In this short paper I illustrate what is mereology for philosophers, and which reasons lead philosophers to discuss mereological theses and disagree one with another about them. I will focus in particular on the role of structure in mereology and propose a rather simple account of what structure is from the viewpoint of mereology. As we are going to see, many philosophical controversies in mereology concern the issue of whether mereology should account for structure or not, and which role (if any) structure plays in mereology. I will also present some examples of philosophical controversies about mereological principles, and of the reasons which might be brought for choosing one side or the other in these controversies.

An additional purpose of the paper is to suggest that structure is a broad topic of common concern between architecture and philosophical mereology. In their discussions about structure, mereologists do often resort to examples involving buildings and villages. These examples – I am going to use some of them in what follows – are extremely simple, to the point of naïveté. The frequent usage of these examples might depend on the rough intuition that there are indeed some connections between mereology and architecture, and the concept of structure might be the link between them.

What Is Mereology About?

Let us begin by asking what is mereology. Mereology is the theory of two (mutually related) relations: parthood and composition. Parthood is a one-one relation between a part and a whole. These are some instances of parthood: 

my left hand is part of my body; 

Portugal is part of Europe; 

an atom of oxygen is part of a molecule of water; 

a handle is part of a door; 

the word ‘salad’ is part of the sentence ‘I eat a salad’.

The other relation which, together with parthood, is the subject matter of mereology is composition, a many-one relation. Composition indeed connects many entities (the components) to a single entity (the composed entity). Consider a rather simplistic house with a base, four outer walls, a roof and nothing else. The base, the four walls and the roof compose the house. Another example is the following: the Netherlands, Belgium and Luxembourg compose Benelux.

Composition is definable in terms of parthood. The composed entity is expected to include all the components as parts and not to include anything extraneous to the components. Also the latter feature of the composed entity can be expressed in terms of parthood, namely by requiring that every part of the composed entity has at least a part in common with at least one of the components. 
In order to express the resulting definition of composition in terms of parthood we can first define the relation of overlap, which holds between two entities if and only if they have at least a part in common (P is the relation of parthood; ⚬ is the defined relation of overlap):

Overlap: x ० y ≡_def ∃z (z P x ⋀ z P y)  

Now, composition can be defined in terms of parthood and overlap (which has in turn been defined in terms of parthood above) as follows (C is the relation of composition; xx is a plural variable for the components; ≺ is the relation of being one of):

Composition: xx C y ≡_def ∀z (z ≺ xx → z P y) ⋀ ∀z (z P y → ∃w (w ≺ xx ⋀  z ० w))

According to this definition, some entities xx compose one entity y if and only if every entity z that is one of xx is also part of y and every entity z that is part of y is such that there is an entity w that is one of xx and overlaps z.

What do mereologists say about parthood and composition? They attribute some features to these relations. The attributed features are generally expected to be formal or topic-neutral, in the sense that the features are not expected to depend on which kinds of entities are part of one another or compose one another. For example, the features of parthood and composition are expected by mereologists to be independent of whether we are considering the parts of a human body, the parts of a chemical molecule, or the parts of a town. Mereology is also formal in a more general sense, inasmuch as mereology never attempts to identify the parts of something or to establish what composes what: it is not the expected duty of mereology to establish what are the parts of a car, or of a building, or what composes a sentence. 

Mereological principles consist in attributions of some formal features to parthood and composition. Let us consider some examples of mereological principles and of formal features which these principles attribute to parthood and composition. In considering these examples, it is important to keep in mind that every mereological principle (any attribution of formal features to parthood and composition) is controversial, including the following examples. It can also be controversial whether the principles at stake are really formal: given a candidate principle, it might be objected that it holds when parthood is instantiated by, say, buildings, while it fails when parthood is instantiated by animal organisms. 

A mereological principle about parthood defended by many philosophers is for example that parthood is a transitive relation (transitivity is the formal feature which this principle attributes to parthood): if a is part of b and b is part of c, then a is part of c. Here are three more or less controversial instances of transitivity (involving various kinds of entities, in coherence with the expected topic-neutrality of mereological principles): 

if a handle is part of a door and the door is part of a building, then the handle is part of the building;

if my left hand is part of my left arm and my left arm is part of my body, then my left hand is part of my body;

if an MP is part of the Italian Parliament and the Italian Parliament is part of the Inter-Parliamentary Association, then the MP is part of the Inter-Parliamentary Association. 

Principles about parthood also concern the problem of whether the chains of parthood terminate or not. Do the chains of parthood terminate downwards? Does everything have a part that is different from itself no matter how deep you go down? Or is there a bottom layer of mereological simples (i.e. entities without further parts)? In the other direction, do the chains of parthood terminate upwards? Is everything part of something different from itself? Or is there a top layer, the mereological universe (i.e. an entity such that everything is part of it, and it is part of nothing different from itself)? Mereological principles can dictate an answer to these questions. 

