The SameP-Relation as a Repsonse to Critics of Baker

Journal of Philosophical Logic (2005) 34: 561–579 DOI: 10.1007/s10992-005-1525-8

© Springer 2005

TOMASZ KAKOL

THE SAMEP-RELATION AS A RESPONSE TO CRITICS OF BAKER’S THEORY OF CONSTITUTION

ABSTRACT. According to the so-called “standard account” regarding the problem of material constitution, a statue and a lump of clay that makes it up are not identical. The usual objection is that this view yields many objects in the same place at the same time. Lynne Rudder Baker’s theory of constitution is a recent and sophisticated version of the standard account. She argues that the aforementioned objection can be answered by defining a relation of being the same P as (sameP). In this paper I shall examine consequences of her response and show that sameP has wrong formal properties, as a result of which this solution cannot be accepted. KEY WORDS: constitution, identity, properties had derivatively/nonderivatively, properties had independently of constitution relations

0. I NTRODUCTION1

This paper is devoted to several formal aspects of a recent and important new contribution to the so-called problem of material constitution: Lynne Rudder Baker’s theory. Baker holds (following David Wiggins,2 Mark Johnston3 and others4 ) that, e.g., a lump of clay is not identical with the statue it makes. The aim of the article is to address to a particular objection to this theory: “the problem of too many objects”. I shall examine her proposal according to which we can define an identity-like relation of being the same P as (in short: sameP). Baker claims that we actually count objects by means of sameP and that this relation can, but need not, entail (classic) identity. I think that Baker’s solution might be successful, provided that the sameP-relation has the right formal properties. Unfortunately, after examination I conclude that this condition cannot be satisfied by the relation in question. The paper is organized as follows: First, the latest version of Baker’s definition of constitution is introduced and shown to need mending. After the corrections are made, an unexpected feature of constitution emerges: its intransitivity.5 Next, we have to deal with “the problem of too many objects”. To begin, I introduce Baker’s notions of properties had independently of constitution relations and properties had (non)derivatively.

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These notions are usually regarded as an immediate cause of the unwelcome “multiplying of beings” in Baker’s theory. I argue that the strategy of defining the sameP-ralation is a certain (neglected) way to resist the aforementioned objection; but the definition of sameP makes the relation inappropriate for counting since it is nonsymmetric and nontransitive. The solution I try to show consists in employing chains of constitution. Unfortunately, there is an argument that it doesn’t help with mending the former formal feature of the relation we consider.

1. T HE D EFINITION OF C ONSTITUTION . C ONSTITUTION I S

I NTRANSITIVE At first glance denying that a lump of bronze is identical with a piece of sculpture it makes seems to be entirely irrational. The fundamental objection is: How two different objects can be in the same place at the same time? The next question is also in order: If what we have here isn’t identity, what do we have? What kind of relation holds between these objects? It is often called “constitution” – but what does it mean? How to define it precisely? Baker’s basic idea is: When an object of one primary-kind property (in short: PKP) is in certain circumstances, another thing, with another PKP appears. By the notion of ‘PKP’ Baker understands something very similar to what is called “dominant kind” (Rea, 2000, pp. 173f, 187), “minimal essential kind” (Swinburne, 1995, p. 324ff) or otherwise.6 Let the latter property be G. In this case, Baker calls the circumstances G-favorable circumstances.7 In the case of our piece of sculpture we obtain the following: When a lump of bronze (according to Baker, it is just an object with PKP of being a lump of bronze) is in certain circumstances, a new object appears – namely, a piece of sculpture. The strict definition of constitution which Baker assumes in one of her latest papers8 can be formulated as follows: (C) x constitutes y iff there are such properties F and G and circumstances D that: [1] [2] [3] [4]

F is x’s PKP, G is y’s PKP and F
= G, D is G-favorable, and x is spatially coincident with y,9 and x is in D, and it is necessary that: every object with PKP of F, being in D, which doesn’t have the property of a higher order than F that doesn’t entail the property G,10 has a spatial coincident object with PKP of G, and

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[5] it is possible that: x exists without any spatially coincident object with PKP of G, and [6] if y is immaterial, then x is immaterial.11 I think it is worth tracing “the history” of this definition since it reveals motivations for introducing some of its parts. Here we shall focus on conditions [4] and [6]. I shall show that (i) the last clause is – on the basis of certain natural assumptions – redundant, and that (ii) the fourth doesn’t fulfill its intended role. Ad (i). The clause [6] was introduced to eliminate the following counterexample: a human person in a Cartesian sense (a composition of an extended body and an immaterial mind) could be – in the light of definition (C) without clause [6] – constituted by the body alone (cf. Baker, 2000, p. 43). Of course, [6] is helpful here provided that something is immaterial when at least one of its parts is immaterial. This strategy has been criticized by Sider and Zimmerman (Zimmerman, 2002, pp. 603–605; Zimmerman, 2002a, p. 298ff; Sider, 2002, p. 46f), but I don’t want to enter into this discussion. I shall show that – given a certain definition of immaterial thing12 and a certain definition of spatial coincidence – if constituting objects are concrete, [2] entails [6]. The definition of immaterial thing is as follows: (IM) x is immaterial := x is a concrete object which doesn’t occupy any spatial point. The definition of spatial coincidence: (SC) x is spatially coincident with y := for all space points z: x occupies z iff y occupies z.13 Informal proof is simple. Assume that x is spatially coincident with y and that x is concrete, whereas y is immaterial. From (IM), y doesn’t occupy any point, and since x and y occupy the same points, x doesn’t occupy any point, either. Now, since x doesn’t occupy any point and is concrete, from (IM) it results that it is immaterial; therefore, if x and y are spatially coincident and x is a concrete thing, then if y is immaterial, x is immaterial as well. Ad (ii). The clause [4] in its original version (i.e. in Baker, 2000) doesn’t have in its antecedent condition expressed by ‘which doesn’t have the property of a higher order than F that doesn’t entail the property G’. We will call this version of the definition of constitution ‘(C*)’, and the relation of constitution described by it ‘C*’. D. Pereboom argues that (informal) proof of asymmetry of C* given by Baker is based on one premise which can lead into contradiction (Pereboom, 2002, p. 618). If kind G is “realizable” only by F (i.e. it is necessary that every object which has PKP

