Divisibility and Extension:

a Note on Zeno’s Argument against Plurality and Modern Mereology

Claudio Calosi, Vincenzo Fano

University of Urbino

Department of Basic Sciences and Foundations

claudio.calosi@uniurb.it

vincenzo.fano@uniurb.it

Final Version published in Acta Analytica

Abstract: In this paper we address an infamous argument against divisibility that dates

back to Zeno. There has been an incredible amount of discussion on how to understand the

critical notions of divisibility, extension and infinite divisibility that are crucial for the very

formulation of the argument. The paper provides new and rigorous definitions of those

notions using the formal theories of parthood and location. Also it provides a new solution

to the paradox of divisibility which does not face some threats that can possibly undermine

the standard Lebesgue-measure solution to such a paradox.

Keywords. (Infinite) Divisibility, Extension, Parthood, Location

1 Introduction

The “Large and the Small” paradox (Makin, 1998, p. 844) is a fragment of an argument

against plurality. It is supposed to show that nothing is infinitely divisible. There has been

an incredible amount of discussion and disagreement in the critical literature on how to

understand some of the crucial notions involved in the paradox, such as divisibility,

extension and infinite divisibility, as it is witnessed in Furley (1967, p. 4), Barnes (1982, p.

277) and Sorabji (1983, p. 352) to name just a few. In this paper we give a new, simple and

rigorous definition of such notions using on the one hand the resources of formal

mereology, and on the other hand that of formal theories of location. This result constitutes

already an improvement on much of the critical literature. It will for example provide a

clear account of how the Eleatic Universe1 can be both indivisible and extended at the same

time. These resources also suggest an entirely novel solution to the paradox. This is because

they suggest a different reading of the notion of infinite divisibility that has been neglected

so far. This solution is different from the standard one in that (i) it does not resort to

Cantorian mathematics and (ii) does not depend on controversial assumptions about the

cardinality of the set of parts of material objects. It is noteworthy that our discussion reveals

some stimulating insights on the relations between the mereological structure of material

objects and that of space or spacetime that the “Large and the Small” paradox is still able to

offer us. The plan of the paper is as follows. In §2 we briefly develop the formal

frameworks that will be used throughout the paper. In §3 we use these frameworks to give

new, simple, and rigorous definitions of the notions of extension and divisibility that are

1 Or the One.

involved in the paradox. Then, in § 4 we review the paradox, its standard solution and we

go on to propose our novel one. These last two sections contain several discussions about

the relation of parthood and location that are of interest independently of the paradox.

2 The Formal Frameworks

In this section we give a very brief introduction to mereology (§ 2.1) and to formal theories

of location (§ 2.2). It is not our purpose to review in detail those formal theories we present.

We refer the reader to Simons (1987) and Casati and Varzi (1999) respectively for that. We

will introduce and discuss only those notions that are relevant for the rest of the paper.

2.1 Mereology

Mereology is the formal theory of parthood relations. Different mereological theories of

different strength can be developed regimenting the primitive notion of parthood2 with

special axioms. First order logic with identity is presupposed throughout and formulas are

universally closed unless otherwise noted. Let

(1) (Parthood) x y

Stand for x is part of y . There are some mereological notions we will be particularly

interested in. These are Proper Parthood, Overlap and Atom:

(2) (Proper Parthood) dfx y x y x y= ∧ ≠

(3) (Overlap) ( , ) ( )( )dfO x y z z x z y= ∃ ∧

2 We follow Varzi (2009) in taking Parthood as primitive. Simons (1987) takes Proper Parthood as a primitive notion, whereas the classic Leonard and Goodman (1940) takes Disjointness where Disjointnss is not Overlap. These notions are interdefinable so that this choice is mostly a matter of preference.

(4) (Atom) ( ) ( )( )dfA x y y x= ∃

A proper part of something is a part of that thing that is distinct from the whole, two things

overlap if they share a part and a mereological atom is an entity without proper parts.

Different mereological theories of different strength can be obtained by regimenting

mereological notions with different axioms. Following literature we call Minimal

Mereology (MM) that mereological theory comprising the following ones:

(5) (Reflexivity) x x

(6) (Transitivity) x y y z x z∧ →

(7) (Anti-Symmetry3) x y y x x y∧ → =

(8) (Weak Supplementation) ( )( ( , ))x y z z y O x z→ ∃ ∧

The first three axioms are quite familiar. They render Parthood a partial order. The Weak

Supplementation axiom informally says that every composite object has at least two disjoint

proper parts. MM, as we have formulated it, is compatible both with an axiom that states

that, at the bottom, everything is made up of atoms, and with one that says that there are no

atoms at all. Adding these axioms yield the Atomistic or the Atomless variants of MM, that

is the mereological theory obtained adding either

(9) (Atomicity4) ( )( ( ) )y A y y x∃ ∧

3 Anti-symmetry is actually redundant within this framework. See §4 for a somewhat detailed discussion. 4 It could be argued that this axiom is too weak to capture the full strength of the idea that something is ultimately made up of atoms. For it does not rule out the possibility of an infinite descent of atomic

Or

(10) (Atomlessness) ( )( )y y x∃

Clearly (9) and (10) are mutually exclusive.

