Found Phys (2011) 41:1740–1755DOI 10.1007/s10701-011-9590-z

Quantum Ontology and Extensional Mereology

Claudio Calosi · Vincenzo Fano · Gino Tarozzi

Received: 1 April 2011 / Accepted: 27 July 2011 / Published online: 2 September 2011© Springer Science+Business Media, LLC 2011

Abstract The present paper has three closely related aims. We first argue thatAgazzi’s scientific realism about Quantum Mechanics is in line with Selleri’s andTarozzi’s proposal of Quantum Waves. We then go on to formulate rigorously dif-ferent metaphysical principles such as property compositional determinateness andmereological extensionalism. We argue that, contrary to widespread agreement, re-alism about Quantum Mechanics actually refutes only the former. Indeed we evenformulate a new quantum mechanical argument in favor of extensionalism. We con-clude by noting that, given the results of the work, Agazzi’s particular attitude towardsQuantum Mechanics is still one of the most promising theoretical perspectives.

Keywords Quantum realism · Properties of composite systems · Mereologicalextensionalism

1 Introduction

Evandro Agazzi has clearly underlined that the neo-positivistic refusal of an au-tonomous philosophical analysis that goes beyond formal clarification of scientificdiscourse is unjustified. For example, philosophical questions that, on the one handlie at the foundations of physical theories, and on the other are implied by them, canbe addressed in non metaphysical terms. This can be done by assuming those veryneo-positivistic criteria of meaning that, even if useless in defining scientific proposi-tions, are perfectly able to guarantee a reformulation of some metaphysical theses interms of philosophical principles endowed with empirical meaning. In this paper wewill in fact address some metaphysical principles such as mereological extensional-ism and property compositional determinateness, to name just two. We will provide

C. Calosi (�) · V. Fano · G. TarozziDepartment of Foundations of Science (Scienze di base e Fondamenti), University of Urbino,Palazzo Albani, Via Timoteo Viti 10, 61029 Urbino, Italye-mail: claudio.calosi@uniurb.it

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their rigorous formulation and we will attribute them empirical meaning by compar-ing them with some ontological intimations that come from a realistic interpretationof Quantum Mechanics.1

Agazzi maintains that recent developments in Quantum Mechanics do not offer adefinitive or exhaustive solution to the philosophical problem of the nature of micro-objects. He underlines, with extraordinary intuition, the need to introduce new con-cepts to analyze the quantum mechanical realm. This novelty should not be just theresult of a combination of classical concepts, but rather, the introduction of an origi-nal set of concepts that could be able, at least in principle, to replace the old ones, aspointed out clearly in Agazzi [2]:

“Only by inventing some new concept, that is new in this fundamental sense,could we possibly overcome the present uneasy state of affairs, which is notrelated to the regret of losing the old concepts but to the lack of new conceptscapable of adequately replacing them”.

In the very same year that Agazzi was advocating the introduction of a new con-cept to solve the wave-particle duality, Franco Selleri proposed a realistic interpreta-tion of QM based on such a new concept in Selleri [21]. It is the concept of empty orquantum wave. This can be thought of as the synthesis of three different conceptionsabout the wave-particle duality held by the fathers of quantum theory.

Coherently with his general perspective, Agazzi [2] observes that the importantconceptual novelty of the quantum wave hypothesis is constituted by the refusal ofthe symmetric nature of wave-particle duality. In fact Selleri endorses neither thoseinterpretations that insist on a unique corpuscular nature, nor those complementaryinterpretations that maintain that the adoption of a unique interpretation over the otheris, in the end, contradictory. Selleri accepts both a realistic interpretation of the dualityand some sort of ontological priority of particles over waves:

“The essential novelty of this concept is represented by the acceptance of thede Broglie realist interpretation of wave-particle duality, but not of symmetricalnature of the dualism. In Selleri’s approach both particles and waves are simul-taneously real, but the latter can be characterized only with relational proper-ties with particles: that is the observables properties of producing interferenceand stimulated emission. Such a possibility would imply an ontological prior-ity of particles over waves, which would therefore belong to a weaker level ofphysical reality, containing objects which are sensible carriers of exclusivelyrelational predicates”.2

Different experiments have been advanced stemming from this new realistic inter-pretation of the wave-function. The interest of these experiments, as Agazzi pointedout, is twofold:

“The interest of these experiments is due to the fact that they would allow onenot only to test this new realist interpretation vs. the Copenhagen one—to dis-criminate experimentally between two different philosophical interpretations of

1QM from now on.2See Agazzi [2], p. 73.

