of being a member of {a} is the property of being such that {a} exists and being identical to a. 19 This relation between the properties in question is different from their being necessarily coextensive. For, assuming that necessarily a exists if and only if so does {a}, the necessary coextension of the properties is symmetrical: nothing can have one of the properties of being identical to a and being a member of {a} without having the other – but although a has the property of being a member of {a} in virtue of having the property of being identical to a, it does not have the latter in virtue of having the former. 20 It is important to emphasise that the non-triviality of PII in these situations would not be due to our ignorance that things have all their properties, or some properties necessarily peculiar to them, in virtue of their identity. Even if we discovered this, through metaphysical argument or any other means, this would be a discovery of a non-trivial fact. 22 It is frequently asserted that a belongs to {a} in virtue of being a rather than being a in virtue of belonging to {a}. My proposed account of the property of being a member of {a} nicely explains why this is so: being a member of {a} consists in satisfying two conditions, one of which is being identical to a, but being identical to a does not consist in being a member of {a}. This also goes some way to dispel the mystery associated with the singleton membership relation. According to David Lewis the relation of singleton to member holds in virtue of qualities or external relations of which we have no conception whatsoever. That is, we do not clearly understand what it is for a singleton to have a member (Lewis 1991: 35). If my account of the property of being a member of {a} is right then we have a conception of the relations in virtue of which a thing is a member of a singleton: these are existence and identity. But there is a sense in which Lewis is right that the singleton membership relation is mysterious. Singletons are atoms and the connection with their members is primitive and thereby in some sense mysterious and opaque. So we do not know in virtue of what a certain singleton has a certain thing as its member. That is, we do not know in virtue of what the property of being a member of {a} is the property of being such that {a} exists and being identical to a rather than the property of being such that {a} exists and being identical to b. After all, if {a} exists, b has the property of being such that {a} exists and being identical to b, but this does not make it a member of {a}. But this is a mystery that we should expect. For there is nothing in virtue of what {a} is the singleton of a as opposed to the singleton of b: it just is. So there is nothing in virtue of what the property of being a member of {a} is the property of being such that {a} exists and identical to a rather than the property of being such that {a} exists and identical to b: it just is one rather than the other. Nothing in my account of the property of being a member of {a} helps with this. All my account says is what the property of being a member of {a} consists in, and makes the property of being identical to a part of that property. So, given that being a member of {a} consists partly in being identical to a, my account makes clear why a is a member of {a} in 23 virtue of being a rather than being a in virtue of being a member of {a}. But nothing in my account explains why being a member of {a} consists partly in being identical to a rather than being identical to b. To understand this we should, I think, know in virtue of what a singleton has its members. But there is nothing in virtue of which a singleton has its members, so that my account does not explain this should not be seen as a problem for it.21 It may be thought that a problem for my account is that it does not make clear why the singleton membership relation has the formal features it has, e.g. irreflexivity, asymmetry, intransitivity. But there is no reason why an account of what the property of being a member of {a} consists in should make clear why singleton membership has those formal features. This is not an account of singleton membership in general: it is an account of what it is for a thing to be a member of its singleton. What matters is simply that my account be compatible with those formal features of the singleton membership relation, and it is. The property of being such that {a} exists and being identical to a is a conjunctive property one of whose conjuncts is the property of being identical to a, and so differing with respect to being a member of {a} may consist in differing with respect to the property of being identical to a. Thus differing with respect to being a member of {a} may be differing numerically. Even more, since whenever two things differ with respect to being a member of {a} both of them are such that {a} exists, differing with respect to being a member of {a} must, and therefore does, consist in differing numerically. So there are properties such that differing with respect to them consists in differing numerically and there are properties such that differing with respect to them may consist in differing numerically. In both cases such properties can be used to establish a trivial version of PII, since establishing a difference with respect to them establishes no more than numerical difference. These are the trivialising properties. But it will be impossible to establish a trivial version of PII by establishing a difference with respect to a property such that differing with respect to it must consist in 21 Here I go beyond Lewis, who seems to think that there may be something in virtue of why singletons have their members. 24 more than differing with respect to a property of identity, and so it must consist in more than differing numerically. These are the non-trivialising properties. 7. We can now define trivialising properties as follows: D5. F is a trivialising property =def. Differing with respect to F is or may be differing numerically. D5 is an intensional definition of trivialising properties: it purports to tell us what to be a trivialising property consists in rather than merely specifying the class of trivialising properties. Since it tells us what is to be a trivialising property, D5 is philosophically illuminating in a way in which a mere specification of the class of trivialising properties is not. By saying what trivialising properties are, D5 specifies a certain class of properties as the class of trivialising properties. D5 is right in this respect to the extent that the class it specifies includes the properties of identity and all the other trivialising properties we have considered. But is D5 extensionally correct? Do all and only trivialising properties satisfy D5? Yes. For, as I have argued, D5 is intensionally correct. So it is extensionally correct. We now know what trivialising properties are. So we know what properties should be not be quantified over in order not to trivialise PII. 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