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of G is constituted by some object which has PKP of F), no problems arise: C* is asymmetric. But if G is “multiply realizable”, i.e. if it is not realizable only by F, one can’t assume (as Baker does) that [4] in (C*) is not satisfied in the purported case of “reversed constitution”: Baker says that – for example – it is necessary that every lump of marble in “statuefavorable circumstances”14 has a coincident statue but it is not necessary that every statue in “marble-favorable circumstances” has a coincident piece of marble (there are bronze statues, for example). Pereboom writes: If a statue can be a bronze statue even when it is in piece-of-marble-favorable circumstances, why can’t a lump be a plant-pot even when it is in statue-favorable circumstances? And if this is possible, then the statue won’t be constituted of the lump, by Baker’s definition. (Pereboom, 2002, p. 618)

In short, it is not necessary that every lump (of marble, of clay etc.) in statue-favorable circumstances has a coincident statue – which leads to the conclusion that constitution doesn’t hold in paradigmatic cases! Having introduced clause [4] in the current shape Baker writes: I think that (revised-c) {i.e. [4] in (C) – TK} will avoid Pereboom’s counterexample. If the lump that constitutes a pot at t is put in statue-favorable circumstances at t, the lump does not thereby come to constitute a statue at t. Clause (ii) of (revised-c) {i.e. clause absent in definition (C*) – TK} is not satisfied in this case. (Baker, 2002, p. 633)

But it is not the case: if a is a lump which constitutes a pot, it can simultaneously constitute a statue – and [4] doesn’t prevent this. We should introduce yet another clause: [7] x doesn’t have any property of a higher order than F that doesn’t entail G. In other words, the constituting object doesn’t have any property of a higher order than its PKP, which doesn’t entail PKP possessed by the constituted object. What benefits does [7] bring about? [7] prevents something we call the branching of constitution. We say that constitution is branchable iff there are cases when one thing constitutes at the same time at least two different things. But a question arises: Is branching an absurd? Let’s imagine the following story.15 In some museum there is a glass box with an exhibit inside which in ancient times was a flower pot. One day, in agreement with those in charge of the museum, experts explored this exhibit and stated that it was in an exceptionally good shape and would be able to serve as a flower pot (the project was accepted but the glass box was replaced with a wooden banister with an alarm and artificial flowers – for safety). From definition (C), the lump of clay constitutes both the exhibit and the flower pot. But clause [7] blocks constituting of the latter since the lump of clay

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already has a property of a higher order than the property of being a lump of clay (that does not entail the property of being a flower pot, of course), namely being an exhibit (although it has it derivatively – see below, (H) and (G)).16 Unfortunately, [7] has a rather strange result: for the same reason, the lump wouldn’t constitute the exhibit if it were a flower pot. Why does it seem strange? It seems so, since it looks as if constitution depended here on temporal order. Imagine a slightly different story: One day, our experts discovered that some flower pot (used only for decoration) came from 100 B.C. . . . (both stories have the same end). Moreover, [7] makes constitution – in the light of Baker’s other assumptions – intransitive. Is it wrong? The answer is: in some sense yes, for, as it will be demonstrated later, it makes it impossible to answer for one of the main objections against Baker’s theory in a promising way. In order to show intransitivity of C after adding [7], we would have to introduce the notions of having properties independently of constitution and having properties derivatively, hypothesis (H) and axiom (G). We shall discuss them shortly. Before discussing it we say that (H) Every PKP had by a given object is had nonderivatively by this object and derivatively by its neighbors (in the chain of constitution). (G) Every object has the property P iff it has it derivatively or nonderivatively. When a, b and c are objects such that F is the PKP of a, G is the PKP of b and H is the PKP of c, whereas a constitutes b and b constitutes c, then if G is the property of a higher order than F that doesn’t entail H, a doesn’t constitute c because of [7]: namely, from (H) and (G), a has the property G (albeit derivatively). Question: Must G always be the property of a higher order than F? Baker writes: If x constitutes y, then y has all the causal powers that x has plus some new kinds of causal powers of its own. (Baker, 2000, p. 25)

The point is that not only if x constitutes y, then y has causal powers of them both, but also that the same goes for x, i.e. x would have weaker causal powers if it didn’t constitute anything (cf. Baker, 2000, p. 41). If this fact wouldn’t be possible if the PKP possessed by the constituted object weren’t of a higher order than the PKP had by the constituting object, then, from this assumption and from (H) and (G), [7] makes constitution intransitive. And it seems that it is unreasonable to claim that x would have weaker causal powers if it didn’t constitute anything, and that the PKP of