Lewis (1986) calls atomless gunk an entity whose parts have further proper parts. Thus we

will call Atomistic Minimal Mereology (AMM) the theory comprising axioms (5, 6, 7, 8, 9)

and Gunky Minimal Mereology (GMM) the one comprising (5, 6, 7, 8, 10) instead. If

spacetime physics is on the right track spatial and spatiotemporal regions are indeed models

of AMM 5. We will therefore assume it is so. The mereological atoms are in this case the

spatial or spatiotemporal points. Though it is a much debated issue in the literature6 we find

possible, and indeed highly plausible, that different ontological domains are models of

different mereological theories. For instance material objects could fail to be a model of any

atomistic mereology.

2.2 Formal Theories of Location

Suppose we can grasp the intuitive notion of Exact Location along the following lines. If a

material object x is exactly located at a spatial7 region R it has the same size, shape, volume

proper parts of something. This distinction will not play a crucial role in our arguments so that we will rest content with this weaker axiom. 5 Actually they are indeed models of a much stronger merological theory namely Atomistic General Extensional Mereology (AGEM). AGEM is the extension of AMM obtained by adding Extensionality, that says that sameness of composition is both a necessary and sufficient condition for identity (or by adding the so called Strong Supplementation Principle that in turns entails Extensionality as a theorem) and the Principle of Unrestricted Composition (UC). Informally UC says that given any non-empty set of objects there exist a mereological sum of them. See Simons (1987) and Varzi (2009) for details. The full strength of AGEM is not needed for our arguments to go through. 6 See Sider (2007) for an overview. 7 We will indeed rest content in talking about spatial, rather than spatiotemporal locations, because we move in a classical, non-relativistic context.

of R. Thus our office could not be the exact location either of the Milky way or of the copy

of the Aristotelian Physics in it. Then let:

(11) (Exact Location) ( , )ExL x R

Stand for x is exactly located8 at region R . From now on , ,...,x y z will be used as

variables and names of material objects, whereas ,...,i kR R will stand for spatial regions.

We will be interested simply in another locative notion9, namely Overfilling:

(12) (Overfilling) 1 1 1( , ) ( )( ( , ) )dfOvF x R R ExL x R R R= ∃ ∧

Something overfills a region if no part of that region is free from that thing. As in the

previous section different theories of location can be obtained by regimenting locative

notions with different axioms. We call Minimal Location (ML) a theory of location that

comprises the following axioms:

(13) (Exactness) ( )( ( , ))R ExL x R∃

(14) (Strong Expansivity)

8 We will make the simplifying assumption that the first argument of this relation is a material object, whereas the second is a spatial region. It should be furthermore noted that this use of exact location seems to assume that material objects and spatial regions are distinct, irreducible entities such that there is no dynamical interaction between the two. The very tenability of this notion would be undermined within the context of the so called background independent theories, such as General Relativity. As we have already mentioned in the previous footnote the paper implicitly assumes a classical framework. We will return to this point later on, when discussing worries raised by Quantum Mechanics. 9 Other locative notions such as Weak Location and Underfilling could be defined via: (Weak Location)

1 1 1( , ) ( )( ( , ) ( , ))

dfWL x R R ExL x R O R R= ∃ ∧ ;

(Underfilling) 1 1 1

( , ) ( )( ( , ) )df

UnF x R R ExL x R R R= ∃ ∧ .

1 1 1( , ) ( )( ( , ) )x y ExL x R R ExL y R R R∧ → ∃ ∧

Informally they say that everything has an exact location and that a composite object cannot

fail to be located where its parts are10.

