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Nature's MetaphysicsLaws and Properties$

Alexander Bird

Print publication date: 2007

Print ISBN-13: 9780199227013

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780199227013.001.0001

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Structural Properties

Structural Properties

Chapter:
(p.147) 7 Structural Properties
Source:
Nature's Metaphysics
Author(s):

Alexander Bird (Contributor Webpage)

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199227013.003.0007

Abstract and Keywords

Geometrical and other properties that may be labelled ‘structural’ are held up as examples of properties that are not potencies, but are categorical properties. The debate between Mellor and Prior is examined in order to shed light on this question. The problem is then related to the question of whether a true physical theory should be background-free, on the grounds that it is the presence of spacetime as a background in intuitive physical theories that is responsible for the appearance that spatial properties are categorical, whereas advanced physical theories tend to be background-free.

Keywords:   spacetime, background-freedom, Mellor, Prior

In this chapter I shall examine the challenge to dispositional essentialism that is presented by natural properties that seem not to be potencies (nor reducible to them) and are held up as paradigms of categorical properties. These are structural, typically geometrical properties. By ‘structural properties’ I mean properties required to account for the structure of things and to explain those things that are explained by the fact that things have structure or are related in a structured manner.103 Take the science of crystallography. The explanation of the properties of a crystal will refer to its structure, which is a matter of the geometrical relations of the ions or molecules that constitute the crystal. Since spatial relations are structural in the current sense, all sciences will depend on structural properties. A categoricalist might think that an object that consists of a set of masses in a particular spatial configuration has just been described in purely categorical terms. Whether dispositional essentialism can account for mass may be up for debate. But the present, greater challenge is to account for the spatial relations in dispositional terms.

In Section 4.1 I defined categoricalism, (CM), as the the view that all natural properties are categorical properties, whereas dispositional monism, (DM) (Section 3.1.1), says that all fundamental natural properties are potencies. This left room for (MV), the mixed view. The latter is a dualistic view of properties, holding that some fundamental natural properties have dispositional essences while others are categorical. The challenge to be faced in this chapter is that some fundamental natural properties seem not to be potencies. Being triangular, for example, seems to bring with it no powers in the way that, say, being negatively charged does; it seems inappropriate to characterize that property in dispositional terms. If that were the case, we would be impelled to give up (DM) in favour of at least (MV).104 That would be damaging to several central claims of this book. (Although, not to all. For one could still be a dis-positional essentialist as regards the source of all laws. Categorical properties could (p.148) feature in the stimulus and manifestation conditions of potencies, and so could figure in our laws, even if all the laws were generated by potencies. The mixed view of properties although dualist about properties does not imply dualism about laws (cf. Section 3.1.3).) Nonetheless, my arguments in Section 4.2 imply that no properties are categorical.105 In which case, how might one defend the claim that structural properties are, despite appearances, potencies also? As in the objection concerning the alleged contingency of laws (see Chapter 8), the dispositional monist must argue that appearances are deceptive. It is not the case that fundamental structural properties do not have dispositional essences.

An old debate between Hugh Mellor and Elizabeth Prior provides a useful starting point in the search for dispositional essences for structural properties. That debate was initiated by Mellor’s (1974: 157) claim that dispositions should not be regarded as ‘shameful in many eyes as pregnant spinsters used to be—ideally to be explained away, or entitled by a shotgun wedding to take the name of some decently real categorical property.’ The supposedly shameful mark of a disposition is that it entails a subjunctive conditional. Mellor argued that this trait was shared by supposedly proper categorical properties too. Specifically he argued that ‘x is triangular’ entails such a conditional. Elizabeth Prior (1982), on the other hand, argues that Mellor’s alleged entailment does not hold. While the motivation for that debate is not identical to that behind this chapter, we can make use of it for our purposes. For if Mellor is right, and we cannot distinguish between dispositional and categorical properties on the basis of their relationship to subjunctive conditionals (because both have such a relationship), then it would appear to follow that there are not really any genuinely categorical properties after all. Allegedly categorical properties, such as structural properties, are potencies after all.

In this chapter I shall first see whether we can use this debate in the service of dispositional monism. My conclusion is that while it does not establish dispositional monism, it does help undermine the view that structural properties are clearly categorical. I then turn to more general grounds, based in recent developments in physics, for thinking that fundamental structural properties such as spatial displacement should have dispositional essences of this kind. I shall argue that our intuition that spatial and temporal properties and relations are categorical is a consequence of our thinking of spacetime as a background to our physical theories and not as an agent or patient of physical causes and effects. Whereas if we follow the lead of certain physicists in demanding that our theories should be background-free, then the way is open to thinking that structural properties do have dispositional essences.

7.1 The Mellor–Prior debate

As we have seen in Chapter 2, disposition talk has traditionally been cashed out in terms of entailing a counterfactual or subjunctive conditional. The debate we are about to consider turns on this. Whether or not structural properties are potencies, (p.149) according to this debate, is a matter of whether those properties are appropriately related to those conditionals. This would seem to provide a test for whether a property is a potency or is a categorical property. However, thanks to the problem of finks and antidotes, application of the test is not as simple as it might at first appear.

7.1.1 Testing for potency

Let us recall the (simple) conditional analysis of dispositions:

(CA) x is disposed to manifest M in response to stimulus S iff were x to undergo S x would yield manifestation M.

Since potencies are defined as natural properties with dispositional essences, (CA) yields the following necessary condition on being a potency:

(P→) If P is a potency then for some S and M and for all x: Px entails (if Sx were the case, then Mx would be the case).

Categorical properties, by contrast, have no non-trivial modal characters:106

(C→) If P is a categorical property then there are no S and M such that for all x:

Px entails (if Sx were the case, then Mx would be the case).

If some property does entail a non-trivial conditional then by (C→) it is not a categorical property. But (P→) does not show that the property is a potency. For that we would need also:

(P←) P is a potency if for some S and M, and for all x:

Px entails (if Sx were the case, then Mx would be the case).

(P←) is contentious. For one thing X may entail Y without Y being part of X’s essence (Fine 1994), even if the entailment is non-trivial. For example, ‘being made of gold’ is not a potency; nonetheless, via the necessity of constitution, it entails having subatomic parts that are positively charged (the atomic nuclei and their protons). And since, according to the dispositional essentialist, being positively charged is a potency, ‘x is made of gold’ will entail a non-trivial modal conditional, for example ‘if the subatomic components of x are individually placed in an electric field, then some of them will experience a force in the direction of the field’. Thus the dispositional essentialist will hold that some properties are not potencies but are nonetheless related to potencies in such a way that they too entail modal conditionals. Even ignoring the claims of dispositional essentialism, the argument of Section 8.2 shows that it is a non-trivial necessary truth that salt dissolves in water. Hence ‘x is salt’ non-trivially entails ‘were x placed in water, it would dissolve’. But being salt is not itself a potency. This line of thought may also seem to put (C→) in question. However, if we restrict ourselves to fundamental properties, these concerns disappear. Fundamental properties will be either categorical properties or potencies. If that is right then we may write: (p.150)

(P↔) A fundamental property P is a potency iff for some S and M and for all x:

Px entails (if Sx were the case, then Mx would be the case);

(C↔) A fundamental property P is a categorical property iff there are no S and M such that for all x:

Px entails (if Sx were the case, then Mx would be the case).