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y needn’t be of a higher order than the PKP of x – keeping in mind how Baker defines higher-order properties (see footnote 10). In private correspondence Baker suggested introducing the clause ‘the property G is of a higher order than F’ directly into the definition of constitution (we will call this clause [8]). I think this modification is worth exploring, because it allows for a very simple proof of asymmetry (hence, irreflexivity) of constitution. And, according to Baker, it doesn’t allow for drawing the conclusion (from our first museum-story) that the exhibit constitutes the flower pot, since, according to her, the property of being an exhibit is of a higher order than the property of being a flower pot.17

2. T HE P URPORTED S URPLUS OF B EINGS AND BAKER ’ S S OLUTION :

S AME P-R ELATION (N EGLECTED BY C RITICS ) In order to show the most serious (in my opinion) objection against Baker’s theory and to show Baker’s own solution, we need two notions: having properties independently of constitution and having properties derivatively. Definitions18 of these notions shall be discussed using a few examples. The definition I use differs from Baker’s in that I ignore indexes of circumstances (which shouldn’t be confused with circumstances we have in the definition of constitution) since they are unnecessary for our purposes (cf. Baker, 2000, p. 52f). The definition of having properties independently of constitution is: (IN) x is H independently of constitution relations to y iff [1] H is not an excluded property, x is H, and: either [2] x constitutes y and [3] it is not necessary that: if x is H, then x constitutes something, or [4] y constitutes x and [5] it is not necessary that: if x is H, then it is constituted by such w that it is not necessary that: if w is H, then w constitutes something. The properties which have to be excluded are, according to Baker, modal properties, properties “rooted outside times at which they are had” (i.e. “properties whose instantiation entails that the bearer existed at earlier or later times” (Baker, 2002a, p. 36)), properties of: existing, being identical with something, constituting something (and being constituted by something) and so-called hybrid properties.19 I shall not consider whether this list is complete since it is also unnecessary for my argument.

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Definitions of having properties derivatively and nonderivatively are as follows: (D) x is H derivatively iff H is not an excluded property and there is something which is H independently of constitution relations to x but it is not the case that x is H independently of constitution relations to this thing. (N) x is H nonderivatively iff x is H but not derivatively. So no property can be had both derivatively and nonderivatively by the same thing. According to Baker, we should assume something more: the following axiom is introduced:20 (G) for all x: x is H iff x is H derivatively or x is H nonderivatively. Now let a be a lump of clay which constitutes statue b. According to Baker, (i) a is21 a lump of clay nonderivetively, whereas (ii) b derivatively. In turn, (iii) a is a statue derivatively, whereas (iv) b nonderivatively. Let’s check shortly (i) and (ii) (as far as (iii) and (iv) are concerned, the reasoning is analogous). a is a lump of clay independently of constitution relations to b, because after suitable substitutions, [1], [2] and [3] are satisfied in (IN). But one can wonder whether [1] is really satisfied here. Now, the essential part of [1], namely ‘a is a lump of clay’, is (from (G)) equivalent to ‘a is a lump of clay derivatively or nonderivatively’. Yet we want to know whether a is a lump of clay derivatively – thus, in our case we assume that a is a lump of clay in general (what seems to be reasonable; although, in consequence, we cannot take Baker’s characterizations as the stages of reductive conceptual analysis, otherwise it would lead to a vicious circle (Zimmerman, 2002a, p. 313). Now, if it is not the case that b is a lump of clay independently of constitution relations to a, then from (D) and (N) we have (ii). After suitable substitutions, [2] and [5] are not satisfied in (IN). Therefore, (ii) is the case. Does (i) also hold? The answer is: It does, provided that there is not any y which is a lump of clay independently of constitution relations to a, and it is not the case that a is a lump of clay independently of constitution relations to y. Yet so far we can conclude only that b can’t be such a y: Baker simply assumes (i) (and does so in each similar example), probably because one doesn’t know indeed what could be such a y. The objection usually formulated here is: We have the surplus of beings with the same property. In our example we have two lumps of clay and two statues – it is so since a is a lump of clay (nonderivatively) and a statue (derivatively), whereas b is also a lump of clay (albeit derivatively) and also a statue (nonderivatively). Since similar effects will arise when

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we take into consideration other pairs of objects than lumps constituting statues, it is hardly surprising that we can hear suggestions like that: Thus, there is a rational, thinking, self-conscious being located where you are, and presumably thinking your thoughts, that is numerically different from you. (. . .) At least two philosophers wrote these comments, and at least two philosophers are sitting in your chair and reading them now. (Olson, 2001b)

Advice given by critics amounts to giving up the theory of material constitution. Since Baker’s theory has been focused on mainly as a theory of personal identity, the problem of surplus has been considered with regard to the pair: an organism – a person. For example, E. Olson thinks that the notion of having properties derivatively not only doesn’t help answering the objection,22 but leads straight to substance dualism. According to him, the only reasonable solution is animalism – the view that a person is identical with an organism, not constituted by it (cf. Olson, 2001a, 2001b). According to D. Zimmerman, either having properties derivatively is “weaker”, “not entirely real”, and then the dualistic interpretation is right, or one should simply accept the surplus with all its absurd consequences (Zimmerman, 2002a, pp. 309, 331).23 It seems that aforementioned critics have not appreciated the following strategy by Baker (already used in Baker, 2000): defining relation being the same P as where ‘P’ represents a common noun denoting a kind. The idea is simple: we count things not by means of the relation of identity, but by means of the relation of identity or constitution. In the paper entitled The Same F (aimed mainly against P. Geach’s theory of relative identity) J. Perry wrote: The view I advocate, and which I believe to be Frege’s, is that the role of the general term is to identify the referents – not to identify the “kind of identity” asserted. (Perry, 1970, p. 185)