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a physical theory—but also the well-known axiom of the reduction of the wavefunction (. . .)”.3

Agazzi was referring to the fact that the properties of quantum waves can be usedto reconstruct photons’ trajectories within an interferometer while at the same timeregistering the interference pattern, contrary to the prediction based on the collapseof the wave-function. This would also establish an important connection between thewave particle-duality on the one hand, and the fundamental quantum measurementproblem on the other. So far, as Hardy [10] points out, no experiment has been suc-cessful in either revealing such quantum waves’ properties or in refuting the collapsepostulate.

However a new thought experiment that could be easily realized has been pro-posed, for example in Auletta and Tarozzi [4, 5], to the point that quantum wavescould produce EPR style correlations. This would actually allow a new and unsus-pected relation to be established between two different realistic interpretations ofQM: Agazzi’s realism about theoretical entities on the one hand and the EPR’s prin-ciple of local realism on the other.

Here is the plan of the paper. In the next section we will discuss broadly Agazzi’sscientific realism. In Sect. 3 we will address ontological issues regarding entangledstates in the tensor product space for composite systems and in Sect. 4 we will givea rigorous formulation of such ontological problems in terms of modern mereology,i.e. the formal theory of parthood relations. This will allow us to discuss in Sect. 5different conceptual relations between various ontological thesis, such as mereologi-cal extensionalism and property compositional determinateness. The closing Sect. 6will be devoted to the formulation of possible future research developments.

2 Agazzi’s Scientific Realism

Agazzi’s conception was elaborated in close connection with the philosophical prob-lems raised by QM, in particular the status and the meaning of the complementarityprinciple and the realistic interpretation of the wave-function.

His realistic demand stems from the problem of the epistemological value of sci-entific theories. The main point is trying to establish whether scientific theories pro-vide any ground for objectivity. Agazzi underlines, in Agazzi [1], three fundamentalsenses of such a term: “objectivity as inter-subjectivity, as invariance and as corre-spondence to objects”.4 Through an effective and detailed analysis he argues thatthese three senses can be identified. The presuppositions grounding the possibility ofsuch an identification, from an epistemological point of view, are essentially three:

(i) The first one is the operational ground of scientific concepts along with thefact that, this grounding notwithstanding, those concepts cannot be reduced to apurely operational dimension.

(ii) The realization that the meaning of scientific terms is context-dependent.

3See Agazzi [2], p. 73.4Original in Italian. Our translation.

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(iii) The fact that scientific objects are constituted by properties established objec-tively through a set of operations and yet they do not only constitute a simplebundle of such properties but rather a well defined structure of relations betweenthem.

We will see that these three aspects are even more inter-related when applied tothose scientific concepts that are expressed through the so-called theoretical terms,i.e. those terms whose denotations are not immediately observable. Let’s considerbriefly each point.

Scientific theories are constructed on the basis of theoretical terms, but their aimis to account for those immediate facts of experience describable using empiricaland observational terms. This raises the infamous problem of how to guarantee thatsuch theoretical terms maintain a link with empirical terms. A theoretical conceptlike “electron” is, we read in Agazzi [1], “a theoretical construction around whichwe gather many properties operationally defined”.5 Actually the operational aspectallows theoretical terms to keep in touch with experience, so to acquire a physicalmeaning. Those theoretical terms cannot however be simply reduced to operationalterms, as Agazzi [1] explicitly claims:

“[. . .] we do not even dream of saying that theoretical concepts can be reducedto operational concepts: those who wanted to do that would do the same thingas someone wishing to reduce his house to the bricks that constitute it”.6

Various combinations of empirical (observational) terms actually form construc-tions (the theoretical terms) that are not themselves directly operational.

From all this, it follows that the meaning of theoretical terms is always contextual,as we underlined in our second point. Agazzi [1] notes that this

“does not mean however that [the meaning of theoretical terms] comes from theobservational terms through a context [. . .], but it comes from the context itself.In this case observational terms are present, but they are not alone in it, for thecontext is actually realized by all those logical and mathematical connectionsthat link together each and every concept, observational and not”. 7

The context within which the theoretical terms acquire their meaning is the phys-ical theory of which they are part and that they help to construct. Only the physicaltheory as a whole can be empirically interpreted and thus can be related to possibleobservations. This point is of decisive importance for a realistic interpretation of thequantum wave. According to Agazzi many interpretative problems of QM stem fromthe attempt to apply classical concepts to quantum objects. The solution cannot sim-ply consist however of a new combination of classical notions of wave and particle,exactly because the meaning of such theoretical terms is context-dependent. This isclearly recognized in Agazzi [1] again:

5Original in Italian. Our translation.6Original in Italian. Our translation. This last remark is of particular importance for the rest of the paper.7Original in Italian. Our translation.