(P↔) and (C↔) provide a test whereby we may decide whether a given fundamental property is a potency or is categorical. Here we are concerned with structural properties, so let it be agreed that P is a fundamental structural property. We then consider whether ‘x is P’ entails some non-trivial subjunctive conditional. If it does not, then we may conclude, via (P→), that P is not a potency, and, via (C←), that it is a categorical property. P would provide a counterexample to (DM), the claim that all properties are potencies. On the other hand, if ‘x is P’ does entail a subjunctive conditional, then P is a potency and not a categorical property. Thus P cannot be employed as a counterexample to (DM); on the contrary, P would be a counterexample to the claim (CM), that all properties are categorical.

The challenge presented to dispositional monism by the property of triangularity and its like is that it seems to satisfy (C↔) and not (P↔). If so dispositional monism is in error. An immediate response would be that we have limited our attention in those principles to fundamental properties, and ‘triangularity’ as such does not seem to be fundamental. Even if fundamental entities might exemplify this property, one would not expect ‘triangular’ to be a predicate in some final physics. Nonetheless, the kind of challenge presented remains. Whatever the fundamental spatial relations are, they would be like triangularity, one might suppose, in not having dispositional essences. If the fundamental spatial relations are potencies, then we would expect a relatively simple derived property such as triangularity to inherit the capacity to entail subjunctive conditionals. Hence the lack of any such entailment would indirectly support the claim that the fundamental spatial properties are categorical. Conversely, if we can find a non-trivial conditional entailed by ‘x is triangular’, then the best explanation of this is that the fundamental spatial properties are potencies. An alternative explanation would be one of the kind referred to above and discussed in 8.2, where the entailment trades on a non-trivial necessity that exists independently of dispositional essences. But such an explanation seems implausible in this case. For these reasons it is instructive as regards fundamental properties to reflect on the status of triangularity.

7.1.2 Complications for the conditional test for potency

Thus far it seems that we have simple test based on (P↔) and (C↔) for whether triangularity is a potency or a categorical property. However, matters are more complex, for two reasons. The first notes that ‘A’ may seem to entail ‘B’ when a de dicto reading of the sentence ‘A’ is given, but not under a de re reading. Hence we need to be careful which we are employing when applying the conditional test. The second reminds us of the problem of finks and antidotes.

(p.151) 7.1.2.1 De re entailment

Success for Mellor’s claim does not necessarily vindicate the dispositional essentialist view. The categoricalist can endorse the claim that some statements asserting the instantiation of a categorical property do entail a conditional. The categoricalist acknowledges that there are dispositional property terms, such as ‘elastic’, ‘irascible’ and so forth. The meanings of these terms, says the categoricalist, may well be conveyed by subjunctive conditionals. Hence there might well be some conditional C such that ‘x is elastic’ entails C. But that will not show that the property we call ‘elasticity’ is essentially dispositional. As I discussed in Section 3.1.1 there is a categorical-dispositional distinction for properties and for predicates, and these need not match up.

There are several ways one can construe the reference of expressions of the form ‘the disposition to M when S’ (’D(S,M)’—where the sans-serif ‘D’ is used to indicate that ‘D(S,M)’ is a singular term, not the predicate ‘D(S,M)’). They are:

  • (i) ‘D(S,M)’ names a property—the property of being-disposed-to-M-when-S. According to this view, in all possible worlds objects with the property D(S,M) are disposed to M when S. Within this view we may take two very different approaches. (a) In the first two sentences of this paragraph ‘the property’ means ‘the sparse property’; (b) ‘the property’ means ‘the abundant property’. If (a) then ‘D(S,M)’ names a potency. If (b) then it seems that ‘D(S,M)’ names an abundant property. But it isn’t clear what it is to name an abundant property. One might reify abundant properties, as entities akin to Fregean concepts. Or one might regard naming an abundant property as being equivalent, in effect, to a simple predication. That is, ‘x has the property D(S,M)’ is equivalent to ‘x is D(S,M)’.

  • (ii) ‘D(S,M)’ is a definite description, equivalent to ‘the (sparse) property that is responsible for something’s being disposed to M when S’. The categoricalist will regard the sparse property thus picked out as being a categorical property. That property will have a contingent dispositional character thanks to the laws of nature of this world. The description will pick out different categorical properties in different possible worlds. In another world the description will pick out a possibly different property that, thanks to the possibly different laws of that world, contingently has this dispositional character (to M when S) in that world. The dispositional monist will take this description to pick out a potency (if it picks out any sparse property)—and the same potency in all worlds (the one whose essence is to M when S).

  • (iii) ‘D(S,M)’ is a rigidified definite description, equivalent to ‘that (sparse) property that is actually responsible for something’s being disposed to M when S’. According to the dispositional essentialist, that property is a potency, and so this is equivalent, as far as the modal characteristics of reference are concerned, to (i) (a). The categoricalist will regard this rigidified definite description as picking out a categorical property. As in (ii) that property will have a contingent dispositional character thanks to the laws of nature of this world. However, in this case, the expression picks out that same categorical property in each possible world, and so in some other possible worlds objects with that property will not be disposed to M when S. The dispositional monist will regard the rigidification as making no difference relative to (p.152) the non-rigidified description in (ii), since if a potency actually has some dispositional character, that very same property has that character, to M when S, in all worlds.

  • (iv) ‘D(S,M)’ names a second-order property—the property of possessing some first-order property responsible for something’s being disposed to M when S. Again the second-order property might be a sparse or an abundant one. An important issue here is whether there are any sparse second-order properties of this kind. They do not appear to have any distinctive explanatory power. There is, I think, one area where this might not be the case, and that is the area of evolved complex systems. A creature’s visual system will have evolved not because of its specific underlying first-order structure but because of its function. The same selection processes might have (and will have in other creatures) brought about the evolution of systems with quite different underlying structures but with the same functionality. Ecologically, therefore, it would be the higher-order property that has explanatory value rather than the diversity of first-order properties (cf. Vicente 2002, 2004). Clearly, however, such cases do not impinge on the current concern with fundamental and near fundamental structural properties. If the second-order property is an abundant property, then ‘x has the property D(S,M)’ is equivalent to a predication: ‘x has some first-order property that is responsible for something’s being disposed to M when S’.