These principles can be applied to matters of potential architectural or urbanistic concern. As regards downward termination, a problem of potential architectural concern is whether there is a bottom layer of parts of a building (say: the layer of bricks, or the layer of the smallest pieces which are visible to the human eye). As regards upward termination, a problem of potential urbanistic concern is whether there is a top layer of entities, over which entities stop being interesting for urbanists and are at best interesting for other kinds of scholars, such as geographists.

Other principles concern composition (the other relation, which – together with parthood – is the subject matter of mereology). The principles about composition mainly provide existence conditions and identity conditions for composed entities. Existence and identity are in general two pivotal concepts in metaphysics, i.e. in the wider branch of philosophy to which mereology belongs. As far as composition is concerned, mereology asks whether, given some entities, there exists something which they compose (in answering this question, mereologists provide existence conditions for composed entities), and whether there may exist two or more entities composed by the same entities (in answering this latter question, mereologists provide identity conditions for composed entities). 

Thus, given some components, how many things do they compose? Consider Barack Obama’s nose, my left shoe and the Great Pyramid of Giza. Do they compose anything? Is there a spatially scattered object (a bit in Washington, a bit here, a bit in Egypt) they compose? 

Consider instead a building entirely made of bricks. Those bricks compose that building. Do the bricks compose only the building? Or do they compose also a different, less structured entity, which we might dub ‘heap of bricks’? Suppose that the building collapses: after the collapse – one might say – the building stops existing, while the heap of bricks continues existing. How can this happen, if they are identical, i.e. if the bricks compose only one entity? A single entity cannot both continue existing and stop existing.

The following are two examples of two rather popular (albeit controversial) mereological principles about composition (their usual label is indicated in the parenthesis):

given some entities – no matter how sparse and heterogeneous they are – there is at least an entity composed by them (Unrestricted Composition); 

given some entities, there is at most an entity composed by them (Uniqueness of Composition). 

Unrestricted Composition provides existence conditions for composed entities, while Uniqueness of Composition provides identity conditions for composed entities.

The Role of Structure in Mereology

In the above example of the brick building and of the heap of bricks, we have seen how the temptation arises to distinguish wholes according to their being structured or non-structured, or according to their being structured in different ways. In order to appreciate the role of structure in mereology, it is very important not to misidentify the subject matter of the philosophical controversies about this kind of example. 

Indeed, no mereologist doubts that, in order to compose a building, the bricks have to be in some mutual relations and that, more in general, not any heap of bricks is a building. There is no interesting philosophical controversy on the fact that the parts of many composed entities are arranged or structured in a certain way. 

The problem mereology is concerned with is different: does structure have any impact on the existence and identity of composed entities? Do some things need to be structured in a certain way (e.g. in a building-like way, or in a car-like way, or in a person-like way, or in a mountain-like way) in order to compose something? There is obviously a difference between the parts of a car on the one hand, and Barack Obama’s nose, my left shoe and the Great Pyramid of Giza on the other. The difference is that the parts of a car are in certain mutual relations and have certain designated functional roles. The relations among the parts of a car and their functional roles are constrained by the nature of cars in general, and by the project of that specific car model. The mutual relations among parts and the roles of the parts are roughly what structure consists in. 

To repeat: nobody doubts that the parts of a car have a kind of structure which, by contrast, disparate, sparse entities lack. Mereology is not about this. The mereological problem at stake is whether, on the basis of this difference between the parts of a car on the one hand, and Barack Obama’s nose, my left shoe and the Great Pyramid of Giza on the other, we should conclude that only the parts of a car compose something, while Barack Obama’s nose, my left shoe and the Great Pyramid of Giza do not compose anything. If this were the case, then, in contrast with the above principle of Unrestricted Composition, it would not be true that, given some entities – no matter how disparate and sparse they are – there exists something they compose: there would be nothing which Barack Obama’s nose, my left shoe and the Great Pyramid of Giza compose. 

Unrestricted Composition is a mereological principle according to which structures have no bearing on the existence conditions for composed entities. According to Unrestricted Composition, composed entities exist irrespective of whether and how their components are structured. 

Another mereological problem concerning structure is: does structure have any bearing on the identity conditions of composed entities? According to Uniqueness of Composition, given some entities there is at most one entity composed by them. Thus, there cannot be two different entities composed by the same entities. This entails that two composed entities cannot have exactly the same parts. If the composed entities are different, then they have a different part, and this different part is their difference maker. The fact that these parts are in different mutual relations and have different roles is not an admissible difference maker for composed entities. If Uniqueness of Composition is true, then the identity conditions for composed entities countenance only their parts, and not their structure. Thus, both Unrestricted Composition (for what concerns the existence conditions for composed entities) and Uniqueness of Composition (for what concerns the identity conditions for composed entities) exhibit a kind of deliberate blindness to structure.

