Divisibility and Extension: a Note on Zeno’s Argument Against Plurality and Modern Mereology

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.

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) it 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 Sect. 2, we briefly develop the formal frameworks that will be used throughout the paper. In Sect. 3, we use these frameworks to give new, simple, and rigorous definitions of the notions of extension and divisibility that are involved in the paradox. Then, in Sect. 4, we review the paradox and 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 (Sect. 2.1) and to formal theories of location (Sect. 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

$$ (Parthood) x\prec y $$

(1)

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:

$$ \left( Proper\ Parthood\right) x\prec \prec y{=}_{df} x\prec y\wedge x\ne y $$

(2)

$$ (Overlap) O\left( x, y\right){=}_{df}\left(\exists z\right)\left( z\prec x\wedge z\prec y\right) $$

(3)

$$ (Atom) A(x){=}_{df}\sim \left(\exists y\right)\left( y\prec \prec x\right) $$

(4)

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:³

$$ (Reflexivity) x\prec x $$

(5)

$$ (Transitivity) x\prec y\wedge y\prec z\to x\prec z $$

(6)

$$ \left( Anti- symmetry\right) x\prec y\wedge y\prec x\to x= y $$

(7)

$$ \left( Weak\ Supplementation\right) x\prec \prec y\to \left(\exists z\right)\left( z\prec y\wedge \sim O\left( x, z\right)\right) $$

(8)

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 yields the Atomistic or the Atomless variants of MM, that is the mereological theory obtained by adding either⁴

$$ \left( A tomicity\right)\left(\exists y\right)\left( A(y)\wedge y\prec x\right) $$

(9)

or

$$ (Atomlessness)\left(\exists y\right)\left( y\prec \prec x\right) $$

(10)

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, and 9) and Gunky Minimal Mereology (GMM) the one comprising (5, 6, 7, 8, and 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 literature,⁶ we find it 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, and volume 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

$$ \left( Exact\ Location\right) ExL\left( x, R\right) $$

(11)

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 R _i , …, R _k will stand for spatial regions.

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

$$ (Overfilling) OvF\left( x, R\right){=}_{df}\left(\exists {R}_1\right)\left( ExL\left( x,{R}_1\right)\wedge R\prec {R}_1\right) $$

(12)

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:

$$ (Exactness)\left(\exists R\right)\left( ExL\left( x, R\right)\right) $$

(13)

$$ \left( Strong\ Expansivity\right) x\prec \prec y\wedge ExL\left( x, R\right)\to \left(\exists {R}_1\right)\left( ExL\left( y,{R}_1\right)\wedge R\prec \prec {R}_1\right) $$

(14)

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 Division ¹² axiom to ML:

$$ (Division) OvF\left( x, R\right)\to \left(\left(\exists y\right)\left( y\prec x\wedge ExL\left( y, R\right)\right.\right) $$

(15)

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 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 Sect. 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:

$$ \left( Eleatic\ Monism\right)\sim \left(\left(\exists x\right)\left(\exists y\right)\left( x\ne y\right)\right) $$

(16)

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 Extension ¹⁵ 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 Monism. Now, everything is ready for our proposed formulations of Divisibility and Extension:

$$ (Divisibility) Div(x){=}_{df}\left(\exists y\right)\left( y\prec \prec x\right){=}_{df}\sim A(x) $$

(17)

$$ (Extension)\begin{array}{l} Ext(x){=}_{df} ExL\left( x, R\right)\to \sim A(R)\hfill \\ {}{=}_{df} ExL\left( x, R\right)\to \left(\exists {R}_1\right)\left({R}_1\prec \prec R\right)\hfill \end{array} $$

(18)

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-dimensionalism 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:

$$ \left( R egion- Extension\right) Ext(R){=}_{df}\sim A(R) $$

(19)

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:

$$ Ext(x)\to Div(x) $$

(20)

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 Simples ¹⁶. An Extended Simple is easily definable within our framework:

$$ \left( Extended\ Simple\right) Ext- S(x){=}_{df} A(x)\wedge Ext(x) $$

(21)