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“not only can we say, but we must say that it is not the same particle, it is notthe same wave as the particle and wave we talk about in classical and quantummechanics. This is because the contexts are different”.8

This raises the need to find truly original concepts in order to escape the traditionalproblems related to a realistic interpretation of QM. And this takes us to our last point.

We have already pointed out that scientific objects, in particular those that aredenoted by theoretical terms, present themselves as relational structures of propertiesdefined operationally. And we have also pointed out that they cannot be reduced tosuch properties. This last point is of fundamental importance in Agazzi’s perspective.And it is deeply related to the contextual character of theoretical terms:

“The object is always a structure, a relational structure. Those relations can be,for the most part, the result of some operations but the ‘holding together’ ofsuch relations cannot be justified in terms of any of those operations”.9

The attempt to reconstruct such a structure is the primary goal of scientific theoriesfor

“structure is not just what lies beneath experimental determinations and objec-tive characteristics, but is what is constituted by them: it is the object”.10

It is this structure that makes the world what it is and that can deem our theorieswrong. For our theories presuppose a structure that is not the structure of the worldand yet they are an attempt to reconstruct such a world structure.

We subscribe to Agazzi’s opinion whereby theoretical terms acquire their empiri-cal meaning through the whole theory in which they are embedded. Nevertheless thisdoes not mean that whenever one endorses a scientific realistic thesis about some the-oretical entities, one is also compelled to accept that the world is exactly as the theorystates. In what follows we adopt a mild form of entity scientific realism, as developedin a semantic approach to theories described for example in Fano and Macchia [7],which is committed to the reality of theoretical entities only in relation to the investi-gated system and to the theoretical framework most suitable for that system.

3 Quantum Mechanics and Properties of Composite Systems

Let x be a quantum particle; x is a model for Quantum Mechanics (QM) in Sup-pes’ [22] sense. Suppose, for the sake of argument, that x is completely describableby the observable O11 with two eigenfunctions |↑〉x and |↓〉x , with eigenvalues 1and −1 respectively. Its state �x is represented by a normalized vector in the two-dimensional Hilbert space Hx spanned by |↑〉x and |↓〉x . This Hilbert space contains

8Original in Italian. Our translation.9See Agazzi [1]. Original in Italian. Our translation.10See Agazzi [1]. Original in Italian. Our translation.11Normally a description of a quantum system is given by at least two observables, but in this context wecan neglect one of them and focus only on the other.

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all linear combinations of such vectors. Hence every possible state of x can be writtenas:

c1|↑〉x + c2|↓〉x (1)

where c1 and c2 are complex coefficients. Let y be another particle, distinct from x.Every possible state of y, �y is represented by a normalized vector in the two-dimensional Hilbert space Hy which can be written as:

c3|↑〉y + c4|↓〉y. (2)

Consider now the system containing both particles. Its associated Hilbert space is afour-dimensional Hilbert space given by the tensor product of the Hilbert spaces ofthe constituting particles12

Hxy = Hx ⊗ Hy. (3)

Such a Hilbert space contains the following vectors representing possible states ofthe pair of particles:

|↑〉x |↑〉y, (4)

|↓〉x |↓〉y, (5)

|↑〉x |↓〉y, (6)

|↓〉x |↑〉y. (7)

Vectors (4)–(7) actually span Hxy . Then all linear combinations of them belong tosuch a space. In general we have

c5|↑〉x |↑〉y + c6|↑〉x |↓〉y + c7|↓〉x |↑〉y + c8|↓〉x |↓〉y. (8)

On the other hand the set of all possible combinations of states (1) and (2) of x and y

is given by:

�x ⊗ �y = c1c3|↑〉x |↑〉y + c1c4|↑〉x |↓〉y + c2c3|↓〉x |↑〉y + c2c4|↓〉x |↓〉y. (9)

Vectors (4)–(7) can be written as (9). For example setting c2 = c4 = 0 andc1 = c3 = 1, (9) will reduce to (4). However it is not possible to write all the vec-tors of form (8) as (9). To see this set c5 = c8 = 1/

√2 and c6 = c7 = 0. In this case(8) reduces to:

1/√2|↑〉x |↑〉y + 1/

√2|↓〉x |↓〉y. (10)

From (10) it is possible to deduce:

c1c3 = 1/√

2 = c2c4, (11)

c1c4 = 0 = c2c3. (12)

12See Jauch [14].