In the rest of the paper we will also be interested in another11 theory of location, that

resulting by adding the following Division12 axiom to ML:

(15) (Division) ( , ) (( )( ( , ))OvF x R y y x ExL y R→ ∃ ∧

Informally (15) says that an object has a part that is exactly located at every region it

overfills. It will follow from Strong Expansivity that it would have a proper part that is

10 We require Exactness because, as we have already pointed out, we move in a classical framework. There the notion of spacetime trajectory is well defined, so that Exactness seems safe. Various interpretations of Quantum Mechanics would however deem it wrong. Bohmian mechanics is a notable exception. Both Casati and Varzi (1999) and Parsons (2006) require Exactness to be an axiom. Two different formal renditions of the driving intuition underpinning Strong Expansivity (14) come to our mind. The first one is a weaker variant of (14) obtained by replacing the notion of proper parthood with that of parthood everywhere in (14), i.e.: (Weak Expansivity):

1 1 1( , ) ( )( ( , ) )x y ExL x R R ExL y R R R∧ → ∃ ∧ . A theory of location with only Weak Expansivity

is probably better suited to regiment the notion of Exact Location for those domains in which the Weak Supplementation principle does not hold true for material objects. To see this consider the case of a material object with a single proper part. Then it could be the case that this object and its unique proper part are exactly co-located. Since we are working with MM as mereological theory (though we will return on this point later on, in §4.3) we prefer to have the full strength of (14). Another formal rendition that would be worth investigating would be the following: (Exact Expansivity)

1 1 1(( )( ( , ) ( )( ( , ) ))x y R ExL x R R ExL y R R R→ ∃ → ∃ ∧ that could also have

a stronger variant obtained by replacing Parthood with Proper Parthood. Such formulations would be better suited in a context of a theory of location that does not feature Exactness among its axioms. Since we have required it, we will not pursue this line of argument here. 11 We will leave aside discussions about another interesting axiom, namely Functionality that states that Exact Location is a function, i.e. everything has a unique exact location. It can be rendered formally via: (Functionality):

1 2 1 2( , ) ( , )ExL x R ExL x R R R∧ → = . Theories that have Functionality

among their axioms can be labeled Functionality theories. Those that don’t can be labeled Multilocation theories instead. Casati and Varzi (1999) and Parsons (2006) are examples of Functionality theories, whereas Balashov (2010) develops a Multilocation theory, to name just one example. 12 Our formulation of Division can be considered a formal rendition of what Van Inwagen (1981) calls the Doctrine of Arbitrary Undetached Parts or DUAP. Parsons (2006) calls a similar axiom Arbitrary Partition.

exactly located at every proper subregion of its exact location. We will call the theory

obtained by adding (15) to ML Divisible Minimal Location (DML). We will see that,

surprisingly, Zeno’s paradox offers stimulating insights on the relations between the

mereological theories developed in section 2.1 and the theories of location of this section. It

is then to the rigorous formulation of some of the notions involved in the paradox that we

now turn to.

3 Divisibility and Extension Defined

The “Large and the Small” paradox is an argument against infinite divisibility. It is

supposed to show that nothing is infinitely divisible. As we have mentioned in the

introduction there has been an incredible amount of discussion and disagreement in the

critical literature on how to understand such notions. It is not our purpose to review such

interpretative literature here and assess its merits and its deficiencies. Rather what we

pursue is a new, clear and rigorous definition of those notions that have fueled such

interpretative work. We intend this as an improvement in itself on much of the critical

literature.

There seem to be at least two fundamental notions of divisibility: physical and conceptual

divisibility. The driving intuitions behind these notions are simple enough. An object is

physically divisible if it is possible to physically separate some of its parts, whereas it is

conceptually divisible if it is possible to individuate some parts of it even in the case in

which it is physically impossible to separate them. Zeno’s argument concerns this last

notion, so we will focus on it. The problem is how to give a rigorous formulation of the

seemingly simple driving intuition behind conceptual divisibility. Such a formulation

should meet at least two requirements. Zeno was concerned with the fact that conceptual

divisibility could have entailed that what might be called Eleatic Monism was wrong.

Eleatic Monism is the thesis, held by Parmenides and Zeno himself, that there are no two

distinct entities. Formally:

(16) (Eleatic Monism) (( )( )( ))x y x y∃ ∃ ≠

In what remains or is reported of Parmenides it is not entirely clear whether he

distinguished between material objects and regions; on the contrary this distinction is surely

presupposed by Zeno. Therefore it is reasonable to assume monism as the thesis according

to which there are not two distinct material objects13.

Actually from now on, when we talk about divisibility or extension, unless otherwise noted,

we mean divisibility and extension for material objects. Thus the first requirement is that

any proposed formulation of conceptual divisibility14 has to imply the negation of (16).

There is furthermore another requirement. Zeno held, along with Parmenides, not only that

there exists only one entity, the Universe or the One, but also that this entity is extended,

i.e. has some spatial extension. In DK, B8, 26 Parmenides states for example that the One is

not divisible because it is everywhere homogeneous (homoìon), not because it is

unextended.