Of these I’ll ignore (iv) and (i). According to (iv), if ‘D(S,M)’ does refer to a sparse property, it is not the sort of sparse property we are dealing with (since it is then a non-fundamental property of a complex system). Otherwise, ‘D(S,M)’ doesn’t refer to a sparse property (it refers to an abundant property or is equivalent to a predication involving quantification over properties). If the reading of the term is such that it does not refer to a sparse property, then we are obliged to ignore that reading as irrelevant, since the conditional test is a test of sparse properties, not of predication. Similarly we can ignore (i)(b). Reading (i)(a) is already to judge that reference is to an essentially dispositional sparse property, and hence begs the question that we are applying the test to answer.

So we are left with readings (ii) and (iii). Which of these we choose will affect what we think is entailed by ‘x has the property D(S,M)’. Crucially, if we choose (ii), the categoricalist will agree that in all possible worlds, if x has the property D(S,M) then were x to receive stimulus S then x would manifest M. In which case the conditional test will not distinguish between categorical properties and potencies. On the other hand, if we choose (iii), the test can make the required distinction. If the property picked out by the rigidified definite description is a potency it will have the same subjunctive character in all possible worlds, whereas if the property is a categorical one, it will not.

This is of course what we want—we want to test for the modal characteristics of one and the same property across possible worlds. In other terms, when testing whether ‘x has D(S,M)’ entails ‘Sx □→ Mx’ we need to ensure that the former is understood de re rather than de dicto.

7.1.2.2 Finks and antidotes again

The second complication for the test of potency notes that its founding principles, (P↔) and (C↔), are derived from (CA). And as we (p.153) have seen we have reason to believe that (CA) is false—in both directions of the biconditional in (CA). Given (P→), the truth of Mellor’s claim is necessary for the truth of dispositional essentialism in this sense. There must be some conditional entailed by ‘x is triangular’; if there is not, triangularity is a counterexample to (DM). (Of course, Mellor’s particular conditional might not be the right one—we shall return to this.) But if (P→) is false, then the non-existence of such a conditional will not show that triangularity is non-dispositional. And (P→) does indeed seem to be false. (P→) is derived from the definitional claim that potencies have dispositional essences plus the left-to-right implication in (CA), (CA→): D(S,M) x→ (Sx □→ Mx). As we saw in Section 2.2.2 finks and antidotes present counterexamples to (CA→). Thus the existence of finks and antidotes means that the conditional test is not definitive, in that a potency need not entail a subjunctive conditional after all.

7.1.3 Rules of the debate

Should the conclusion of the previous section be that the truth of subjunctive conditionals is a red herring as regards testing for potency? No, I think not—but we should be careful. There is, as Charlie Martin has said, clearly some sort of connection between dispositions and conditionals, even if it is not one of straightforward entailment (Armstrong etal. 1996:178). So the debate can proceed, only we need to ensure that the problems identified are not exploited. On the one hand, we need to ensure that circumstances involving finks and antidotes are not employed to show that a structural property is not a potency. On the other hand, de dicto entailments may not be used to show that a property is a potency. That is, we must ensure that any entailment claim holds in virtue of a de re reading, because only the de re reading ensures that we are testing for the modal characteristics (i.e. those relating to the subjunctive conditional) of one and the same property across possible worlds. Put another way, we want the entailment to reflect the metaphysics of the properties, not the merely analytic features of the terms used to denote them

Thus the debate should be governed by two rules:

Rule 1: Any alleged entailment between the possession of a property and a subjunctive conditional should hold in virtue of a de re reading of the former claim; more generally, any link established between a property and a conditional must be a metaphysical rather than a merely analytic one.

Rule 2: The existence of a link between a property and a conditional may not be refuted by appeal to finks or antidotes (or established by appeal to finks or mimics).107

With these rules or caveats in place we may use the conditional test, i.e. the existence or otherwise of a relation between properties and conditionals, to decide (p.154) whether a structural property such as triangularity is a potency or a categorical property.

7.2 The case of triangularity

The challenge to the dispositional monist is the claim that geometrical shape entails nothing as regards counterfactual or subjunctive conditionals. There is no C, it is asserted, such that C is a genuine, non-trivial, modal conditional and ‘x is triangular’ entails C. Mellor, however, states that there is just such a C. His candidate is ‘if someone were to count x’s corners correctly, then the result would be three’. That is, he asserts:

(T) That x is triangular entails that if someone were to count x’s corners correctly, then the result would be three.

Hence triangularity is at least no proven counterexample to dispositional monism. (C↔) does not show that triangularity is a categorical property. And to the extent that (P↔) can be relied upon, (T) seems to show that triangularity is a potency.

The subsequent debate hinged on the interpretation of ‘correctly’. Prior held that Mellor’s claim acquired prima facie plausibility only because of the use of this word. For without it we would see that the entailment does not hold—people frequently count things and get the wrong answer. More significantly, we are entitled to consider another possible world in which the laws of nature are different so that its inhabitants make systematic errors in counting. (Prior suggests perceptual errors, but one could imagine deeper neurophysiological interference also.) The inclusion of ‘correctly’ is significant because it seems to rule out such cases. But, says, Prior, it does so only because we take the claim that a task was carried out ‘correctly’ as meaning that it was performed successfully, that it got the right result. Since it is analytic that triangles have three corners, it is also analytic that someone who counts the corners of a triangle correctly gets the answer three. And so the entailment does not seem to reflect the metaphysics of the property of being a triangle. Rather it depends only on analytic relations and so Mellor’s argument falls foul of Rule 1.

Prior (1982) notes that Mellor states in a footnote that ‘correctly’ is intended to refer not to the result of counting but rather to the manner of counting. But she thinks that if this is so, then the entailment fails, since if it is only the manner of counting that is invoked, then counting in the unusual world with systematic error may be carried out in the correct manner without getting the correct result.

Prior has a second argument that invokes a different unusual world, in which the laws of nature are such that when one starts to count the corners of a triangular object, the object is caused to change the number of corners it has. Hence, if one counts as well as one can one will get an answer other than three.

What defence has Mellor against these two objections? He does not address the second. But he does not need to. For it is clear that in the world considered, triangularity is finkish, in that the stimulus, counting, causes an object to lose its triangularity. Consequently, this objection to the entailment contravenes Rule 2 and may be ignored.

(p.155) As regards the first objection, here the accusation is that Mellor has broken Rule 1. Mellor responds that he can spell out precisely what counting correctly is without referring to the correctness of the result: it is to count the items in question once each (and once only), which is to put them in a ‘1-1 correspondence with an initial segment of the sequence of positive integers 1, 2, 3 .... The highest number in the segment is the result of the counting’ (Mellor 1982: 97). Does this reply block Prior’s appeal to Rule 1?

Let us compare ‘x is even’, which entails ‘if x were to be divided by 2, then the result would be an integer’. On one understanding, where dividing is understood as an abstract mathematical operation, this is clearly true. Does this make ‘being even’ a dispositional property? If so, it would be difficult to deny that being triangular or any other property is dispositional. If Mellor’s claim is understood analogously, with ‘counting correctly’ taken to be an abstract mathematical operation, it might well be regarded as analytically trivial, and so outlawed by Rule 1. It is analytic that the set of corners of any triangle has three members. It is analytic that when any three-membered set is put into 1-1 correspondence with an initial segment of the positive integers, the highest number in the segment is three. So Mellor’s entailment is analytic. But is it merely analytic?