The relation being the same F as . . . should share some of the properties ordinarily attributed to identity: transitivity, symmetry, and substitutivity. Reflexivity is lost: every object need not be the same F as itself, for all objects are not F’s. But these relations should be at least weakly reflexive: any object that is the same F as some object must be the same F as itself. (Perry, 1970, p. 186)

The relation is: x is the same P as y := x is P and x = y It is easy to prove that such a relation is weakly reflexive, symmetric, transitive and “respects identity”,24 i.e. satisfies the following condition: if x is P and x = y, then x is the same P as y. In (Baker, 2000) Baker defines relation being the same P as (in short: sameP) in the following way:25

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x sameP y := x is P and: either x = y or they are in constitution relation (i.e. either x constitutes y or y constitutes x) On p. 177 she writes: If x and y have constitution relations and x is an F, then x is the same F as y.

Of course, it holds immediately in virtue of the definition. In a footnote she adds that it is the case only when we consider “nonexcluded properties”. In (Baker, 2002a) she defines sameP in a different way: (sP) x sameP y := either [1] P is an excluded property, x is P and x = y, or [2] P is not an excluded property, x is P, and: either x = y or they are in constitution relation It is easy to prove that (sP) is equivalent to: (sP*) x sameP y := x is P and: either x = y or: P is not an excluded property, and x and y are in constitution relation It seems that it would be desirable if sameP had formal properties mentioned by Perry. For example, H. Deutsch has constructed the theory of identity where the relation being the sameP as is weakly reflexive, symmetric, transitive and “respects” ordinary identity.26 Common to Baker and Deutsch is at least that both of them think that . . . there are cases in which strictly distinct [i.e. not identical when ordinary identity is taken into account – TK] objects are counted as one K, where K is an Nt-noun [i.e. a common noun representing a kind of things – TK] . . . (Deutsch, 1998, p. 185)

Both of them want to have ordinary identity in their theories, too. Now we should answer the following question: What formal properties does sameP have?

3. F ORMAL P ROPERTIES OF S AME P: T WO G OOD AND T WO BAD

O NES . T HE R EMEDY FOR O NE OF T HEM The answer is: (1) sameP is weakly reflexive, nonsymmetric and nontransitive, and respects identity. The first and the last property are easily provable. Let ‘EX(P)’ denote that P is an excluded property; ‘xCRy’ denote that x and y are in constitution relation. First, x sameP y entails P(x), whereas applying definition to x sameP x results in P(x) ∧ (x = x ∨ (¬EX(P) ∧ xCRx)), i.e. P(x) and tautology, what is equivalent to P(x). Therefore, from x sameP y we have x sameP x.

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Secondly, from P(x) ∧ x = y we have (P(x) ∧ x = y) ∨ (P(x) ∧ (¬EX(P) ∧ xCRy)), which is equivalent to P(x) ∧ (x = y ∨ (¬EX(P) ∧ xCRy)), i.e. we have x sameP y. Therefore, P(x) ∧ x = y entails x sameP y. Nonsymmetry and nontransitivity can be demonstrated in the following way. Following Baker, we have discussed some examples of PKPs had by objects standing in constitution relation. They suggest putting forward the following hypothesis: (H) Every PKP had by a given object is had nonderivatively by this object and derivatively by its neighbors (in the chain of constitution). Now take the model we have talked about: a lump of clay a constitutes a flower pot b, which in turn constitutes an exhibit c. Let ‘1’, ‘2’ and ‘3’ denote suitable PKPs; underline denotes that a given property is had nonderivatively, whereas lack of underline – derivatively. ‘1’ and ‘2’ simply represent that 2 is the property of a higher order than 1. Arrows denote relation of constitution. From (H) and the definition of constitution (clause [7] and [8]) we have:

From (IN) it follows that it is not the case that a is a lump of clay independently of constitution relations to c and vice versa, and that it is not the case that a is an exhibit independently of constitution relations to c and vice versa. Now ask whether b is a lump of clay independently of constitution relations to c. From (IN); [1], [2] and [3] are satisfied, therefore b is a lump of clay independently of constitution relations to c. Does it hold in reverse, i.e. is c a lump of clay independently of constitution relations to b? Now it seems that [5] in (IN) is not satisfied: if c is a lump of clay, then it has to be constituted by such a thing, that if this thing is a lump of clay, the very thing doesn’t have to constitute anything. Therefore, c is not a lump of clay independently of constitution relations to b, hence, from (D) we have that c is a lump of clay derivatively:

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However, the definition of constitution (clause [7]) implies that a cannot have the property denoted by ‘3’. Now from (sP*) it is clear that sameP is not symmetric since we have in our model: b is the same exhibit as a (although a is not at all an exhibit), yet a is not the same exhibit as b. The relation is not transitive, either, for we have in our model: a is the same lump of clay/flower pot as b, b is the same lump of clay/flower pot as c, yet a is not the same lump of clay/flower pot as c. It looks as if we had two lumps (and two flower pots as well). The remedy for the nontransitivity can be as follows. All the examples of properties (not only PKPs) that, as Baker argues in (Baker, 2000), are had derivatively/nonderivatively, suggest putting forward a hypothesis more general than (H): (H*) Every nonexcluded property had nonderivatively by a given object, is had derivatively by its neighbors (in the chain of constitution). Now if we assume that constitution is a one-to-one relation, and if in the definition of sameP we put the relation C2 defined below instead of constitution, we can prove that sameP is transitive: (C2) x is in relation C2 to y := x constitutes y or there exist (finitely many) z1 , z2 , . . . , zn such that x constitutes z1 , z1 constitutes z2 , . . . , zn constitutes y In the proof of the transitivity of sameP the following fact is essential: (2) C2 is transitive One-to-oneness of constitution means of course that: (3) for every x, y, z: if y constitutes x and y constitutes z, then x = z (4) for every x, y, z: if x constitutes y and z constitutes y, then x = z (3) says that constitution cannot be branchable. Our discussion of the clause [7] in the definition of constitution implies that this is the case. (4) also seems to be the case, although the definition of constitution doesn’t entail it.27 As far as symmetry of sameP is concerned, one could relatively easily prove symmetry of sameP putting – as above – relation C2 in the definition of sameP instead of constitution, and using (G) and two following intuitive premises:28 (H1*) Every nonexcluded property had nonderivatively by a given object is had derivatively by all the other objects in the chain of constitution.