The second requirement is then that any proposed formulation of Conceptual Divisibility

and Spatial Extension15 should not imply that everything that is extended is divisible, for if

it did, the existence of any spatially extended object would entail the falsity of Eleatic

13 We will omit such a specification from now on. 14 The notion of divisibility is worth an independent, careful and detailed discussion which is not possible to provide here. For interesting suggestions see Fano (2012, p. 63). 15 We will drop both the Conceptual and the Spatial qualifications from now on. No confusion can arise from that.

Monism. Now everything is ready for our proposed formulations of Divisibility and

Extension:

(17) (Divisibility) ( ) ( )( ) ( )df dfDiv x y y x A x= ∃ =

(18) (Extension) 1 1

( ) ( , ) ( )

( , ) ( )( )df

df

Ext x ExL x R A RExL x R R R R

= →

= → ∃

Let us see what these definitions claim. The first one says that something is divisible if it

has proper parts, the second one that something is extended if it is exactly located at a non-

atomic region.

Note that this definition of extension is applicable only to located objects. This might raise

worries about having different extension predicates that are applicable to different

ontological categories, such as material (located) objects on the one hand and spatial

regions on the other.

It is worth spending a few words on such a worry. First of all this is what many

philosophers think about other related predicates and ontological categories. Consider

extension in time. Many philosophers think that different time-extension predicates apply to

different ontological categories such as material objects and events. Many philosophers for

example maintain that an event is extended in time by having a part that is present at every

instant of its occurrence. On the other hand they think that material objects do not extend in

time in the same way, but rather by being wholly present at each instant of their existence.

This is actually the main metaphysical tenet of one of the most widely held metaphysics of

persistence, namely Three-dimensonalism or Endurantism. We cannot do better than to

refer the interested reader to Sider (2001) for a review. All we want to suggest here is that it

is not oddly suspicious to hold that different extension predicates are applicable to different

ontological categories. Furthermore it might very well be that this is exactly what we

should do in this particular context if we are to make sense of Zeno’s arguments. We could

define for example spatial extension for regions in mereological terms, for example via:

(19) (Region-Extension) ( ) ( )dfExt R A R=

Claim (19) entails that, for spatial regions, extension boils down to divisibility. And this is

exactly what we don’t want in the case of material objects, exactly for the arguments we

just discussed. We now show that the definitions we have provided meet the necessary

constraints we required.

The first requirement is that the notion of Divisibility should imply the negation of (16). To

see this assume that x is divisible. Then, by (17) ( )( )y y x∃ , and it will follow from

the definition of proper part (2) that x y≠ , against (16) as required. Thus the first

requirement is met.

The second requirement is that, for material objects, being extended should not imply being

divisible. In other words it should not be the case that:

(20) ( ) ( )Ext x Div x→

But (20) does not hold in ML. It actually holds true only if we require Division to be an

axiom of our formal theory of location, that is, it holds true if our location theory is DML

rather than ML. To see that (20) holds in DML consider the following argument. Suppose

x is extended according to definition (18). Then it is exactly located at a region R that

has some proper parts, i.e. it has proper subregions. By Division x will have some parts

that are exactly located at those proper subregions, for x overfills them. Those parts would

have to be proper parts given Strong Expansivity (14). Thus x would be divisible

according to definition (17).

Makin (1998: p. 844) has to resort to a highly artificial premise in order to account for the

separability of extension and divisibility, an assumption that informally reads “if there is

something divisible, then what is extended is divisible”, whereas in our novel approach this

separation is much more natural and straightforward. It is worthy to spend a few words

more on this. The Division axiom is usually challenged by those philosophers who believe

either in the possibility, or even in the existence, of the so called Extended Simples16. An

Extended Simple is easily definable within our framework:

(21) (Extended Simple) ( ) ( ) ( )dfExt S x A x Ext x− = ∧

It is not a coincidence then that the Parmenidean Universe is exactly an example of an

Extended Simple.

We have then argued that (20) does not hold simply in virtue of the location theory we have

assumed. A substantive addition, such as the axiom of Division, is required in order to

enforce that problematic entailment. Thus, also the second requirement is met.

Our formulations are simple, coherent and rigorous. They have also proved to be effective

in meeting all the necessary requirements. This first new result concludes this section. With

these clear formulations at hand it is now the time to turn to the paradox itself.

4 The “Large and the Small” Paradox

16 See for example Markosian (1998), McDaniel (2007) and Scala (2002).

We have given clear formulations of some problematic notions17 that feature prominently

in the so called “Large and the Small” paradox. In this last section we briefly review such a

paradox (§ 4.1) and its traditional solution (§ 4.2). We then point out three threats that such

a solution has to face. These threats do not undermine the validity of such a solution, but

they suggest that a novel way of resolving the paradox that does not have to face them

would be worth. We (§ 4.3) set up to provide such a novel solution. It stems from the

possibility of a neglected reading of a crucial notion involved in the paradox, a reading that

is suggested by the very analysis we have developed so far. It turns out that the solution we

provide reveals deep consequences about the relation between the mereological structure of

material objects on the one hand and that of space on the other. These consequences are

interesting even independently from Zeno’s argument.