As we have seen, the distinction between a merely analytic entailment and one that reflects the metaphysics of the entities involved is the distinction between an entailment that holds de dicto only and one that holds de re. Thus if the entailment is not merely analytic it should continue to hold when we employ any rigid designator to pick out the entity in question. So ‘S is the inventor of bifocals entails S invented bifocals’ is a merely analytic entailment, since ‘Sis Benjamin Franklin’ does not entail ‘S invented bifocals’. While even if one thought that being H2O is part of the definition of water, ‘x is water entails x is H2O’ would not be merely analytic, since, for example, ‘x is that substance which, in the actual world, is the main component of living things on earth entails x is H2O’ is also true (but not analytically).

By this test Mellor’s entailment will not come out as merely analytic, since for any rigid designator ‘D’ that picks out the property of triangularity, ‘x is D’ entails ‘if someone were to count x’s corners correctly, then the result would be three’ (where ‘counting correctly’is still understood abstractly). Yet we should note that the efficacy of the way of distinguishing merely analytic and metaphysical relations depends on the difference in modal properties between definite descriptions and rigid designators. But there is no such difference between mathematical definite descriptions and corresponding rigid designators. So the test does not seem to be applicable here, and it is not clear that Mellor’s entailment does not infringe Rule 1.

However, a different reason for dismissing Mellor’s claim, on this understanding, is that it is in conflict with the thought that the stimulus of a disposition is a cause of the manifestation—dropping the fragile vase caused it to break, pulling the elastic caused it to stretch, and so forth. Although this is contentious in the eyes of some, we could add a third rule. Rule 3 would state that there must be a causal or nomic connection between the antecedent of the conditional and its consequent. Mellor’s claim would be outlawed by Rule 3 on the mathematical interpretation.

(p.156) On the other hand, we might understand the operation of dividing as an intellectual, psychological operation, not as an abstract mathematical one. This allows the stimulus (i.e. dividing) to cause the manifestation (getting an integer as the answer). If we regard the process of counting the corners of the triangle in this way, then Mellor’s claim looks to be a substantial one. However, we might then ask, can we be sure that his entailment holds under this interpretation? One might argue on Prior’s behalf that it does not. For now there is a gap between the fact of the corners of the triangle having been correlated with the set of numbers {1, 2, 3} and the fact of the subject’s being in the mental state of getting the answer three. In normal cases this gap is traversed without difficulty. But in unusual cases it need not be. Where environmental conditions or the laws of neurophysiology are different, the counting may have been carried out correctly, the appropriate correlations having been made, yet the answer achieved is a number other than three. For example, we may imagine a ‘killer triangle’ whose particular size and angles interact with a subject’s neurophysiology to kill them instantly or to cause mental aberration. Or we could take the case of a triangle painted Kripke’s killer yellow (cf. Lewis 1997: 145).

Hence, the conditional is not entailed by the ascription of triangularity. However, this does not prove that Prior is right. We already know that in general disposition ascriptions do not entail the corresponding conditional, because of finks and antidotes. We saw that Prior’s case of a world where triangles ceased to be triangles when counting began is an invocation of a fink. The cases considered in the previous paragraph do not invoke finks (the triangles remain the same), but they do involve antidotes, since they interfere with the normal operation of the stimulus. Hence this response breaks Rule 2 again.

As we shall see, the debate is by no means concluded. Nonetheless, at this point Mellor’s claim that triangularity is no less related to its conditional than potencies in general are related to theirs, has not been refuted.

7.2.1 Locating dispositions

The debate is not concluded since one might reject Mellor’s entailment as proving that triangularity is a potency on the following ground. While the entailment discussed seems to indicate a dispositional character somewhere, it is not, on reflection, clear that the character lies with the triangle rather than the counter. Consider the following:

(N) S is a normal observer entails if S were to count the corners of a triangle correctly then S would get the answer three.

Modulo finks and antidotes, this seems to be true. Given the link between dispositions and conditionals upon which we have been trading, this suggests that ‘normal observer’ is at least a dispositional concept, which is plausible enough. Note however that this entailment is equivalent to (T), if finks are excluded. So it looks as if we have two dispositions for the price of one. In this case the entailment is perhaps de dicto rather than de re. Nonetheless, the question is raised, whether it isn’t the dispositionality of the concept of ‘normal observer’ that is doing the work in generating the conditional (T).

(p.157) It is worth noting, as Martin does (Armstrong et al. 1996:135-6), that dispositions frequently come in pairs of ‘reciprocal disposition partners’. The negatively charged electron is disposed to attract the positively charged proton; the proton is disposed to be attracted to the electron (and also to attract the electron towards it). In fact our discussion suggests that dispositional concepts might always come in reciprocal pairs. For the following are typically equivalent:108

X entails were it the case that Y, then Z would be the case; and

Y entails were it the case that X, then Z would be the case.

So it seems too hasty, simply because there is dispositionality in the subject (the counter), to exclude triangularity from having the dispositional nature that makes it a potency. However, the resulting position remains unsatisfactory from the points of view of both the categoricalist and the dispositional monist. On the one hand the dispositional reciprocity between the triangle and the observer that is suggested by Mellor’s account makes triangularity look like a secondary property, akin to a colour. But there is a clear disanalogy between structural properties such as triangularity and secondary properties such as colour, in that the latter have an explanatory role only in a limited portion of science, primarily the behavioural sciences. That is as it should be, since the manifestations of colours and all other secondary properties are the mental states of sentient observers. Yet structural properties play a role at the most general and basic level in science. And their doing so is independent of any power to produce effects in human observers. This does not show but does suggest that the reciprocity between triangle and observer is one-sided, that the dispositionality comes primarily or even completely from the observer and not from the triangle.

Furthermore, the same line of reasoning will suggest to the dispositional monist that Mellor’s conditional does not show which potency triangularity is. The existence of a property may be related to a wide range of conditionals. But not all of them reflect the essence or nature of that property109 In this case (T) seems to make being triangular a secondary property, a property whose nature is to be a disposition to cause a certain effect in a human observer. One can deny that triangularity is a secondary property without asserting that it is a categorical property. Perhaps it should be understood as a genuinely tertiary property, one that is a disposition which is manifested not in human subjects especially but in some other, broader, class of entities, a class specifiable at a more general level in science.

(p.158) Insofar as we are still employing subjunctive conditionals as a sign of dispositionality, we should look for a conditional that reflects the nature of the (alleged) disposition, and a sign of this will be that the stimulus and manifestation reflect the role of the property in scientific explanation. In effect, both sides should accept this as Rule 4. Triangles may exist in pretty well any possible world that has a physical component. It would be odd, if triangularity is a potency, that it should be one whose dispositional essence, if it can be specified, is specifiable only in terms of entities (things that can count) that exist at a very limited range of possible worlds. Rule 4 says that if triangularity is to be shown to be genuinely essentially dispositional, we should look for a conditional characterization that has appropriate generality.