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

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 (Sect. 4.1) and its traditional solution (Sect. 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 (Sect. 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 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, which resorts to Cantorian mathematics and the Lebesgue measure theory, let us give a somewhat more detailed formulation¹⁹ of it, which is reminiscent of Grünbaum (1968). Here it is.

$$ \mathrm{Suppose}\ \mathrm{something}\ \mathrm{is}\ \mathrm{infinitely}\ \mathrm{divisible}. $$

(22)

$$ \mathrm{Then}\ \mathrm{it}\ \mathrm{is}\ \mathrm{the}\ \mathrm{case}\ \mathrm{that}\ \mathrm{it}\ \mathrm{is}\ \mathrm{composed}\; either\;\mathrm{of} $$

(23)

$$ \mathrm{an}\ \mathrm{infinite}\ \mathrm{sum}\ \mathrm{of}\ \mathrm{extended}\ \mathrm{entities}, or\;\mathrm{of} $$

(i)

$$ \mathrm{an}\ \mathrm{infinite}\ \mathrm{sum}\ \mathrm{of}\ \mathrm{unextended}\ \mathrm{entities}. $$

(ii)

$$ \mathrm{If}\ \left(\mathrm{i}\right)\ \mathrm{the}\ \mathrm{entity}\ \mathrm{we}\ \mathrm{started}\ \mathrm{with}\ \mathrm{should}\ \mathrm{have}\ \mathrm{an}\; infinite\ extension. $$

(24)

$$ \mathrm{If}\ \left(\mathrm{ii}\right)\ \mathrm{the}\ \mathrm{entity}\ \mathrm{we}\ \mathrm{started}\ \mathrm{with}\ \mathrm{should}\ \mathrm{have}\; no\ extension\;\mathrm{at}\ \mathrm{all}. $$

(25)

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 the 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 an 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 of 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 an 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:²⁰

$$ \left( Inf inite\ Divisibilit{y}_1\right)\ \mathrm{Something}\ \mathrm{is}\ Infinite\ divisibil{e}_1\left( Inf- Div\right)\ \mathrm{iff}\ \mathrm{it}\ \mathrm{has}\ \mathrm{an}\ \mathrm{infinite}\ \mathrm{number}\ \mathrm{of}\ \mathrm{proper}\ \mathrm{parts}. $$

(26)

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:²¹

$$ \left( Inf inite\ Divisibilit{y}_2\right)\ \mathrm{Something}\ \mathrm{i}\mathrm{s}\ Infinitely\ divisibil{e}_2\left( Inf- Di{v}_2\right)\mathrm{iff}\ \mathrm{i}\mathrm{t}\ \mathrm{i}\mathrm{s}\ \mathrm{gunky},\ \mathrm{i}.\mathrm{e}.\ \mathrm{every}\ \mathrm{part}\ \mathrm{of}\ \mathrm{i}\mathrm{t}\ \mathrm{has}\ \mathrm{further}\ \mathrm{proper}\ \mathrm{part}\mathrm{s}. $$

(27)

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 divisible ₂ , 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 divisible _{1 (2)} without thereby being infinitely divisible _{2 (1)}, that is we have to show that the following does not hold:

$$ Inf- Di{v}_1(x)\leftrightarrow Inf- Di{v}_2(x) $$

(28)

First, we show that the right-to-left direction of (28) does not hold without further assumptions. In particular, it does not hold if the Anti-symmetry of parthood is violated²². If Anti-symmetry is given up, then a composite entity could be infinitely divisible ₂ without thereby being infinitely divisible ₁ , 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 of an infinite number of atomic proper parts. In this case, it would count as infinite divisible ₁ without thereby being infinite divisible ₂ . 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 (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 divisible ₂ entity that is unextended, that is, is exactly located at a region R which is atomic:

$$ A(R) $$

(29)

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 R ₁. It follows from Strong Expansivity that:

$$ {R}_1\prec \prec R $$

(30)