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(H1**) Every property had derivatively by a given object is had derivatively or nonderivatively by every object in the chain of constitution. The problem is that our model of lump/pot/exhibit falsifies both (H1*) and (H1**). That is why (although we are replacing the relation of constitution with C2 in the definition of sameP) we obtain: b is the same exhibit as a (although a is not at all an exhibit), but not vice versa; and c is the same exhibit as a, but not vice versa. It appears that demonstrating that sameP has the required properties eliminates or at least substantially weakens the force of “the surplus of beings” objection. The analysis I propose also reveals that Zimmerman is not right when he refers to the definition of sameP in the following way: Notice that there is no significant use of the notions of derivative and nonderivative exemplifications in this characterization; all that is needed is a distinction between the “excluded properties”, the ones that are not automatically shared by constitutionally related things. (Zimmerman, 2002a, p. 323)

It is true that the notions he mentions are missing from the definition of sameP. However, from the premises required for the proof of symmetry of sameP it is clear that the idea of having properties derivatively/nonderivatively is a key one. Counting “via sameP” is not surprising, either. M. Rea writes: We count one object (and only one object) in every region that is filled by matter unified in some object-constituting way. (Rea, 1998, p. 321) When we sell our dining room furniture, for example, we don’t charge people for the table, the chairs, and the pieces of wood that constitute them. (Rea, 1998, p. 321)

Critics of the theory of constitution don’t intend to deny the cited remarks; but – according to Baker – erroneously assume that counting “via identity” is right, but “via sameP” not. Unfortunately, nonsymmetry of sameP should make us find another candidate for the relation “by means of which we could count” – just to make the theses of the theory of constitution less paradoxical. To sum up: I argued that the clause [7] should be added to Baker’s definition of constitution in order to block Pereboom’s counterexample. I also argued that [8] seems to be essential in Baker’s theory – given her views on the role of constitution in bringing about irreducible causal powers. However, [7] and [8] made constitution intransitive. Next I showed that sameP-relation was not transitive and not symmetric, either. Subsequently, I suggested using the ancestral of constitution in the definition of sameP and assuming one-to-oneness of constitution. This amendment rendered

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sameP transitive. The problem is that sameP is nonsymmetric because of [7] and [8]. The upshot is that [8] belongs, I believe, to the core of Baker’s theory, and [7] is needed to guarantee that constitution is not branchable, which in turn is essential in the proof of transitivity of sameP (see next section), yet we cannot have them both. In the next section we prove that if (2), (3) and (4) hold, and if the definition (sP*) is modified in such a way that we have C2 instead of constitution, then sameP is transitive. Next, we prove that through having modified this definition in the aforementioned way and using (G), (H1*) and (H1**), we obtain symmetry of sameP. Nevertheless, as I pointed out above, (H1*) and (H1**) failed in the theory we examined, and the result is that sameP is nonsymmetric.

4. F ORMAL P ROOFS

The proofs will be made according to the following scheme. First, (i) via dictum de omni, we omit (all or some) general prefix quantifiers, then (ii) using metatheorem (M) we deduce from such open formulas theorem which is to be proved, and which will be also an open formula, and finally (iii) we use the rule of generalization. Since the relation of deducibility is transitive, from premises in a closed form, we obtain the theorem that is to be proved. Now the aforementioned metatheorem (M) is: If z1 , . . . , zn are all the free variables (different from each other) in the formula A, and b1 , . . . , bn are different individual terms which don’t occur in formulas from set X ∪ {A, B}, and B(z1 /b1 , . . . , zn /bn ) ∈ CnPC-I (X ∪ {A(z1 /b1 , . . . , zn /bn )}), then ‘A → B’ ∈ CnPC-I (X).29 Where CnPC-I is the consequence operator of (classic) first-order predicate calculus. For brevity’s sake, we omit in proofs part (i) and (iii), and we will write simply about substitutions (of individual terms), but such substitutions are indeed made on open counterparts of premises. Let ‘xC2Ry’ denote that xC2y or yC2x, ‘xCy’ denote that x constitutes y, ‘EX(P)’ denote that P is an excluded property. First, we prove two lemmas: LEMMA 1. ∀x, y, z (yC2x ∧ yC2z → xC2Rz ∨ x = z) LEMMA 2. ∀x, y, z (xC2y ∧ zC2y → xC2Rz ∨ x = z)