We should look therefore for a much greater level of generality in trying to understand what kind of potencies structural properties are. I shall outline two ways of achieving this. The first approaches this question head on and seeks an appropriately general conditional. The second approach is more oblique and makes us ask why we expect that the sparse structural properties of physics should be thought to be categorical. The answer is that in classical physics these properties characterize a background. But contemporary physics seeks background-free theories, in which the structural properties do have powers.

7.2.2 Properties and geometries

Plausibly there are conditionals for structural properties that come much closer to obeying Rule 4 than (T). Sungho Choi has suggested to me that we could generalize the notion of counting corners. All we would need is a counting machine that can distinguish travelling along a geodesic from not doing so. If it did not do so at any point, then it would add one. Such a machine, travelling along a triangular path, starting at any non-apex point, would count to three on returning to its starting position. Even so, one might hope to find an essence constituted out of properties that one might expect to find in a fundamental theory. In Bird (2003 b) I suggested the following as a starting point:

(T) The paths AB, BC, and AC form a triangle entails if a signal S travels along AB then immediately along BC, and a signal S* travels along AC, starting at the same time and travelling at the same speed, then S* will reach C before S.

The problem I raised for this suggestion was that this is false for many non-Euclidean triangles. I therefore proposed that the following is true (again barring finks and antidotes):

(TE) The paths AB, BC, and AC form a Euclidean triangle entails if a signal S travels along AB then immediately along BC, and a signal S* travels along AC, starting at the same time and travelling at the same speed, then S* will reach C before S.

One could then regard ‘triangle’ as ambiguous, or generic, across a range of triangle properties, each for different kinds of geometry, and each of which has a different essence of this kind. I suggested triangles in Riemannian geometry or Lobatchevsky (p.159)

                   Structural Properties

FIG. 7.1

Bolyai geometry might have different (Ti), although in fact (T) will do for many geometrical contexts. In spherical geometry one may consider the figure whose vertices are the north pole, N, and two nearby points, A and B, on the equator and whose sides are the longitudinal arcs NA and NB and the equatorial arc AB that goes the long way round the equator (see Fig. 7.1). AB is a little less than 2π ×NA, and so the signal (such as a pulse of light) along AB will take longer to reach B than the signal passing from A to N and thence to B. Whether this counts as a counterexample to (T) rather depends on the definition of ‘straight line’ in the context of defining a triangle as a ‘figure with three vertices joined by straight lines’. For if a straight line is the shortest path between two points, the longer part of the great circle will not be a straight line and our figure is not a triangle, and so no counterexample to (T).110 On the other hand, if we remove from the sphere the points other than N on a line of longitude that passes through the narrow gap between A and B, then our figure is a counterexample. On this view, there is no (sparse) property of triangularity in general. Triangularity is a portmanteau term covering different kinds of triangularity. The different kinds have dispositional essences relating to some variant on (T). It is (T) and its family of variants that define triangularity in general.

One drawback for this approach is that it does not demonstrate that the dispositional monist is correct. For where the latter sees a specific and allegedly dispositional property (’being a Euclidean triangle’) the categoricalist will see a conjunctive property consisting of a general categorical property plus a specification of the space it is in (’being a triangle in Euclidean space’). The approach considered does not show that the former is correct, at most only that is it is an option. (That may be enough for the dispositional monist given that the properties in question are raised as counterexamples.)

It is in any case far from clear that there is some clearly defined family of variants on (T) that will pick out all and only the triangles in various geometries. Furthermore, it seems a rather convoluted way of characterizing something that can (p.160) be so easily defined in non-dispositional terms (’a closed figure bounded by three straight line segments’). The Mellor–Prior debate was about whether being triangular entailed any subjunctive conditional, and happened to focus on one concerning the counting of vertices. But such a conditional would never have sufficed to characterize the essence of triangularity if triangularity has an essence that is sufficient for something’s being a triangle, because many figures have three vertices that are not triangles (not having straight edges). Rather better than either Mellor’s suggestion or my (T) is the following:

The straight line segments AB, BC, and AC form a triangle entails were a signal to pass along AB it would not pass through C (and similarly for the other two permutations of A, B, and C).111

Even so, I am inclined to think that such conditionals fail to get at the heart of the problem, for two reasons. First, it is difficult to see that anything like a causal or nomic role is being assigned to triangularity. In Bird (2003 b) I claimed that the connection is causal, while admitting that this could be disputed. I am now less sure. The mere fact of (T) being a counterfactual may confer a spurious appearance of causality. I don’t take counterfactuals to be definitive (?a Lewis) of causality. A dispositional essentialist could accept a Lewisian account of causality and add that the counterfactuals arise because of the presence of dispositions. However, it would then seem to make sense to cut out the middle man, counterfactuals, and to regard causal relations as instances of dispositional relations. There is in any case good reason to do so, since both the counterfactual analysis of causation and the counterfactual analysis of dispositions have counterexamples. Perhaps both sets of counterexamples could be eliminated by bypassing counterfactuals altogether?

Matters are, however, not quite so simple. The counterfactual analysis of the basic causal relation is: C causes E iff ¬C□→¬ E. However, the most obvious dispositional analysis of causation says that C causes E iff E is the manifestation of some disposition whose stimulus is C. If we apply the simple conditional analysis of dispositions to this, we have: C causes E iff C□→E & C & E.112 This reveals a contrast: whereas Lewis’s counterfactual analysis of causation focuses on (counterfactually) necessary conditions, the dispositional analysis identifies causation with sufficient conditions. (A dispositional account of causation suggests that Lewis’s approach, along with Hume’s claim from which it originates, was misguided from the very start.)

We should remember, furthermore, that ‘triangular’ is unlikely itself to name a fundamental structural property, and the dispositional essentialist is therefore not required to find a dispositional essence for it. The dispositional monist ought instead to focus attention on the fundamental structural (primarily spatial and temporal) properties and argue that these have dispositional essences.

(p.161) 7.3 Background-free physical theories

I will now sketch an alternative view of how dispositional essentialism may be reconciled with structural properties at the fundamental level. I shall concentrate on spatial separation (displacement), but the argument carries over to temporal relations also. Our knowledge of the nature of space and time is in a state of flux and we do not know what the role of fundamental spatial and temporal properties will be in the final theory of everything. Note that it is not a priori that such a theory would refer to spatial and temporal properties at all, nor, if it does, that the fundamental ones neatly mirror the role of such properties in folk physics or even classical physics.