Principles of Structure Obliteration

Before considering how it is possible to argue in favour or against this blindness to structure, it is useful to be a bit more precise on how structure is construed in this context. A theory of parthood and composition can countenance or obliterate various aspects of structure. This depends on whether, in a certain theory of parthood, certain principles of structure obliteration hold or not. We will consider four principles of structure obliteration.[1] These principles are interesting because they are of help in distinguishing various aspects of structure. These aspects of structure are obliterated in the identity conditions for composed entities, if the respective obliteration principle holds. They are by contrast countenanced in the identity conditions for composed entities, if the respective principle fails. 

The first principle is Absorption and claims that the repetition of parts is not a difference maker for composed entities. If Absorption is true, then the repetition of parts has no impact on the identity conditions for composed entities. Absorption can be formalised as follows (Σ is an operation of composition, whose inputs are the components and whose output is the composed entity; the formula expresses the fact that the multiple occurrences of the inputs – expressed by the multiple occurrences of the variables x and y in the left part of the formula – make no difference for the identity conditions of the output): 

Absorption: ∑ (…, x, x, …, y, y, …) = ∑ (…, x, …, y, …)

The second and the third principles are Collapse and Levelling and jointly claim that the stratification and the groupings of parts at different levels are not difference makers for composed entities (i.e. they have no impact on the identity conditions for composed entities). 

Collapse: ∑ (x) = x

Levelling: ∑ (…, ∑(x, y, z, …), …, ∑ (u, v, w, …), …) = ∑ (…, x, y, z, …, …, u, v, w, …, …)

The fourth principle is Permutation and claims that the order of the parts is not a difference maker for composed entities (i.e. it has no impact on the identity conditions for composed entities).

Permutation: ∑ (…, x, …, y, …, z,…) = ∑ (…, y, …, z, …, x, …)

This approach manages to differentiate various theories of parthood, according to their degree of blindness with respect to structure. What is usually called (for historical reasons) Classical Mereology abides by all the above four principles of structure obliteration and is, as a result, deliberately blind with respect to the repetition, to the stratification, to the groupings at different levels and to the order of the components in a composed entity. Only the parts matter for the identity conditions of composed entities, according to Classical Mereology. By contrast, the ways in which the parts are arranged/structured (e.g. repeated, stratified, grouped or ordered) do not matter. 

It is noteworthy that the four principles of structure obliteration are mutually independent. If you adopt a Non-Classical Mereology, you are not thereby forced to reject all the four principles as a single package. You can reject one or more of them, while keeping the others. In so doing, you sometimes end up adopting a theory which is no less well-established than Classical Mereology. For example, suppose that you think that the stratification and groupings at various levels of parts matter for the identity conditions of composed entities, while their order and repetition do not matter. What you obtain is Set Theory, a well-established theory, with a pivotal role in the foundations of mathematics.

Controversies in Mereology (and How to Argue About Them)

How should we argue about the formal features of parthood and composition? What reasons can be brought in favour of or against the mereological principles which attribute formal features to parthood and composition? And what reasons can – in particular – be brought in favour of or against the principles we have introduced above, such as Transitivity of Parthood, Unrestricted Composition, Uniqueness of Composition and the four principles of structure obliteration? 

Mereologists mainly proceed either by analysing and assessing alleged counterexamples to the mereological principles, or by analysing and assessing a priori arguments in support of or against them. As far as counterexamples are concerned, let us focus on Uniqueness of Composition and on its radical blindness to structure (i.e. – as we have seen in § 3 – blindness to repetition, stratification, groupings at different levels and order of components in a composed entity). One might be tempted to dismiss Uniqueness of Composition rather quickly, on the basis of the fact that some prima facie unavoidable counterexamples might seem fatal to it. 

Consider the components of the sentence ‘Gina loves Mario’: the words ‘Gina’, ‘loves’ and ‘Mario’. The same components can also form the sentence ‘Mario loves Gina’. ‘Gina loves Mario’ and ‘Mario loves Gina’ might seem two composed entities with the same components, in contrast with Uniqueness of Composition. The order of words (an aspect of structure, which belongs to the subject matter of syntax in linguistics) in sentences seems to matter for the identity conditions of sentences, in contrast with Uniqueness of Composition and with Permutation. 

Consider also a very small village, composed by a square, two streets and four buildings. It seems prima facie plain that Uniqueness of Composition fails for those components: the disposition of the streets with respect to the square, and the location of the buildings with respect to those of the square and of the streets (in general: the way in which the components of the village are arranged) would seem to matter for the identity of the village. 