4.1 The Paradox, Briefly

The “Large and the Small” paradox has been passed on by Simplicius (Phys.139, p. 27),

probably discussed by Democritus and then given its first classical formulation in Aristotle

(De Gen. et Corr., 316a, p. 14-35). We have shown that divisibility dims Eleatic Monism

wrong. Thus Zeno set out to prove that nothing is divisible. First he argues that if

something is divisible, then it is infinitely divisible (Simplicius, Phys. 139, p. 19), then that

infinite divisibility entails a contradiction. This second part of the argument is the so called

“Large and the Small” paradox. Roughly the argument is the following.

Suppose something is infinitely divisible. Then there are two cases. Either we end up with

an infinite sum of extended entities or we end up with an infinite sum of unextended

17 Though we would need to give a rigorous formulation of yet another notion, namely that of Infinite Divisibility.

entities. In the first case the object we started with should have an infinite extension, in the

second case it should have no extension at all. Thus the divisible object we started with

would have been either too large or too small, hence the name of the paradox. As

Simplicius (Phys. 139,p. 9) puts it: “If there is a plurality, things are both large and small,

so large as to be infinite in magnitude, so small as to have no magnitude at all.”(English

translation: Lee, 1936, p. 19)

The first thing to note is that the paradox features the notion of infinite divisibility rather

than divisibility simpliciter. As it was expectable there has been again a lot of controversy,

at least after Aristotelian physics (Fano, 2012, § II.4.), on how to understand the notion of

infinite divisibility and whether this is compatible with the fact that we seem to be given a

result of an infinite process. This last point is clearly tackled in Grünbaum (1968, pp. 130-

132)18.

Before we review the by now standard solution to the paradox, that resort to Cantorian

mathematics and Lebesgue measure theory, let us give a somewhat more detailed

formulation19 of it, that is reminiscent of Grünbaum (1968). Here it is.

(22) Suppose something is infinitely divisible.

(23) Then it is the case that it is composed either of

(i) an infinite sum of extended entities, or of

(ii) an infinite sum of unextended entities.

18 Our reformulation of the paradox does away with this problem, so we will not enter in these details here. 19 In this formulation we are using the notion of composition (and also entity) in an entirely non-technical sense. We will give some technical details in §4.3.

(24) If (i) the entity we started with should have an infinite extension.

(25) If (ii) the entity we started with should have no extension at all.

Let us call claims (24) and (25) the “Large” and the “Small” horn of the paradox

respectively. The classical solution solves, so to say, the “Small” horn. Our proposed one

does away with that horn and solves the “Large” instead.

4.2 The Standard Solution, Very Briefly

The classical solution to the paradox lies in Cantorian mathematics and Lebesgue measure

theory. It can be summed up, very roughly, in the following two steps: (i) arguing that if

something is infinitely divisible it has uncountable parts and (ii) in noting that (24) is valid

only if the set of entities into which the division process resolve the object we started with

is a countable set. It does not hold if the set is uncountable, as it happens in some cases.

This solution has been advanced for the first time in Grünbaum (1952), proposed again in a

different way in Grünbaum (1968), Salmon (1975, pp. 52-58) and defended very recently in

Huggett (2010). As we have already mentioned it can be consistently applied in a lot of

interesting cases, such as points on the real line, that constitute effectively an uncountable

set of unextended entities.

It is not our purpose to challenge the validity of such a solution. But we do want to point

out some possible threats it has to face. Three such threats come to our mind.

The first threat deals with the definability of an additive measure for uncountable sets. The

standard solution uses implicitly the mathematical fact that additivity for Lebesgue measure

holds only for countable sets. It has been however pointed out, most notably by Massey

(1969, p. 337), Skyrms (1983) and White (1992, pp. 9-10), that a ultra-additive measure

can be defined which will allow for example to sum an uncountable number of lengths.

The second threat deals with uncountability per se. The problem is that it is already

contentious whether physical space(time) is really composed by an uncountable set of

unextended entities such as spatial points. It is true that the vast majority of spacetime

theories make such an assumption and some might object at this point that we ourselves

have explicitly assumed a somewhat classical framework. Never mind then the spatial case.

The uncountability assumption is particularly controversial in the case of material objects.

It is in fact far from clear whether material objects are constituted by an uncountable set of

unextended atomic parts. This seems a controversial and substantive empirical question and

there is no current physical theory that seems to back it up definitively. Thus the very

applicability of the standard solution, not its validity, to the domain of material objects

seems questionable.