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The proof of Lemma 1. 1. bC2a ∧ bC2c (assumption with substitution) 2. (bCa ∨∃z1 , . . . , zn (bCz1 ∧· · ·∧zn Ca))∧(bCc ∨∃w1 , . . . , wm (bCw1 ∧ · · · ∧ wm Cc)) (def. of C2 with substitution, 1) 3. (bCa ∧ bCc) ∨ (bCa ∧ ∃w1 , . . . , wm (bCw1 ∧ · · · ∧ wm Cc)) ∨ (bCc ∧ ∃z1 , . . . , zn (bCz1 ∧ · · · ∧ zn Ca)) ∨ ((∃z1 , . . . , zn (bCz1 ∧ · · · ∧ zn Ca)) ∧ (∃w1 , . . . , wm (bCw1 ∧ · · · ∧ wm Cc))) (2, classic propositional calculus [= CPC]) 4. bCa ∧ bCc → a = c ((3), substitution) 5. bCa ∧ ∃w1 , . . . , wm (bCw1 ∧ · · · ∧ wm Cc) → aC2c (Lemma 1a) 6. bCc ∧ ∃z1 , . . . , zn (bCz1 ∧ · · · ∧ zn Ca) → cC2a (Lemma 1b) 7. (∃z1 , . . . , zn (bCz1 ∧· · ·∧zn Ca))∧(∃w1 , . . . , wm (bCw1 ∧· · ·∧wm Cc)) → aC2Rc ∨ a = c (Lemma 1c) 8. (bCa ∧ bCc) ∨ (bCa ∧ ∃w1 , . . . , wm (bCw1 ∧ · · · ∧ wm Cc)) ∨ (bCc ∧ ∃z1 , . . . , zn (bCz1 ∧ · · · ∧ zn Ca)) ∨ ((∃z1 , . . . , zn (bCz1 ∧ · · · ∧ zn Ca)) ∧ (∃w1 , . . . , wm (bCw1 ∧ · · · ∧ wm Cc))) → aC2Rc ∨ a = c (3, 4, 5, 6, 7, def. of C2R, CPC) 9. aC2Rc ∨ a = c (3, 8, CPC) Now we prove Lemmas 1a–1c. For the proof of Lemma 1a observe that when bCw1∗ (where w1∗ is some value of the variable w1 ) and bCa, suitably substituted (3) entails: w1∗ = a. Therefore, either ∃w2 , . . . , wm (aCw2 ∧ · · · ∧ wm Cc) or aCc (if m = 1), hence, from the definition of C2, aC2c. Similarly, for the proof of Lemma 1b observe that when bCz1∗ (where z1∗ is some value of the variable z1 ) and bCc, suitably substituted (3) entails: z1∗ = c. Therefore, either ∃z2 , . . . , zn (cCz2 ∧· · ·∧zn Ca) or cCa (if n = 1), hence, from the definition of C2, cC2a. Lemma 1c can be proved in the following way. Let bCz1∗ and bCw1∗ . Therefore, from (3) suitably substituted we obtain z1∗ = w1∗ . Next, let z1∗ Cz2∗ and z1∗ Cw2∗ . Similarly, from (3) suitably substituted we have z2∗ = w2∗ . If n is finite and n = m, then using (3) n times results in a = c; if n > m, then using (3) m times results in cC2a; finally, if m > n, then using (3) n times results in aC2c. Therefore, we have: aC2Rc ∨ a = c.

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Lemma 2 can be proved in a similar way to Lemma 1 (except using (4) instead of (3)). THEOREM. sameP is transitive. Proof. 1. a sameP b ∧ b sameP c (assumption, substitution) 2. P(a) ∧ (a = b ∨ (¬EX(P) ∧ aC2Rb)) ∧ P(b) ∧ (b = c ∨ (¬EX(P) ∧ bC2Rc)) (def. of sameP, substitution) 3. (a = b ∨ (¬EX(P) ∧ aC2Rb)) ∧ (b = c ∨ (¬EX(P) ∧ bC2Rc)) (2, CPC) 4. (a = b∧b = c)∨(a = b∧(¬EX(P)∧bC2Rc))∨(b = c∧(¬EX(P)∧ aC2Rb)) ∨ ((¬EX(P) ∧ aC2Rb) ∧ (¬EX(P) ∧ bC2Rc)) (3, CPC) 5. a = b ∧ b = c → a = c (elementary theory of identity, substitution) 6. a = b ∧ (¬EX(P) ∧ bC2Rc) → ¬EX(P) ∧ aC2Rc (as above) 7. b = c ∧ (¬EX(P) ∧ aC2Rb) → ¬EX(P) ∧ aC2Rc (as above) 8. aC2b ∧ bC2c → aC2c ((2), substitution) 9. cC2b ∧ bC2a → cC2a (as above) 10. bC2a ∧ bC2c → aC2Rc ∨ a = c (Lemma 1, substitution) 11. aC2b ∧ cC2b → aC2Rc ∨ a = c (Lemma 2, substitution) 12. (bC2a ∧ bC2c) ∨ (aC2b ∧ cC2b) → aC2Rc ∨ a = c (10, 11, CPC) 13. (aC2b∧bC2c)∨(cC2b∧bC2a)∨(bC2a ∧bC2c)∨(aC2b∧cC2b) → aC2Rc ∨ a = c (8, 9, 12, def. of C2R, CPC) 14. (aC2b ∨ bC2a) ∧ (bC2c ∨ cC2b) → aC2Rc ∨ a = c (13, CPC) 15. ((¬EX(P) ∧ aC2Rb) ∧ (¬EX(P) ∧ bC2Rc)) → a = c ∨ (¬EX(P) ∧ aC2Rc) (14, def. of C2R, CPC) 16. P(a) ∧ (a = c ∨ (¬EX(P) ∧ aC2Rc)) (2, 4, 5, 6, 7, 15, CPC)