However, the evaluation of these alleged counterexamples is not as easy as it seems, and Uniqueness of Composition should not be dismissed so quickly. Why? Because whether two composed entities (two sentences, two small villages) can be different while having the same parts and whether they can – as a consequence – be different only in virtue of their structure depends on a controversial identification of the entities at stake. 

In the case of the sentence, it depends on whether the entities at stake are linguistic tokens or types. Consider only single concrete tokens or inscriptions of ‘Mario’, ‘loves’ and ‘Gina’: sequences of sounds, stains of ink or groups of pixels on a screen. These concrete tokens are always in a single, specific order. Some of these inscriptions are ordered in a way such that ‘Mario’ is the first inscription (counting from the left) and ’Gina’ is the last inscription (this is the case of the word inscriptions in ’Mario loves Gina’). Others of these inscriptions are ordered in a way such that ’Gina’ is the first inscription (counting from the left) and ’Mario’ is the last inscription (this is the case of the word inscriptions in ’Gina loves Mario’). It never happens that the same inscriptions are arranged in two ways and thereby compose two different wholes. The initial impression that it is clear that two sentences can be different simply due to the arrangement of their parts (even if they have exactly the same parts) depends on seeing words not as tokens but as abstract word types, which occur in many different sentence types. 

Consider also the case of the small village. Given a specific small village, at a specific time, the square, the two streets and the four buildings are arranged in a single way (they have a single structure). There are not two small villages composed by that square, those streets and those buildings at that time. 

These considerations about the sentence and the small village can, as a matter of fact, be generalised. Whenever we are tempted to dismiss those principles of Classical Mereology which express its deliberate blindness to structure, it turns out that the temptation depends on a controversial characterisation of the involved entities.[2]

Please note that the philosophers objecting to Uniqueness of Composition might refine their counterexamples, and the defenders of Uniqueness of Composition might refine their analysis in order to deal with these counterexamples: the purpose of the above analysis is not to resolve the philosophical disputes about the role of structure in mereology in favour of blindness to structure, but to exemplify the way in which philosophers argue one with another about mereology and structure. The exemplifications are also meant to suggest that these controversies are unlikely to be easily solvable by adducing counterexamples: the analysis of these counterexamples is often arduous and depends on controversial assumptions. 

Finally, I would like to discuss an example of the second main way of arguing about mereological principles, the one which involves general, a priori arguments in support of or against mereological principles. Let us focus in this case on Unrestricted Composition. Suppose that you deny Unrestricted Composition. This denial will be based on the intuition that there is a patent difference between – say – the parts of a car on the one hand, and Barack Obama’s nose, my left shoe and the Great Pyramid of Giza on the other. What does this difference consist in? In order to obtain an alternative to Unrestricted Composition, this difference should correspond to a general condition which a plurality of entities should satisfy in order to have a fusion. 

We might try to extract this general condition from the examples, e.g. by observing that the parts of a car are spatially close one to another, while Barack Obama’s nose, my left shoe and the Great Pyramid of Giza are not; or by observing that there are causal links among the parts of a car (e.g. a movement in the steering causes a movement in its anterior wheels), while Barack Obama’s nose, my left shoe and the Great Pyramid of Giza are in no direct causal connection. On this basis, one might propose the following criteria for the restriction of composition: only mutually close entities compose something; only entities which move together (or act together) compose something. However, these criteria are unavoidably vague. There is no such thing as being definitely close in space or as being definitely causally connected. Every two parts of the universe are at some distance and have some kind of more or less remote causal connection. 

How is it possible to fix a threshold, i.e. a minimal degree of proximity or of causal connectedness? The threshold should be such that: entities above that threshold compose something; entities below that threshold do not compose anything. For example, we should determine once and for all at which maximum distance some buildings should be in order to contribute to compose a certain town, instead of belonging to two different towns. Any such threshold would be arbitrary, and would risk making arbitrary our existence claims about composed entities (such as towns) as well. 

A famous argument in support of Unrestricted Composition is based on the thesis that existence claims cannot be either vague or arbitrary. This means that no compelling motivation for restricting composition can be satisfied, and that, as a consequence, composition cannot be restricted. Thus, composition would be unrestricted. 

The general idea behind this famous argument for Unrestricted Composition (of which I have presented a simplified version)[3] is that existence conditions for composed entities should not be constrained by structural considerations about the mutual unity of parts. Why? Because it is arbitrary to delimit the domain of what is unitary. Every plurality of entity can be seen as unitary (or as non-unitary) according to certain criteria and/or from a certain standpoint. This is a general motivation why classical mereology is deliberately blind to structure: because the attribution of roles to structure risks introducing arbitrary and hardly justifiable thresholds. Thus, also, when mereologists proceed by analysing and assessing a priori arguments – as much as when they proceed by analysing and assessing counterexamples, as we have seen above – the problem whether structure has any role in mereology is pivotal, and has no easy solution.