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It follows from (12) that at least one between c1 and c4 and one between c2 and c3is 0, which is impossible according to (11). This shows that in general vectors thatcan be written as (8) cannot be reduced to vectors that can be written as (9).

Suppose now the paper we are using is 0.1 mm thick. Then this paper has theproperty of being 0.1 mm thick. In general, in classical physics, a property13 can betruly ascribed to a system when a physical variable assumes a determinate value forthat system. In QM the situation is quite different. If the system x is in the eigenstate|↑〉x of the observable O , then we can say that x has the property that the observableO has value 1. Nonetheless, in general, with respect to the observable O , x is ina superposition state such as (1). Since QM is our best description of micro-objects,and it considers superposition states as pure states, not as mixtures, we can reasonablymake the following assumption:

Quantum realism (QR): For each vector state (modulo superselection rule), in theHilbert space of the system x associated with one of its observables, there exists aproperty of the system x that corresponds to it.

QR states the realistic assumption whereby not only are eigenvectors of an ob-servable the properties of a system, but every linear combination as well. QR mustnot be confused with Einstein’s realism, as described in Tarozzi [23], since the lat-ter ascribes a classical property before the collapse of the superposition. QR simplyamounts to the denial that wave function is just information. Hughes [13] discussesthis problems of quantum properties at length. The view that the wave function is justinformation is for example defended in Fuchs [8] and criticized in Hagar [9].

It is clear that Quantum Realism commits us to Property Realism. In fact the veryformulation of QR quantifies over properties.

Moreover let us call “Property Compositional Determinateness” the following the-sis:

Property Compositional Determinateness (PCD): Let X be a composite system suchthat its component parts x1, . . . , xn are pairwise disjoint14 and let p1, . . . , pn be setsof monadic properties defining x1, . . . , xn. Then, for every property P of X, P iscompletely determined by p1, . . . , pn.

According to PCD it is not possible to change the properties of a composite systemwithout changing the properties of its component parts. Here the term “possible” mustbe intended in a nomological sense, that is, there are scientific laws that forbid thispossibility. As far as we know PCD holds, both in classical and relativistic physics.For instance the gravitational mass of a composite body is completely determined bythe masses of its composing parts. The same holds for non additive properties, such asvelocity and temperature. Classical electromagnetic theory is no exception. The totalvalue of the electromagnetic field at a single spacetime point is in fact completelydetermined by the different values of different fields acting on that point.

13Now we do not ascribe any ontological commitment to the notion of “property”. We will come back tothe subject later.14Disjoint can be defined rigorously as not overlapping via D(x,y) = df ∼ O(x,y), where Overlap isdefined by (17).

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Note that according to radical nominalism PCD is trivially true. Realism aboutproperties is a necessary condition for PCD to be non trivially problematic.

Given QR, quantum mechanics provides a strong argument against PCD. Here isa sketch of such an argument. Given QR the vector (10) represents a property of thesystem S composed by particles x and y. But (10) cannot be reduced to vectors ofthe form (9), i.e. vectors that represent a combination of properties of particles x andy taken separately. So there is at least one property of the composite system that isnot determined by the properties of its component parts, which contradicts PCD.

Before passing to the next section let us briefly address some worries about thisfirst argument of ours. Orthodox quantum theorists could argue that if S is in anentangled state it is not possible to ascribe the entangled property to any of its com-ponent parts x and y taken separately. This is not an objection to our argument. Itwill in fact still be the case that there is a property, namely the entangled property, ofthe composite system that fails to be determined by those of the component parts. Inthis case it will fail to be determined by those properties for the very reason that therearen’t any, but PCD will still be violated.

Orthodox theorists could advance however a more radical objection. They couldargue that if S is in an entangled state, it is not just the case that there aren’t propertiesof the component parts that are able to determine those of S, but that the componentparts themselves do not exist. If so S is not a composite system after all and PCDcannot even apply. However in general the component parts of a composite systemare individuated by many other properties beside the entangled one. Therefore wewill be able to ascribe properties that are different from the entangled one to distinctparts. If this is the case, it will be difficult to maintain that they do not exist at all.

4 Extensional Mereology

It is often claimed, implicitly or explicitly, that the argument in Sect. 3 is not just anargument against PCD, but an argument against extensionalism, as in Maudlin [17],in particular mereological extensionalism (ME). We want to check whether this istrue. It will become clear that the situation is much more subtle. In order to do that,we have to provide a clear formulation of what mereological extensionalism amountsto. In this section we sketch a brief formal development of extensional mereologies.