The last threat is of historical interest. If we choose to discard the first two threats and still

advance the standard solution when it comes to material objects it seems that we get

dangerously close to the endorsement of some sort of Division axiom. Note in fact that that

axiom would indeed guarantee that, if a spatial region is constituted by un uncountable set

of spatial points any material object that is exactly located at that region would have

uncountably many proper parts. But we have already argued that Division should not be

admitted as an axiom in an Eleatic Universe. Thus, any solution which entails it, would beg

the question against Zeno.

As we have said many times already we do not take these threats to undermine the

mathematical validity of the standard solution. However as long as they are taken to be real

threats they suggest that other solutions to the paradox that do not face them would be

worth exploring. In the following section we propose one such solution. It depends on an

alternative reading of the notion of infinite divisibility which stems from the analysis we

put forward. In order to appreciate clearly the difference between these two solutions note

that the standard one seems to understand infinite divisibility along the following lines:

(26) (Infinite Divisibility1) Something is infinitely divisible1 1( )Inf Div− iff it

has an infinite number of proper parts20.

Actually, as we have already pointed out, (26) is even too weak. The standard solution

understands infinite divisibility as having not just infinite, but uncountably many proper

parts.

4.3 A Novel Solution

In this section we propose a solution that (i) does not face the threats of the standard one,

and (ii) sheds new light on the relation between the notions of parthood and location. To

skip a little bit ahead we will argue that the “Small” horn was not a metaphysical possibility

to begin with and the “Large” horn can be solved. This solution depends upon another

possible reading of the notion of infinite divisibility that is different, and logically

independent from (26). Here it is:

(27) (Infinite Divisibility2) Something is infinitely divisible2 ( 2Inf Div− ) iff it is

gunky, i.e. every part of it has further proper parts21.

20 We could give a (rather inelegant) formalization via:

1 ( , , )( )( ) ( ( , )( )df n n i j

i j n x x x i j O x xInf Div x ∀ ∈ ∃ ∧ ≠ →− =

Some comments are in order. Why should (27) even be considered a possible reading of

infinite divisibility? This is because if we take an object that is infinitely divisible2 we will

always be able to divide it into further proper parts, that is every part of it is itself divisible

given our definition of divisibility (17). And this seems to capture at least some sense of the

driving intuition behind infinite divisibility.

Then we should address the question of the logical independence of notions in (26) and

(27). To do that we only have to show that something is infinitely divisible1(2) without

thereby being infinitely divisible2(1), that is we have to show that the following does not

hold:

(28) 1 2( ) ( )Inf Div x Inf Div x− ↔ −

First we show that the right to left direction of (28) does not hold without further

assumptions. In particular it does not hold if Anti-symmetry of parthood is violated22. If

Anti-symmetry is given up then a composite entity could be infinitely divisible2 without

thereby being infinitely divisible1, that it is it could be the case that it has only a finite

number of proper parts. As far as the left to right direction goes note that something could

be composed by an infinite number of atomic proper parts. In this case it would count as

infinite divisible1 without thereby being infinite divisible2. Note that this is exactly what

happens in the standard solution of the “Large and the Small” paradox.

Recall Zeno’s paradox. It stems from the fact that we have two possibilities, namely (i) an

infinite sum of extended entities that somehow compose an entity with infinite extension or

21 A formal rendering would read:

2( ) ( ) ( )( ( ))

dfInf Div x A x y y x A y− = ∧ ∀ →

22 We will return to this issue later on.

(ii) an infinite sum of unextended entity that compose something with no extension at all.

First we want to argue that the latter is not a metaphysical possibility after all, that is we

want to argue that if something is infinitely divisible, according to our reading (27), it does

not have unextended proper parts. Here is the argument.

Suppose this is not the case. Then there is a proper part y of our alleged infinitely divisible2

entity that is unextended, that is, is exactly located at a region R which is atomic:

(29) ( )A R

This proper part, given (27) would have some further proper parts. Call one of them z . The

proper part z has an exact location, by Exactness. Call it 1R . It follows from Strong

Expansivity that:

(30) 1R R

But, clearly (29) entails that:

(31) 1 1( )( )R R R∃

Thus resulting in a contradiction. We have just proven that an infinitely divisible2 entity

does not have unextended proper parts. Thus, the “Small” horn of Zeno’s paradox was not a

metaphysical possibility after all. The same argument also shows that Division does not

hold. For surely an infinitely divisible2 extended object overfills a spatial point, yet it does

not have any proper part that is exactly located there. Hence our solution does not face the

third historical threat we discussed in the previous section.

We just dispensed with the “Small” horn. The “Large” one still lingers.