Nonetheless, we can make some prognostications that suggest that a final theory would treat all fundamental properties dispositionally. I will first mention a brief response by Stephen Mumford (2004: 188) to the current problem. The gravitational force on an object is sensitive to both the masses of it and of other massy objects and its displacement from those other objects; looking at Newton’s law: F = Gm 1 m 2/r 2, the force F is a function of the masses m 1 and m 2 and also of their displacement r. Mathematically speaking mass and displacement are on a par—there is no way for Newton’s law itself to distinguish between the two quantities as regards dispositional (causal or nomic) priority. In which case why should we not regard the force as a manifestation of the displacement, in which case displacement is characterized dispositionally: the displacement r between two points is the disposition whose manifestation, when masses m 1 and m 2 are located at the points, is a force between those masses with magnitude F = Gm 1 m 2/r 2?

While I think this is along the right lines, it needs supplementation. There are two issues to be addressed. First, we need some explanation as to why it seems so much more natural to regard the force as a manifestation of the masses rather than of their displacement. Speaking figuratively we are inclined to think of the force as being generated by the masses, not by their displacement. Secondly, displacement crops up not just in the law of gravitation, but also in Coulomb’s law and elsewhere. Thus it would appear that we could characterize displacement dispositionally with respect to a variety of different and seemingly independent manifestations. If so, then either (i) displacement is a multi-track disposition (one with more than one kind of manifestation); or (ii) one of these manifestations (e.g. gravitational rather than electric force) is privileged over the others.

7.3.1 Displacement as a multi-track disposition

The problem with regarding displacement as a multi-track disposition is that multi-track dispositions are not pure dispositions (see Section 2.2.1. The conjunction of two dispositional essences is not itself a dispositional essence, just as the conjunction of two counterfactuals is not itself a counterfactual. So we are unable to characterize the nature of displacement in terms of a dispositional essence. The dispositional essentialist is required to see such properties such as displacement as non-fundamental (along with paradigm multi-track dispositions such as the ability to speak French).

This conclusion is not itself inevitably problematic—it is not a priori that spatiotemporal properties and relations are fundamental. But it would mean that the de (p.162) bate is off. If they are not fundamental properties, then having dispositional essences or not does not directly bear on the truth of dispositional monism. As already remarked, it is the fundamental properties and relations that are held to be essentially dispositional. If it turns out that spatial separation is not a fundamental relation but supervenes on some other as yet unknown property or relation, then spatial separation provides no counterexample to dispositional monism and an investigation by inspection of whether the truly fundamental properties and relations are essentially dispositional must await further developments in physics.

The alternative is to regard one of the dispositions as privileged in characterizing the essence of displacement, and given the general theory of relativity it is natural to see gravitational force as participating in the essence of spatial properties and relations. If we do take this view, the first question remains to be addressed. Why do we tend not to regard gravitational effects as equally the effect of displacement as of mass? This question is significant, because if we are right not to regard spatial displacement as causally efficacious (or more loosely as a potential agent) then displacement cannot be characterized with a dispositional essence.

7.3.2 Background structures and substantivalism versus relationalism

In order to respond to that question, we must take a short detour via our conceptions of space and time, in particular the view of the nature of space (and time) associated with the conception of spatial properties as causally inert. The classical conception of spacetime has been that of a stage or container within which things and laws act, but which is not itself involved in the action. It is a mere background. As such, although space and time are a part of the natural world, they are certainly not patients, that is to say recipients of effects, in any cause-and-effect relation, or more generally subject to change according to natural law. Their status as causes or agents of law-governed change is ambivalent. On the one hand, terms for spatial and temporal dimensions appear in the laws. On the other hand, we do not classically regard these terms as indicating action on the part of space and time. One reason for this is what Harvey Brown (Anandan and Brown 1995, Brown and Pooley 2006) among others calls the ‘action–reaction’ principle. Something can only be an agent if it is also a potential patient; something may only be a cause if it is also potentially the recipient of effects. According to substantivalism spacetime is a background entity, and so the displacement r between two objects is only indirectly a relation between them. It is primarily a relation between spacetime points. The objects inherit that relation by being located at spacetime points that are a distance r apart. On the classical view the structure of spacetime is also fixed and unchanging. Since spacetime points do not change their relations with one another, it is difficult to see how they and their properties can contribute causally to the behaviour of objects located in spacetime. Thus the displacement r between spacetime points is inert, and so, in consequence, is the supervening displacement r between the objects.

In the light of the forgoing, the relationalist, non-substantivalist conception of spacetime ought to be more congenial to the dispositional monist. On the simplest version of this view spatial relations are directly relations between objects (not be (p.163) tween spacetime points). This reverses the absolutist/substantivalist view according to which relations between object supervene on relations between spacetime points. The relationalist takes all the facts concerning space and time to supervene on facts about relations between objects. The laws of nature mention only spatial and temporal relations and these can be accounted for. They are at least in a position to obey the action–reaction principle, since spatial relations appear both as sources of change (e.g. in the gravitational law) and as objects of change (as in Newton’s second law). Since space and time just are the sum of spatial and temporal relations, there is nothing more to be explained than has been explained. While it hasn’t yet been shown that spatial relations really are agents of change, the possibility is now open that they are.

The obvious problem with the simple relational view is that it fails to account for the full range of spatio-temporal possibility. There seem to be times and places where objects and events could be but are not. Hence Leibniz extends the set of relations to both actual and possible spatio-temporal relations. But then we must ask, what grounds such possibility? Furthermore the set of spatio-temporal relations is found to have a metric structure and we may ask for an explanation of that fact too. If spatial and temporal relations are fundamental, then we should expect them, according to a dispositional essentialist view of their essences, to generate the laws that underlie the structure of spacetime, including facts about its metric.

In classical physics understood in an absolutist, substantivalist sense, however, this seems not to be the case. We have discussed that fact that laws such as Newton’s law of gravitation and Coulomb’s law mention spatial relations. These laws cannot be serious candidates for expressions of the essence of spatial relations since they tell us nothing about the structure of space. They tell us how the magnitudes of forces of certain kinds depend on spatial relations. But they do not tell us what spatial relations are possible and they do not tell us what metric the set of points in space possesses. It is telling that one response to this is conventionalism about spacetime, ?a Poincar?Schlick, or Duhem, for example. According to views of this sort, a choice of geometry and metric is conventional. We typically choose our geometry in such a way as to make the laws of physics expressible in a convenient form. The choice does not reflect some fact concerning the real structure of space and time, there being no facts of that sort. While one is not obliged by classical physics to be a conventionalist about the geometry of space and time, the fact that conventionalism is an option shows that no laws in classical physics determine that geometry. Either way, whether one prefers Newtonian substantivalism or conventionalism, there is no room for laws of the sort (ones telling us about the structure of spacetime) that the dispositional essentialist about spatial and temporal properties seeks.

In summary the dilemma is this. The general problem in classical physics is that relationalism posits too little structure, not enough to explain empirically revealed aspects of space and time, while substantivalism posits too much, and in particular makes spacetime a background structure. Scientifically, there is a problem in not having enough structure—excess structure is preferable. From the metaphysical point of view (that of the dispositional monist at least) matters are reversed. The thinner commitments of relationalism are prima facie acceptable. If space is nothing (p.164) but the spatial relations between objects, then we do have a law telling us how space changes, Newton’s second law. On the other hand, if impelled by the requirements of physics to posit more structure, so that space is the fixed structure of spatial points, not the changeable structure of objects, then we have introduced a mere background that is not subject to any law. Such background structures of substantivalism, being inert, cannot be accommodated within the dispositional essentialist viewpoint.