Mereology is the theory of parthood relations, i.e. the relation of a part to thewhole and the relations between parts within a whole. First-order logic with identityis assumed throughout. The primitive notion is that of parthood. We will write x < y

for “x is part of y”.15 The first three axioms that regiment such primitive notion aresupposed to constitute the lexical core of any mereological theory. They are supposedto capture what we mean when we use the notion of mereological part. They arereflexivity, i.e. everything is part of itself, transitivity, i.e. any part of any part of athing is part of that thing, and antisymmetry, i.e. two distinct things cannot be part of

15We follow Varzi [26] in using the relation of part as primitive, though we are not following his notation.

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each other.16 Formally we have

(Reflexivity) x < x, (13)

(Transitivity) (x < y) ∧ (y < z) → x < z, (14)

(Anti-Symmetry) (x < y) ∧ (y < x) → x = y. (15)

The basic mereological theory that comprises just (13)–(15) is called “Mereology”tout court, M.

Given (13)–(15) different mereological notions can be introduced by definition.Here is a brief list:

(Proper Parthood) x y = x is a proper part of y = df(x < y)∧ ∼ (y < x), (16)

(Overlap) O(x, y) = x overlaps y = df(∃z)(z < x ∧ z < y), (17)

(Underlap) U(x, y) = x underlaps y = df(∃z)(x < z ∧ y < z). (18)

It follows from definitions (16)–(18) and axioms (13)–(15) that proper parthood istransitive, irreflexive and anti-symmetric, while overlap is reflexive, symmetric butnot transitive.

Thus developed M embodies the common core of every mereological theory.Adding different principles to M will result in different mereological theories of dif-ferent strength. It is common practice to divide such principles into Decompositionand Composition principles. Decomposition principles are those principles that gofrom the whole to its constituent parts. Composition principles are those principlesthat go from constituent parts to the whole they constitute. Our focus here is on exten-sional mereologies. These are obtained by adding a particular decomposition princi-ple to M. The principle in question is called the “Strong Supplementation Principle”.Informally it says that if x fails to include y as part, then there is something which ispart of the latter that does not overlap the former. Formally

(Strong Supplementation Principle) ∼ (y < x) → (∃z)(z < y∧ ∼ O(x, z)). (19)

The mereological system that comprises axioms (13)–(15) and (19) is known as Ex-tensional Mereology, or briefly, EM. Its extensional character is due to the fact thatthe following theorem can be proved in EM:

(Extensionality) ((∃z)(z x) ∨ (∃z)(z y)) → (x = y ↔ (∀z)(z x ↔ z y).

(20)Note the antecedent. It claims that x and y are not both mereological simples. Ifthey both are such simples in fact (20) will be trivially true and extensionalism willfollow.17 It will in fact be reduced to identity. This is the core of Mereological Exten-sionalism (ME). It is the endorsement of (20) and nothing else. This should be duly

16We will drop any universal quantification. Unless otherwise specified every formula has to be intendedas universally closed.17This will play a role in one of our arguments on Mereological Nihilism.

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noted. Informally it can be captured by one simple claim: sameness of compositionis both a sufficient and necessary condition for identity.

It is important to emphasize that we are speaking about tokens and not types, i.e.about singular individuals.

To our knowledge the only possible violation of ME in the ontology of physicscould be given by Bose-Einstein statistics. Imagine two photons x1 and x2 sentagainst a beam-splitter. One of them is sent to screen 1, S1, and the other one toscreen 2, S2. QM cannot distinguish between the situation (S1 − x1, S2 − x2) and(S1 − x2, S2 − x1). Therefore, applying the identity of indiscernibles, it will followthat the two situations are one and the same. Thus it seems that ME has been vio-lated, since one and the same individual, namely the final state, can have differentparts, either S1 − x1 or S1 − x2. But this argument depends crucially on the applica-tion of identity of indiscernibles. If this principle holds true it will yield that S1 − x1

is identical to S1 − x2 as well. Hence there is no violation of ME.We conclude that ME is not a metaphysical principle violated by contemporary

physics.

5 Quantum Mechanics and Mereological Extensionalism

Having defined the notions of PCD and ME, the significant question at hand is thefollowing: does Mereological Extensionalism entails Property Compositional Deter-minateness? For, if it is so, then by the argument in Sect. 3, the endorsement of QRdoes actually refute the former. To see this consider the following argument:

(i) ME → PCD (hypothesis)(ii) QR (assumption)

(iii) QR → ∼PCD (argument in Sect. 3)(iv) ∼PCD (modus ponens, ii and iii)(v) ∼PCD → ∼ME (contraposition, i)

(vi) ∼ME (modus ponens, iv and v).