However it can be solved. Suppose we want to compare the extension of different spatial

regions. In certain cases we could simply do that in mereological terms. For example it is

reasonable to stipulate the following:

(32) If regions 1,R R are such that 1R R , then region R is more extended

than region 1R 23.

We could then go on to compare the extension of material objects, in certain cases, simply

by comparing the extension of their exact locations, i.e.:

(33) If a material object x is exactly located at xR and a material object y is

exactly located at yR and y xR R then x is more extended than y .

Actually we could probably compare their extension directly, just by saying that if

x y then y is more extended than x . We are not sure whether this avoids any

reference whatsoever to spatial regions so we will rest content with (33). All this will help

us solve the Large Horn. Let us see how. Consider an infinitely divisible2 entity x exactly

located at R . Consider now one of its proper parts 1x . It will be exactly located at 1R such

that 1R R by Strong Expansivity. Consider now a proper part 2x of 1x . By the same

argument it will be exactly located at 2 1R R R since proper parthood is transitive.

For the n-th proper part we will then have that it will be exactly located at nR such that:

23 Note that the other direction of the conditional would be far too strong, for it will entail that any two regions 1,R R with different extension would be related via proper parthood, which is not always the case.

(34) 1... ...nR R R

Claim (34) shows that the proper parts of the infinitely divisible2 entity x we started with

will have different and decreasing extensions, given the way we compared extensions in

(32). If you now take the sum of all this extensions this sum will never exceed the extension

of R , let alone become infinite. Let us see why while at the same time dispelling one last

worry. The notion of sum we have just mentioned might lead to suspect that we are

introducing some way of adding extensions that is more or less equivalent to that of

Lebesgue measure theory. And we have not justified any such introduction so far. But the

argument can be put in pure mereological terms. Let ( )xϕ be a well-formed formula of our

language. The mereological sum z of the ϕ − ers is defined24 via:

(35) ( , ( )) ( )( ( , ) ( )( ( ) ( , ))dfSum z x y O y z x x O y xϕ ϕ= ∀ ↔ ∃ ∧

In other words the mereological sum of the ϕ − ers is that entity that overlaps all and only

those things that overlap a ϕ − er. Now, let’s go back to our infinitely divisible2 entity x

exactly located at R . We argued that the exact locations of the proper parts of x are proper

subregions, i.e. proper parts, of R . Now let all those subregions be our ϕ − ers. We want

to show that the mereological sum of those ϕ − ers cannot be a region that is more

extended than R , given (32). Suppose that this is not the case. Then given Strong

Expansivity and (32) it will be a region R+ such that (i) R R+ . By Weak

24 There is (at least) another stronger definition of mereological sum in the literature:

*( , ( )) ( ) ) ( )( ( )( ( ) ( , ))dfSum z x x x z y y z x x O x yϕ ϕ ϕ= → ∧ ∀ → ∃ ∧ . To see

that this is indeed a stronger definition note that it is possible to derive only the right-to-left direction of the following biconditional given only the partial ordering axioms for parthood:

( , ( )) *( , ( )Sum z x Sum z xϕ ϕ↔ . This argument is due to Hovda (2009).

Supplementation we get (ii) ( )( ( , ))R R R O R R− − + −∃ ∧ . By the first conjunct of (ii)

it follows trivially (iii) ( , )O R R− + , whereas from the second conjunct it follows that (iv)

( , )i iR R O R R−→ for each proper subregion iR of R . But (iii) and (iv) together

imply that R+ cannot qualify as the mereological sum of the proper subregions of R ,

according to definition (35) because it overlaps something, namely R− that does not

overlap any of them. This proves that the mereological sum of proper subregions of R will

never be more extended than R 25, let alone infinite. We can then go back to the way we

actually understood exact location and claim that x has the same extension as R , and so it

does not have an infinite extension. This solves the Large horn of the paradox.

Note that this solution works also for entities that are infinitely-divisible1, and so for entities

that have infinite proper parts. It does not however require that they have uncountably many

proper parts. As such it is immune from the criticisms we raised against the standard

solution in §4.2.

Furthermore note that our solution, as we have just briefly mentioned, solves the Large

horn rather than the Small one. This is a significant difference with the standard solution.

And the Large horn is indeed very difficult to solve within the standard framework. Recall

what the situation amounts to in that framework. After the division process we end up with

uncountably many parts with finite extension. Now, it is true that we cannot add those

extensions, for as we said additivity for Lebesgue measure fails for uncountable sets.

However it is possible to find a countable proper subset of that set such that all elements of

25 Given the way we have compared extensions of spatial regions in (32).

it have small yet finite extension. Then we can sum up those extensions according to

Lebesgue measure and we end up with an entity with infinite extension.