But, clearly, (29) entails that:

$$ \sim \left(\exists {R}_1\right)\left({R}_1\prec \prec R\right) $$

(31)

thus resulting in a contradiction. We have just proven that an infinitely divisible ₂ 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 divisible ₂ 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:²³

$$ \mathrm{If}\ \mathrm{region}\mathrm{s}\ R,{R}_1\ \mathrm{are}\ \mathrm{s}\mathrm{uch}\ \mathrm{that}\ {R}_1\prec \prec R,\ \mathrm{then}\ \mathrm{region}\ R\ \mathrm{is}\ \mathrm{more}\ \mathrm{extended}\ \mathrm{than}\ \mathrm{region}\ {R}_1. $$

(32)

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.:

$$ \mathrm{If}\ \mathrm{a}\ \mathrm{material}\ \mathrm{object}\ x\ \mathrm{is}\ \mathrm{exactly}\ \mathrm{located}\ \mathrm{a}\mathrm{t}\ {R}_x\ \mathrm{a}\mathrm{nd}\ \mathrm{a}\ \mathrm{material}\ \mathrm{object}\ y\ \mathrm{is}\ \mathrm{exactly}\ \mathrm{located}\ \mathrm{a}\mathrm{t}\ {R}_y\ \mathrm{a}\mathrm{nd}\ {R}_y\prec \prec {R}_x\ \mathrm{t}\mathrm{hen}\ x\ \mathrm{is}\ \mathrm{more}\ \mathrm{extended}\ \mathrm{t}\mathrm{han}\ y. $$

(33)

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 divisible ₂ entity x exactly located at R. Consider now one of its proper parts x ₁. It will be exactly located at R ₁ such that R ₁ ≺ ≺ R by Strong Expansivity. Consider now a proper part x ₂ of x ₁. By the same argument, it will be exactly located at R ₂ ≺ ≺ R ₁ ≺ ≺ R since proper parthood is transitive. For the nth proper part, we will then have that it will be exactly located at R _n such that:

$$ \dots \prec \prec {R}_n\prec \prec \dots \prec \prec {R}_1\prec \prec R $$

(34)

Claim (34) shows that the proper parts of the infinitely divisible ₂ 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 the Lebesgue measure theory. And, we have not justified any such introduction so far. But, the argument can be put in pure mereological terms. Let φ(x) be a well-formed formula of our language. The mereological sum z of the φ − ers is defined²⁴ via:

$$ Sum\left( z,\varphi (x)\right){=}_{df}\left(\forall y\right)\left( O\left( y, z\right)\leftrightarrow \left(\exists x\right)\left(\varphi (x)\wedge O\left( y, x\right)\right.\right) $$

(35)

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 divisible ₂ 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 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) R _i  ≺ ≺ R → ∼ O(R _i , R ⁻) for each proper subregion R _i 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 be 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 divisible ₁ 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 Sect. 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 the Lebesgue measure fails for uncountable sets. However, it is possible to find a countable proper subset of that set such that all elements of it have small yet finite extension. Then, we can sum up those extensions according to the Lebesgue measure and we end up with an entity with infinite extension.

We have argued that something can be infinitely divisible ₂ without thereby being infinitely divisible ₁ . 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 divisible ₂ but not infinitely divisible ₁ , 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 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 Supplementation ²⁷. 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 the 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:

$$ \left( Proper\ Parthood*\right) x\prec \prec \prec y{=}_{df} x\prec y\wedge \sim y\prec x $$

(36)

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

$$ x\prec \prec y\leftrightarrow x\prec \prec \prec y $$

(37)

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:

$$ \left( Weak\ Supplementation*\right) x\prec \prec \prec y\to \left(\exists z\right)\left( z\prec y\wedge \sim O\left( x, z\right)\right) $$

(38)

Thus, we could (i) replace proper parthood (2) with proper parthood* (36), (ii) drop Anti-symmetry, and (iii) replace Weak Supplementation (8) with Weak Supplementation* (38). But, Weak Supplementation* is enough to warrant the 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 2,000 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.