We have argued in Sect. 4 that ME is a thesis about identity and composition.But PCD is, at first sight at least, a reductionist thesis about property instantiation.Therefore a robust argument is required to show that the first commits to the latter. Itis possible to run a strong argument in favor of the fact that it is the case. Here is asketch of it.

Mereological Extensionalism implies that a composite system is nothing over andabove its constituent parts. But then the properties of the composite system must bedetermined by those of the component parts. And this is exactly PCD.

We believe this argument, as it stands, fails. Let’s consider it more carefully, refer-ring to the case we presented in Sect. 3 of two quantum particles x and y. Grant forthe sake of argument that there exists a mereological sum z of x and y. In mereolog-ical terms this object would satisfy

(∃z)(∀w)(O(w, z) ↔ (O(w,x) ∨ O(w,y)). (21)

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It can be proved that, in the presence of the strong supplementation principle (19), theentity whose existence is asserted in (21) is unique. Let’s then introduce the followingdefinition

x + y = dfιz(∀w)(O(w, z) ↔ (O(w,x) ∨ O(w,y)), (22)

where the symbol “ι” is a descriptive operator meaning “the only object such that”.Then (21) can be phrased more perspicuously as:

(∃z)(z = x + y). (23)

In our opinion the previous argument rests on a misunderstanding of the notion ofmereological sum. This notion is the one defined in (22). Definition (22) just saysthat the sum of x and y is the object that overlaps all and only those things that x andy overlap. It does not tell us anything else. It does not say what kind of an object thesum is and does not specify all of its properties. The latter is an empirical questionand should be answered case by case through scientific investigation. The same pointcould be made even more perspicuously introducing a little more mereological for-malism. Suppose you have a set of ϕ-ers, entities that have the property ϕ. Let z bethe mereological sum of these ϕ-ers, defined via:

(∃z)(∀w)(O(z,w) ↔ (∃v)(ϕv ∧ O(v,w))). (24)

Then the mereological definition by itself does not specify what kind of object themereological sum z is. For instance it does not tell us whether z is itself a ϕ-er. Forexample, let us suppose xi (i = 1, . . . , n) are connected neurons and let z be the mere-ological sum of such neurons. Given Mereological Extensionalism z is the only thingthat overlaps all and only the xis. According to many physicalistic philosophers18 z

can have a mental property P . However, at the present status of neuropsychologicalknowledge, P does not always supervene on the properties of the xis. But then, theproperties of the composite system do not need to be determined by those of its con-stituent parts. So, Mereological Extensionalism, as a thesis about sameness of compo-sition, does not entail the thesis about property instantiation that we labeled PropertyCompositional Determinateness. Further empirical investigations are needed.

There is another partly controversial way to see that Mereological Extensionalismis not incompatible with QR. Suppose you endorse Mereological Nihilism, as definedin Rosen and Dorr [19], i.e., the controversial thesis whereby given two simple ob-jects,19 a sum of those objects never exists. Thus, according to Mereological Nihilismall the existing things are mereological atoms. In this case too Mereological Exten-sionalism is not threatened by QR. For, suppose that x and y are atoms. The problemwith QR was that properties of the composite system are not determined by thoseof its constituent parts. But there is no such a thing as the composite system. So theargument does not have any bite. Mereological Extensionalism is however triviallytrue, as we have already noted in our discussion of the extensionality theorem (20).

18Apart from those who endorse a radical form of reductionism such as eliminativism.19For the sake of completeness a mereological simple or atom is defined via A(x) = df ∼ (∃y) (y x).

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The antecedent of that conditional is false, then the theorem holds true. Thus, in thiscase, you can have Mereological Extensionalism and QR, since mereological nihilismrounds off the latter, so to speak.

We have argued so far that the failure of PCD does not entail the failure of Mere-ological Extensionalism. QM does not provide a direct argument against extension-alism. This is not to say that the argument we have presented in Sect. 3 does not poseany problems whatsoever to ME.

Arguments against mereological extensionalism focus on the right to left directionof the biconditional in (20). They are purported to show that sameness of compositionis not sufficient for identity. Typical examples go from the same words constitutingdifferent sentences in Hempel [11] and Rescher [18], to the same flowers constitutinga scattered whole on the floor and a nice bunch in Eberle [6], to the same mechanicalparts constituting a pocket watch or a pile of springs and screws in Maudlin [17]. Canwe build a somehow indirect quantum mechanical argument against ME on the basisof some of the considerations presented in Sect. 3?