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17. a sameP c (16, def. of sameP) DER

Let ‘P x’ denote that x is P derivatively, whereas ‘PNDER x’ denote that x is P nonderivatively. At first glance, propositions (H1*) and (H1**) are: (H1*+) ∀x (PNDER x ∧ ¬EX(P) → ∀y(xC2Ry → PDER y)) (H1**+) ∀x (PDER x → ∀y(xC2Ry → PDER y ∨ PNDER y)) Notice that if there is no free variable y in the antecedents of implications, (H1*+) and (H1**+) are respectively equivalent to: (∗)

∀x, y (PNDER x ∧ ¬EX(P) → (xC2Ry → PDER y))

(∗∗)

∀x, y (PDER x → (xC2Ry → PDER y ∨ PNDER y))

In the case of the excluded properties, our two pairs of formulas are trivially true. Now, because we want to allow relational properties, yet some realizations of the schemas (H1*+) and (H1**+) yield in this case false formulas, (H1*) and (H1**) should be read as (∗) and (∗∗). Observe that as far as properties such as being a lump, being a pot, being an exhibit are concerned, we can choose any reading of (H1*) and (H1**).

THEOREM. If (∗) and (∗∗) hold, sameP is symmetric. Proof. 1. a sameP b (assumption, substitution) 2. P(a) ∧ (a = b ∨ (¬EX(P) ∧ aC2Rb)) (def. of sameP, substitution) 3. (P(a) ∧ a = b) ∨ (P(a) ∧ ¬EX(P) ∧ aC2Rb) (2, CPC) 4. (P(a) ∧ a = b) → P(b) (elementary theory of identity, substitution) ((G), substitution) 5. P(a) ↔ PDER a ∨ PNDER a ((G), substitution) 6. P(b) ↔ PDER b ∨ PNDER b NDER DER a ∧ ¬EX(P) → (aC2Rb → P b) ((∗), substitution) 7. P 8. PDER a → (aC2Rb → PDER b ∨ PNDER b) ((∗∗), substitution) (7, CPC) 9. PNDER a ∧ ¬EX(P) ∧ aC2Rb → PDER b (8, CPC) 10. PDER a ∧ aC2Rb → PDER b ∨ PNDER b DER DER NDER b (10, CPC) 11. P a ∧ ¬EX(P) ∧ aC2Rb → P b ∨ P 12. (PDER a ∨ PNDER a) ∧ ¬EX(P) ∧ aC2Rb → PDER b ∨ PNDER b (9, 11, CPC) 13. P(a) ∧ ¬EX(P) ∧ aC2Rb → P(b) (5, 6, 12, CPC)

THE SAMEP-RELATION AS A RESPONSE

14. P(b) 15. (b = a ∨ (¬EX(P) ∧ bC2Ra)) 16. P(b) ∧ (b = a ∨ (¬EX(P) ∧ bC2Ra)) 17. b sameP a

577

(3, 4, 13, CPC) (2, symmetry of identity and C2R, CPC) (14, 15, CPC) (16, def. of sameP)

ACKNOWLEDGEMENTS I would like to thank Professor Lynne Rudder Baker for discussing the details of her theory with me (by means of email), support and the idea of publishing formal remarks concerning the Constitution View. I am also grateful to mgr Slawomir Wacewicz, my university colleague, for making linguistic correction of the paper. Special thanks to an anonymous referee who helped me make several points clearer.

N OTES 1 Paragraphs 1 and 2 are based on my earlier paper O niektorych trudnosciach teorii

materialnej konstytucji Lynne Rudder Baker in Kognitywistyka i Media w Edukacji 7(1–2) (2003). 2 See Wiggins (1968) (also in: Rea, 1997, pp. 3–9) and Wiggins (1980). 3 See Johnston (1992) (also in: Rea, 1997, pp. 44–62), Johnston (1997) and Johnston (2002). 4 See, e.g., articles collected in (Rea, 1997) in Part I (entitled “Coincident Entities”). Among recent papers favorable to the constitution view, (Fine, 2003) is worth mentioning. 5 For the reader’s convenience I shall repeat usual definitions of several relations: R is symmetric iff ∀x, y (xRy → yRx) R is nonsymmetric iff R is not symmetric R is asymmetric iff ∀x, y (xRy → ¬yRx) R is transitive iff ∀x, y, z (xRy ∧ yRz → xRz) R is nontransitive iff R is not transitive R is intransitive iff ∀x, y, z (xRy ∧ yRz → ¬xRz) 6 Baker doesn’t give the definition of PKP, but from her examples it follows that it is (at least) the narrowest essential property. 7 The term ‘circumstances’ has somewhat misleading connotations, but here it has a technical sense. 8 See Baker (2002a, p. 34). Indeed, the condition expressed by ‘which doesn’t have the property of a higher order than F that doesn’t entail the property G’ is only mentioned in footnote 6. I would like to emphasize that I omit everywhere time variables since in the current paper I consider mainly the synchronic aspect of constitution. 9 Where Baker writes about space, she understands by it physical space.