We have argued that something can be infinitely divisible2 without thereby being infinitely

divisible1. In such a case the standard solution could clearly not be applied. Would our

solution fare better? There is indeed a problem with our solution too, as it stands. We

already argued that if we want something to be infinitely divisible2 but not infinitely

divisible1 we should give up Anti-symmetry of parthood. And this causes some problems for

the application of our preferred solution to the paradox, in particular when it comes to

discarding the Small horn. This is due to the fact that Anti-symmetry follows from

Transitivity and Weak Supplementation. Since giving up Transitivity seems a high cost, this

would leave us without Weak Supplementation. But without that mereological axiom

Strong Expansivity seems rather strong, as we suggested in footnote 10. It will follow that

the Small horn of the paradox could not be precluded after all, for our argument crucially

depends on that locative axiom26. On the other hand the Large horn would be radically

solved, even without invoking the solution we have presented. This is because the

26 This does not mean that another argument cannot be found. A possible one would be the following. Recall that we are assuming that x has some proper parts. If x is exactly located at R then x and R have the same relevant geometrical properties. This intuition could even be regimented via the following axiom: ( , ) (( )( ( ) ( ))ExL x R G G x G R→ ∀ ↔ , where G stands for the relevant geometrical properties in question. Then, it could be pointed out that x and its proper parts do not have the same geometrical properties and so they could not be exactly co-located. So, where should x ’s proper parts be located? They could not be located at R for the argument we have just given. They could not be exactly located at subregions of R for R is atomic by assumption. They could be located neither at superregions of R nor at disjoint regions from R for it would entail that they are located outside x , thus violating even a weak principle of Expansivity (see footnote 7). So it has to be the case that x does not have proper parts, and this contradicts our assumption. This argument seems at first sight

compelling. However there is a substantive question left open. In a mereological theory that does not have Weak Supplementation objects could have a unique proper part. Suppose this is the case and that x has only y as its proper part. Is the claim that x and y do not have the same geometrical

properties, that is crucial to our argument, that unproblematic? It is not possible to enter this discussion here.

composite object would be constituted by a finite number of parts. And this will be enough

to resist the paradox.

Before concluding let us take a step back for a moment and consider carefully this last

argument we put forward. We claimed that Anti-symmetry follows from Transitivity and

Weak Supplementation27. And, given the formal framework we have developed so far, it

does. To see this consider the following argument. Suppose it does not. Then the

antecedent of (7) will be true, whereas its consequent would be false. i.e. we will have (i)

x y , (ii) y x and (iii) x y≠ . From (i), (iii) and definition of proper parthood in (2)

we will then have (iv) x y . By Weak Supplementation (v) ( )( ( , ))z z y O z x∃ ∧ .

Given (ii) and the first conjunct of (v) we get by Transitivity (vi) z x , and thus (vii)

( , )O z x , contra the second conjunct of (v).

This argument depends crucially upon the definition of proper parthood given in (2). This

is not however the only definition of proper parthood on the market. Cotnoir (2010)

suggests the following28:

(36) (Proper Parthood*) dfx y x y y x= ∧

Given anti-symmetry proper parthood and proper parthood* are equivalent, i.e. it can be

proven that:

27 The following discussion is much indebted to Cotnoir (2010) and Cotnoir and Bacon (2012). 28 Cotnoir (2010) actually argues that we should actually prefer (33) over (2). His primary concern is extensional mereology, in particular its defense in Varzi (2008). He argues convincingly that Varzi’s defense presupposes the definition of proper parthood given in (2) and thus Anti-symmetry. He then goes on to argue that this definition is unfriendly to anti-extensionalists. In a nutshell this is because Anti-symmetry can be thought of as a ban on coinciding entities, such as a statue and the lump of clay it is made of, that seem at first sight counterexamples to extensionality.

(37) x y x y↔

However only the left-to-right direction of (37) is derivable without Anti-symmetry. And

the possibility of abandoning Anti-symmetry is exactly what we are considering here. With

definition (37) in hand we could go on to formulate an alternative principle of Weak

Supplementation* via:

(38) (Weak Supplementation*) ( )( ( , ))x y z z y O x z→ ∃ ∧

Thus we could (i) replace proper parthood (2) with proper parthood* (36), drop Anti-

symmetry and (iii) replace Weak Supplementation (8) with Weak Supplementation* (38).

But Weak Supplementation* is enough to warrant to endorsement of Strong Expansivity.

Our argument discarding the Small horn of the paradox would now still go through29.

As we have seen Zeno’s argument against infinite divisibility still offers us, after more than

two thousand years, precious insights on important philosophical problems, as all his other

arguments do. It may very well be that there isn’t a clearest sign of philosophical depth.

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