First, let us introduce another well known metaphysical principle, namely Leib-niz’s law (LL). LL informally says that x and y are one and the same object if andonly if they have the same properties.

Go back to our main argument of Sect. 3. Suppose x and y are two not entangledparticles in states represented by (1) and (2) respectively, where every coefficientc1, . . . , c4 �= 0. Then the composite system will be in the state given by the tensorproduct of such states, namely it will be in a state of type (9) where, again c1, . . . , c4 �=0. Let us call z1 the composite system in such a state.

Suppose now two measurements of the observable O are carried out on x andy and suppose that after the measurement the particles are in state (25) and (26)respectively:

|↑〉x, (25)

|↓〉y. (26)

Then the composite system will be in state (6). Call the system in such a state z2.Note that, given QR, both (6) and (9) represent properties of the composite system.But, given our assumption that in the first case all coefficients are different from 0,it follows that they do represent different properties. So we are forced to conclude,via LL, that

z1 �= z2. (27)

But, the argument goes on, z1 and z2 are composed of the same parts. So samenessof composition is not sufficient for identity, against (20). It does seem that we haveindeed found a strong quantum mechanical argument against mereological extension-alism.

This is however not the end of the story. There is a serious reply to be made onbehalf of the extensionalist. He/she will argue that this is just a sophisticated vari-ant of the classical arguments we were talking about earlier. He/she will point outthat the quantum mechanical particles x and y will be able to compose z1 and z2only at different times. But in this case it is well known that the appeal to LL is at

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least problematic. A strict application of Leibniz’s law in these cases will in factyield the controversial consequence that objects do not persist through change. Thesealleged counterexamples are not, at the bottom, counterexamples to extensionality.They are just another formulation of the so called “puzzle of change”. In a nutshellthis amounts to finding a satisfactory account of how one and the same object caninstantiate different incompatible properties. In other words, the extensionalist main-tains that z1 and z2 are indeed the same object that has changed some of its properties,contrary to (27). Then, the extensionalist goes on, a solution to this puzzle will pro-vide the required revisions to save (20) on the face of the aforementioned allegedcounterexamples.

There is moreover a way to save (20) on the face of the previous considerationsthat does not even require a solution to that puzzle. To see this consider that z1 and z2in our argument will constitute a counterexample to (20) if, and only if, they have thesame mereological parts. Suppose now that one endorses the following metaphysicalthesis:

Instantion is Parthood (IP): The relation of property instantiation is simply mereo-logical parthood.

In other words, properties are literally mereological parts of those individuals thatdo instantiate them. The endorsement of such a thesis will indeed save (20). In factthe composite systems z1 and z2 do have different properties, in particular those rep-resented by (6) and (9) respectively. But these properties, given IP are parts of z1and z2. Moreover, since these properties are different from the composite objects thatactually instantiate them, it follows from (16) that they are proper parts of z1 and z2.Hence those composite systems have different proper parts and so do not have thesame mereological structure. It follows that they cannot constitute a counterexampleto (20).

To sum up we have argued that, contrary to the widespread agreement, QM doesnot provide an argument against Mereological Extensionalism per se. It does so onlyif one accepts both (i) a controversial strictly application of LL in diachronic contextsand (ii) the denial of IP.

Moreover, ironically enough, within a realist framework about properties, if othercontroversial assumptions about property instantiation are accepted, QM even pro-vides an argument favoring ME.

Suppose you hold a reductionist view about relations, along the following lines:

Relation Supervenience (RS): For every relational property R holding among the in-dividuals x1, . . . , xn, there is a set S of monadic properties Pi such that R superveneson S.

It is not necessary to give a complete account of what supervenience amounts to inthis context. It is enough to say that given RS you cannot have a change in R withouta change in S.

Note that such formulation of RS does not require that the properties of the setS are instantiated by x1, . . . , xn taken individually. In particular it leaves open thepossibility that some of the Pis in S are instantiated by different mereological sumsof x1, . . . , xn, even by the whole x1 + · · · + xn. That is, RS does not assert that all

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relations are reducible to monadic properties of the base individuals x1, . . . , xn takensingularly.