7.3.3 Dispositional essences and background-free physical theories

In a dispute between physicists and metaphysicians, it would be wise to take the side of the physicist. And so the preceding discussion might seem to put dispositional monism at a disadvantage. Recently, however, physicists such as John Baez (2001), Lee Smolin (1991), and Carlo Rovelli (1997) have advocated the view that a good physical theory should be background-free. Thus either space and time should be eliminated from our theories (although an unlikely prospect, this is not impossible). Or they should be shown not to be merely background. Either way the grounds for spatial and temporal properties and relations being clear exceptions to dispositional monism would be removed—in the first case because the properties no longer figure in fundamental science at all, and so are not fundamental, natural properties; and in the second case because space and time would no longer be mere background but instead are fully fledged agents, capable of acting and being acted upon. This would permit spatial and temporal properties to be understood dispositionally.

It should be noted, however, that the motivation behind the drive for background-free physical theories is not exactly the same as that which seeks dispositional fundamental properties. In causal terms, the latter is concerned to show how space and time can be genuine causes. While spatial and temporal relations occur in physical laws, they seem, as presented classically, not to be entirely genuine causes. But the search for background-free theories is principally a matter of showing how space and time can be affected by other causes.

Let us put this a little more precisely. The dispositional essentialist wants the following to be true of any fundamental spatio-temporal property or relation P:

P has a dispositional essence, viz. P can be characterized in terms of some stimulus and manifestation, S and M, such that it is essential to P that if Px, then (ceteris paribus) Sx□→ Mx.

The problem is that although we may be able to find an entailed counterfactual for certain spatio-temporal properties, it is unclear that this must be seen as flowing from the essence of those properties. One reason for this is that when:
  • (A) Qx entails Sx□→Mx

it is typically also true that:113
  • (B) Sx entails Qx□→Mx

Thus if (A) shows us the essence of Q, it looks as if we have another entailment of a counterfactual, (B), which seems to indicate the essence of S. We should not exclude (p.165) the possibility that (A) and (B) both characterize the essences of Q and S respectively, in which case they are instances of Martin’s reciprocal disposition partners. Equally we should not assume automatically that (A) and (B) do both characterize the essences of Q and S. And in certain cases there seems to be an asymmetry; our initial problem was that the relationships between the various sources of force (mass, charge, etc.) and displacement seem to be examples of such an asymmetry.

I take it that something such as the following characterizes what is meant by a background:

If K is a background structure in a theory T, then

(i) K is not subject to change and is not affected by changes elsewhere;

(ii) the laws of T refer to properties and relations of elements of K or properties and relations defined on K.

It is the first clause, in particular the phrase ‘not affected by changes elsewhere’ that characterizes the ‘backness’ as it were of the background.

The claim therefore that theories should be background-free, or that there is no background, is tantamount to saying:

(B-F) In a true theory, any structure appearing in the laws of that theory is subject to being affected by changes elsewhere.

Thus, if L is a structure in a true theory T, then for certain stimuli S, the following is true:

(C) S□→a change in L.

Does (C) help the dispositional monist, where L is spacetime? The dispositional monist seeks dispositional essences for spatial and temporal properties. If (C) provides them, then the essences of the spatio-temporal properties in question are such that under certain stimuli the structure of spacetime is itself changed. Now this is somewhat different from what we started looking for, which was spatio-temporal properties being responsible for changes rather than spacetime being the recipient of change. Nonetheless, (C) could do perfectly well in providing a dispositional essence. We may distinguish active from passive dispositions; active dispositions are those that have manifestation in entities other than the possessor of the disposition, while passive dispositions have manifestations in the possessor of the disposition. Some favourite dispositions, such as fragility, are passive.

In general, however, the manifestation of a disposition should be of a kind such that it can itself be responsible for changes. Otherwise we would find that the properties in question are merely epiphenomenal. In the case of (C) that means that changes in L (here spacetime) must themselves be capable of being responsible for changes of certain kinds. If so, those dispositions may be the ones most appropriately regarded as the essences of spatio-temporal properties. Nonetheless, (C), even if it does not constitute a solution to our problem, can show how our concerns maybe addressed. One reason why it is difficult to see space and time as causes on a classical substantivalist conception, is that it is difficult to see them as in any way being effects. The (p.166) background is unchanging. But if it is unchanging how can it generate any effects? On the other hand, if it is subject to change, then it is easier to see how it might itself be a cause of change. According to the action–reaction principle, something is a potential cause only if it is a potential effect also. Thus (C), which reflects the requirement of background-freedom, (B-F), tells us that spacetime and its properties may be affected by changes elsewhere; and the action–reaction principle tells us that since spacetime and its properties may be the recipients of change they may also be causes of it. In dispositional essentialist terms, we can see that by being potential manifestations of dispositional essences, spatial and temporal properties may also have dispositional essences themselves.

That perspective is precisely that endorsed by General Relativity. Each spacetime point is characterized by its dynamical properties, i.e. its disposition to affect the kinetic properties of an object at that point, captured in the gravitational field tensor at that point. The mass of each object is its disposition to change the curvature of space-time, that is to change the dynamical properties of each spacetime point. Hence all the relevant explanatory properties in this set-up may be characterized dispositionally. And furthermore, this relationship helps address the second question raised above, by explaining why gravity is privileged over other forces in characterizing the essence of spatial relations.

7.4 Extrinsic structural properties?

Molnar (2003: 158-62) argues that some structural properties cannot be potencies because they are extrinsic properties. Thus the spatial position of an object is, Molnar argues, extrinsic since it is a relation between an object and spacetime. But extrinsic properties cannot be potencies. So spatial position is not intrinsic.

One could resist this argument by rejecting the premise that potencies are always intrinsic. After all, as McKitrick (2003) has shown, not all dispositions are intrinsic. Nonetheless, I maintained above that McKitrick’s argument does not refute the more restricted claim that sparse dispositions are intrinsic. A better line of attack is to argue that the properties in question are either not fundamental or are intrinsic after all. For example, one might argue, for the reasons relating to relationalism about space-time or the acceptance of background-freedom, that spatial position as conceived by Molnar is not fundamental whereas spatial displacement, considered as a direct relation between objects, is fundamental. In which case the relevant question is whether displacement is an intrinsic or an extrinsic relation. The latter question is not entirely straightforward, since discussions of intrinsicness typically focus on monadic properties of individuals, rather than on dyadic relations among pairs of individuals. Nonetheless, the idea of intrinsicness is extendible. To say that Jack and Jill are lovers implies nothing about the existence of anything else, but to say that they are co-authors implies the existence of some third thing, the work they wrote together.

