## 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.

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## Notes

- 1.
Or the One

- 2.
We follow Varzi (2014) in taking

*Parthood*as primitive. Simons (1987) takes*Proper Parthood*as a primitive notion, whereas the classic Leonard and Goodman (1940) takes*Disjointness*where*Disjointness*is not*Overlap*. These notions are interdefinable so that this choice is mostly a matter of preference. - 3.
*Anti*-*symmetry*is actually redundant within this framework. See Sect. 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 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 exists a mereological sum of them. See Simons (1987) and Varzi (2014) 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.

- 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*)*WL*(*x*,*R*) =_{ df }(∃*R*_{1})(*ExL*(*x*,*R*_{1}) ∧*O*(*R*,*R*_{1}));(

*Underfilling*)*UnF*(*x*,*R*) =_{ df }(∃*R*_{1})(*ExL*(*x*,*R*_{1}) ∧*R*_{1}≺*R*). - 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*) :*x*≺*y*∧*ExL*(*x*,*R*) → (∃*R*_{1})(*ExL*(*y*,*R*_{1}) ∧*R*≺*R*_{1}). 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 to this point later on, in Sect. 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*)*x*≺*y*→ ((∃*R*)(*ExL*(*x*,*R*) → (∃*R*_{1})(*ExL*(*y*,*R*_{1}) ∧*R*≺*R*_{1})) 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*) :*ExL*(*x*,*R*_{1}) ∧*ExL*(*x*,*R*_{2}) →*R*_{1}=*R*_{2}. 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.
- 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. - 16.
- 17.
Though we would need to give a rigorous formulation of yet another notion, namely that of

*Infinite Divisibility*. - 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 Sect. 4.3.

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

*Inf*−*Div*_{1}(*x*) =_{ df }(∀*i*,*j*,*n*∈*ℕ*)(∃*x*_{ n })(*x*_{ n }≺ ≺*x*) ∧ (*i*≠*j*→ ∼*O*(*x*_{ i },*x*_{ j }) - 21.
A formal rendering would read:

*Inf*−*Div*_{2}(*x*) =_{ df }∼*A*(*x*) ∧ (∀*y*)(*y*≺ ≺*x*→ ∼*A*(*y*)) - 22.
We will return to this issue later on.

- 23.
Note that the other direction of the conditional would be far too strong, for it will entail that any two regions

*R*,*R*_{1}with different extension would be related via proper parthood, which is not always the case. - 24.
There is (at least) another stronger definition of mereological sum in the literature:

*Sum** (*z*,*φ*(*x*)) =_{ df }*φ*(*x*) →*x*≺*z*) ∧ (∀*y*)(*y*≺*z*→ (∃*x*)(*φ*(*x*) ∧*O*(*x*,*y*)). 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*,*φ*(*x*)) ↔*Sum** (*z*,*φ*(*x*)). This argument is due to Hovda (2009). - 25.
Given the way we have compared extensions of spatial regions in (32)

- 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. - 27.
- 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. - 29.
It is actually worth noting that we could even endorse what is known as the

*Strong Supplementation Principle*: ∼*y*≺*x*→ (∃*z*)(*z*≺*y*∧ ∼*O*(*x*,*z*)) and get*Weak Supplementation** (along with*Proper Parthood**) without thereby having*Anti*-*symmetry*. Our proposed solution will apply again.

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Calosi, C., Fano, V. Divisibility and Extension: a Note on Zeno’s Argument Against Plurality and Modern Mereology.
*Acta Anal* **30, **117–132 (2015). https://doi.org/10.1007/s12136-014-0233-9

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### Keywords

- (Infinite) divisibility
- Extension
- Parthood
- Location