It seems reasonable that RS holds, since relational quantum phenomena, as bosonsand their collective effects, like super-fluidity, Bose-Einstein condensate, lasers,masers etc. probably could be interpreted as monadic properties of a sum of particles.The same holds for collective fermion effects like conductivity in metals (Sommer-feld’s model) and superconductivity (BCS theory). This topic clearly deserves carefulattention that is beyond the scope of the present work.

Suppose moreover that you endorse some form of instantiation principle, that wecould label the Aristotle-Armstrong Principle, and that we have adapted from Arm-strong [3]:

Aristotle-Armstrong Principle (AAP): For any n-place property P there are n objectsthat do instantiate such a property.

Note that IP entails AAP for monadic properties.RS is different from PCD, evenif failure of PCD in a quantum mechanical context could be problematic for RS. For,suppose you treat quantum states represented by vector (8) as representing relationalproperties of x and y. Call this point of view Quantum Relationalism. It is describedfor example in Jauch [14] and developed in Rovelli [20] and Laudisa [16], to mentionjust a few. Then, according to the argument in Sect. 3, there is no set of properties of x

and y taken singularly such that this relation supervene on it. So RS fails. Thereforequantum relationists are compelled to abandon RS.

Those who want to retain RS could however reply that this argument takes forgranted that there are just x and y. But, suppose now that there is another objectbeside x and y, namely the mereological sum of those particles. Then (8) easilysupervenes on the monadic properties of such a sum. Now the question is: does thissum exist? If AAP holds then it must be the case.

In classical mereology there is a controversial principle that actually guaranteesthat such a sum always exists. It is called the “Unrestricted Composition Principle”.This principle informally says that given any two non empty aggregates of objects x

and y, there exists an object z, usually called “the sum” or “the fusion of x and y”,that has x and y as proper parts and nothing else. It can be written as:

(Unrestricted Composition Principle)

(∃w)ϕw → (∃z)(∀w)(O(z,w) ↔ (∃v)(ϕv ∧ O(v,w))). (28)

Varzi [25] and Hovda [12] argue that the unrestricted composition principle entailsthe strong supplementation principle,20 that is (19).21 And so we reach our ironicconclusion. As we have already mentioned in Sect. 3, the Strong SupplementationPrinciple in turn implies the extensionality theorem (20). Thus, in a certain sense QM

20The proof depends on the so called Weak Supplementation Principle.21The mereological system obtained by adding the strong supplementation principle and the unrestrictedcomposition principle to the lexical axioms (13)–(15) is known as General Extensional Mereologyor GEM.

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does not provide an argument against Mereological Extensionalism, but rather can beused to offer one in favor of it.

To sum up, even if we abandon PCD we are not forced to drop ME. Not only, wecould even build a quantum mechanical argument in favor of it. Which is roughly thefollowing. Assuming QR, if we want to preserve at least RS and AAP, due exactly tothe failing of PCD, we are compelled to admit that there always exists a mereologicalsum of simple particles. And this leads to extensionalism.

6 Concluding Remarks

We have presented arguments in favor of the following thesis:

(A) Quantum Realism entails the negation of Property Compositional Determinate-ness, that is QR → ∼PCD.

(B) Mereological Extensionalism does not imply Property Compositional Determi-nateness, therefore Quantum Realism alone is not incompatible with Mereolog-ical Extensionalism.

(C) We have shown that Quantum Realism together with Relation Supervenienceand Aristotle-Armstrong Principle imply Mereological Extensionalism, that isQR ∧ RS ∧ AAP → ME.

In general we can conclude that Quantum Realism is more at variance with Prop-erty Compositional Determinateness than with Mereological Extensionalism.

Our analysis makes four different kinds of research programs possible:

(i) Abandoning property realism for radical nominalism. This amounts to an instru-mentalist perspective about scientific theories, in line with Van Frassen [24] andLaudan [15].

(ii) Endorsing QR together with RS and AAP, abandoning PCD and saving ME.(iii) Endorsing QR abandoning both PCD and RS. In such an approach the case for

ME should be made on independent grounds.(iv) Abandoning QR. This however is not based on the endorsement of nominalism

as in (i), but rather on the suspicion that quantum mechanical description ofthe world is incomplete, hoping that the completion will not violate either PCDor ME.

To the present state of our knowledge option (ii) seems the best, because, althoughwe have lost PCD, we will be able to save at least ME and RS. But, in line withAgazzi [1]’s suggestions, we should not forget that perspective (iv) could reserve themost rewarding theoretical surprises.

Acknowledgements We are indebted to Bas van Fraassen, Xavier de Donato and an anonymous refereefor helpful suggestions on previous drafts.

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