The details of the analysis of intrinsicness are still under debate. The intuitive idea is that the possession of intrinsic properties should not depend on what else there is. One way to spell this out is in terms of independence of accompaniment: there can be an F that is lonely; there can be an F that is accompanied; there can (p.167) a non-F that is lonely; and there can be a non-F that is accompanied (Langton and Lewis 1998). A similar but nonetheless distinct approach is to take the world with the object in question and to contract (or expand) it by removing (or adding) objects; if the property can remain despite contraction or expansion then the property is intrinsic (Vallentyne 1997, Yablo 1999). Lastly, one might appeal to the idea that perfect duplicates share intrinsic properties (Lewis 1983 a: 111). (In fact Langton and Lewis need to combine the first and third approaches in their account.) Spatial displacement comes out as intrinsic on the first and second approaches. It does seem to fail on the third, since we could duplicate two objects without placing them the same distance apart. But the third approach is little use to an account of intrinsicness that covers dyadic properties, since a dyadic property of a pair of objects will be retained by their duplicates only if it supervenes on those properties retained by the duplicates individually; i.e. on this approach a relation is intrinsic only if it supervenes on intrinsic monadic properties of its relata.

I think that even Molnar’s spatial position comes out as intrinsic. Arguably an object can be in the place that it is independently of accompaniment and independently of any contraction or expansion of its world. A relationalist might question whether the position of another object is independent of other things. But then the relationalist should deny that position is fundamental—separation is. Molnar thinks that position is extrinsic because it is a relational property. But it is an error to infer extrinsicality from a property’s being relational (Weatherson 2006). Position is a relational property, according to Molnar, since it involves a relation between the object and spacetime. But that will make position extrinsic only if the existence of space-time is independent of the object. A material object has to exist in spacetime—a lonely material object is not one that exists outside space; it’s the only thing in space. Once again, the duplication approach gives a different answer since duplicates do not have to be in the same place. However, we may in any case regard the duplication approach as too restrictive with its attributions of intrinsicness: being Socrates would not be an intrinsic property of Socrates; nor is the property of undergoing a change whereby some part is exchanged for a duplicate of that part.

7.5 Conclusion

If spatial and temporal properties and relations are fundamental natural properties and relations, then the dispositional monist must provide reasons for thinking, contrary to a common intuition, that they too have dispositional essences. One approach is to take a familiar geometrical property (such as triangularity) and show that its instantiation entails some counterfactual. But ultimately this turns out to be indecisive. The dispositional monist will expect dispositional essences to be reflected in the laws of nature. And since triangularity is not a fundamental property of the kind that appears in the laws of nature, strictly speaking it is irrelevant to the dispositional monist’s argument, in that it is not the sort of example that would provide a coun (p.168) terexample to the claim that fundamental sparse properties are essentially dispositional.114

We looked therefore at the seemingly basic property of spatial separation and its relationship to the laws of nature. On a classical substantivalist conception of space, spatial separation is a relationship between points in an unchanging spatial background, and thus incapable of acting as a cause, and so also incapable of having a dispositional essence. A relationalist conception of space may seem more accommodating to dispositional monism, but was scientifically problematic in the classical era. Nonetheless contemporary physicists have resurrected relationalism in the form of the requirement that theories should be background-free. If that requirement is correct, then structures in true theories will not be mere backgrounds, but will be capable of being the recipients of effects. Add to that the action–reaction principle, then such structures, including spacetime, become potential causes also. In the light of the action–reaction principle it is the fact that in classical physics space is a mere background that prevents us from being able to regard it as having a causal role and so prevents us from seeing it as having a dispositional essence. Thus it is the requirement of background-freedom that makes room for dispositional essences for spatial (and likewise temporal) properties and relations. And that space is occupied by the relationship between spatiotemporal properties and mass in General Relativity.

Notes:

(103) Note that in talking of ‘structural’ properties I am referring to something somewhat different from (but related to) the structural universals discussed by Lewis (1986 a), Armstrong (1978 b), and Bigelow and Pargetter (1989). Nor am I discussing properties as may be conceived of by structuralists of various kinds. A structuralist may maintain that all there is to some set of entities is the set of more or less formal relations between them. On such a view the essence of a property might just be its relations with other properties. Certainly dispositional monism should be regarded as a structuralist account of properties in that sense— cf. Chapter 6; Hawthorne (2006) calls dispositional monism ‘causal structuralism’ for this reason. But I am not begging the question in this chapter by thinking of ‘structural’ properties in this sense; rather, they are the properties of objects that exist in virtue of their spatial relations or in virtue of the spatial relations of their parts.

(104) Brian Ellis (2005) makes the objection that structural properties must be categorical on behalf of the mixed view. A similar point is made by George Molnar (2003).

(105) In Section 4.2.3 I showed why these properties do not somehow escape the objections to categoricalism.

(106) The qualification ‘non-trivial’ is supposed to exclude, for example, necessarily true conditionals that are entailed by every proposition.

(107) One could reformulate the conditional analysis (CA) so as to exclude finks and antidotes, and so remove the need for Rule 2. This is in effect what Mellor (2000) proposes. It is contentious whether the reformulation still constitutes an analysis. Either way it is more convenient for the following discussion, but equivalent to Mellor’s proposal, to keep the simple conditional analysis and to exclude finks, antidotes, and mimics via Rule 2.

(108) But, as Hawthorne and Manley (2005: 185) appear not to notice, these entailments are not equivalent simpliciter (cf. Bird 2003 b : 163 and Cross 2005: 336-7). The following are equivalent: X ⊧(X&Yֺ→ Z) and Y ⊧ (Y&X□→ Z). So the non-equivalence between X ⊧ Y□→ Z and Y ⊧ Y□→ Z arises thanks to worlds (taking the first entailment) such as w: at w X holds but at the nearest world to w where Y holds, X does not hold. Now in the cases we are considering, X will be a statement asserting that a dispositional property holds and Y asserts that some stimulus property holds. Thus at w the disposition exists but at the nearest world where the stimulus exists, the disposition does not—i.e. the dispositional property in question is finkish.

(109) Thus the equivalence, modulo finks, of the two conditionals may indicate that dispositional concepts come in reciprocal pairs, but does not suggest that dispositional essences come in such pairs.

(110) But one could define a straight line as the set of points L = {a + tb; tS}where a and b are vectors and S is a closed segment of R. In which case both great circle paths are straight lines between A and B.

(111) This derives from a suggestion made by Philip Welch.

(112) Note that we need an analysis of counterfactuals for which C&E does not suffice for C□→E—which it does according to Lewis and Stalnaker. But we need this in any case if the counterfactual analysis of dispositions is to be acceptable. Nozick’s treatment of counterfactuals, for example, is such that C&E does not entail C□→E.

(113) Typically, but not always; cf. footnote 108.

(114) On the other hand, a convincing argument that triangularity does have a dispositional essence would provide strong indirect evidence that the fundamental structural properties have dispositional essences also.