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10 That PKP of H is of a higher order than PKP of G means that H and G are such PKPs

that H confers on its bearers higher-order causal powers than G. Cf. Baker (2002, p. 632f). We will see later the importance of this definition in Baker’s theory. 11 If fulfillment of [4] guarantees that D is G-favorable, then the condition ‘D is G-favorable’ in [1] is redundant. 12 The definition was introduced by Baker in private correspondence. 13 Cf. Baker (2000, p. 210). It should be pointed out that later in her text Baker proposes another definition of spatial coincidence (x and y coincide spatially iff they have the same closed outer boundary), but finally the definition is: x and y coincide spatially iff they have the same closed outer boundary and it is possible that for all space points z: x occupies z iff y occupies z (see Baker, 2000, p. 211). 14 I shall use the form instead of saying in a prolix way: “every object that has the PKP of being a lump of marble in circumstances favorable to the property of being a statue has a coincident object that has the PKP of . . .” etc. 15 The story is a little bit crazy but “a lot more possible” than stories about zombies, brain transplant and the like. I wonder if something similar to my story has actually taken place somewhere. 16 An anonymous referee wrote that it is doubtful whether “the relation between a flower pot and an exhibit is one of constitution”. I agree entirely, since it is not clear whether being an exhibit can be regarded as a PKP in Baker’s sense (see above). Although Baker didn’t raise similar doubts in our correspondence concerning this example (from which I concluded that she accepted that being an exhibit was the PKP of the object in question), I don’t want to attribute to her definitely this view. Nevertheless, even if one cannot invent a similar example (whose point is to show dependency of constitution on temporal order due to the “rivalry” between appropriate PKPs), the avoidance of branching is worth pursuing for reasons which will be explained later. 17 Private correspondence. See also footnote 16. 18 More precisely, I offer a kind of formal interpretation of informal but precise definitions given by Baker, cf. (Baker, 2002a, p. 36f). Recall that I omit everywhere time variables. All the definitions and theorems of that kind will be de facto schemas of definitions and theorems. They can be obtained by substituting predicates for H (or for a different variable – the context will usually determine for which one). 19 Cf. (Baker, 2000, p. 100). 20 Explicite in (Baker, 2002a). 21 Of course, ‘is’ doesn’t express here identity. We could say: a has the property of being a lump of clay. 22 Baker seems to think it does. See (Baker, 2000, pp. 54f, 98, 102f, 172f; Baker, 2002a, p. 38). Notice, however, that we have formulated the charge using the notions of properties had derivatively/nonderivatively. 23 Zimmerman writes that Baker “. . . can always either accept that there are many minds, and learn to live with the results (. . .); or she can deny that coincident entities besides the person really have mental states while affirming that the person really has the properties of its constituters. But Baker does not seem to me to have decisively chosen one of these alternatives rather than the other, and the result is a theory afflicted with creeping dualism.” (Zimmerman, 2002a, p. 331) But the latter amounts to holding that we have two coincident organisms. So one should either accept the surplus of beings or accept Cartesian dualism of two substances with radically different properties. 24 I use this expression following H. Deutsch. See (Deutsch, 1998, p. 183).

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25 (Baker, 2000, p. 174). Here I also omit time variables. 26 See Deutsch (1998). Deutsch proposes an original semantics for his theory but we

needn’t focus on it here. 27 Formal proof of the transitivity of sameP is in the next section. 28 It will also be proved in the next section. 29 See (Batóg, 1994, p. 164).

R EFERENCES Baker, L. R. (2000) Persons and Bodies. A Constitution View, Cambridge University Press, Cambridge. Baker, L. R. (2002) Replies, Philosophy and Phenomenological Research 64(3), 623–635. Baker, L. R. (2002a) On making things up: Constitution and its critics, Philosophical Topics 30(1), 31–51. Batóg, T. (1994) Podstawy Logiki, Wydawnictwo Naukowe UAM, Pozna´n. Deutsch, H. (1998) Identity and general similarity, Philosophical Perspectives 12, 177– 199. Fine, K. (2003) The non-identity of a material thing and its matter, Mind 112(446), 195– 234. Johnston, M. (1992) Constitution is not identity, Mind 101(401), 89–105. Johnston, M. (1997) Manifest kinds, The Journal of Philosophy 94(11), 564–583. Johnston, M. (2002) Parts and principles: False axioms in mereology, Philosophical Topics 30(1), 129–166. Olson, E. (2001a) Persons and bodies: A constitution view (review), Mind 110(438), 427– 430. Olson, E. (2001b) Thinking animals and the constitution view, e-symposium devoted to Persons and Bodies (www.uniroma3.it/kant/field/bakersymp.htm) Pereboom, D. (2002) On Baker’s Persons and Bodies, Philosophy and Phenomenological Research 64(3), 615–622. Perry, J. (1970) The same F, Philosophical Review 79, 181–200. Rea, M. C. (ed.) (1997) Material Constitution. A Reader, Rowman and Littlefield. Rea, M. C. (1998) Sameness without identity: An Aristotelian solution to the problem of material constitution, Ratio (new series) 11(3), 316–328. Rea, M. C. (2000) Constitution and kind membership, Philosophical Studies 97, 169–193. Sider, T. (2002) Persons and bodies: A constitution view (review), The Journal of Philosophy 99(1), 45–48. Swinburne, R. (1995) Spójno´sc´ teizmu, tł. T. Szubka, Wydawnictwo Znak, Kraków (original version: The Coherence of Theism, Oxford University Press, Oxford, 1993). Wiggins, D. (1968) On being in the same place at the same time, Philosophical Review 77, 90–95. Wiggins, D. (1980) Sameness and Substance, Basil Blackwell, Oxford. Zimmerman, D. (2002) Persons and bodies: Constitution without mereology?, Philosophy and Phenomenological Research 64(3), 599–606. Zimmerman, D. (2002a) The constitution of persons by bodies: A critique of Lynne Rudder Baker’s theory of material constitution, Philosophical Topics 30(1), 295–338.

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