From Extensional to Classical Mereology

A second way that MM has been extended is with the aim to provide a number of closure operations to the mereological domain. As discussed, for example, in (Varzi, 1996, 2003) and (Simons, 1987), theories named CMM (Closure Minimal Mereology) and CEM (Closure Extensional Mereology) can be obtained by extending MM and EM with the following operations:

a) Sum: also named mereological fusion (or juxtaposition in the case of Assembly Theory). The sum z of two objects x and y, symbolized as Sum(z,x,y), is the entity such that every object that overlaps with z, overlaps either with x or with y (or with both);

30An analogue for the improper part-of is already provable in M due to the reflexivity and antisymmetry of ≤.

Figure 5-4 Two distinct entities that are composed of the same parts; example of a model of minimal mereology that is not a model of extensional mereology

(19) Sum(z,x,y) =_def ∀w((w • z) ↔((w • x) ∨ (w • y)))

b) Product: also named superposition (in Assembly Theory). The product of two objects x and y is the entity z such that every part of z is either part of x or y;

(20) Pro(z,x,y) =_def ∀w((w ≤ z) ↔ ((w ≤ x) ∧ (w ≤ y)))

c) Difference: the difference of two objects x and y is the entity z such that every part of z is part of x and does not overlap with y;

(21) Dif(z,x,y) =_def ∀w((w ≤ z) ↔ ((w ≤ x) ∧ ¬(w • y)))

d) Complement: the complement of an entity x is the entity z such that every part of z does not overlap with x;

(22) Comp(z,x) =_def ∀w((w ≤ z) ↔ ¬(w • x))

These operations are the mereological counterpart of the set theoretical operations of union, intersection, set difference and complement of a set, respectively. In the presence of the extensionality principle, the z’s that are the results of these operations are unique. Thus, for example, in an extensional mereology, if two objects x and y overlap then there is a unique entity z that is composed of the common parts of x and y. In particular, the existence of the product and difference of two individuals x and y, and of the complement of an individual x, are only guaranteed in certain cases.

These conditions are expressed in formulas (23-25) below, respectively.

(23) ∀x,y (x • y) → ∃z∀w((w ≤ z) ↔ ((w ≤ x) ∧ (w ≤ y))) (24) ∀x∃y ¬(y ≤ x) → ∃z∀w((w ≤ z) ↔ ((w ≤ x) ∧ ¬(w • y))) (25) ∀x∃y ¬(x • y) → ∃z∀w((w ≤ z) ↔ ¬(w • x))

The mereological sum (Sum), conversely, is guaranteed by the presence of an entity, termed the Universe of which everything is part (Simons, 1987):

(26) Universe(z) =_def ∀x (x ≤ z)

Once more, in an extensional mereology, the universe z is unique. The existence of a “null individual” that is part of everything would also guarantee the existence of the product and difference for any two

FORMAL THEORIES OF PARTS 149

individuals, and the existence of a complement for any individual. However, most theories do not define such an entity. An exception is Bunge’s Assembly theory (Bunge, 1977).

As demonstrated in (Simons, 1987), if the product operator is functional then the strong supplementation axiom (16) is implied by the weak supplementation principle (15). As a consequence, CMM and CEM collapse in one single theory.

Finally, traditionally, unrestricted operations of fusion and product are also defined for closure mereologies. For the (unrestricted) mereological sum we define the following formula schema:

(27) ∃xF(x) → ∃z∀y((y • z) ↔ ∃w(F(w) ∧ (y • w))

This expresses that for every satisfied predicate F there is an entity consisting of all those things that satisfy F or, to put it differently, z is the sum of the arbitrary non-empty set of entities w_i such that F(w_i) holds.

Once more, in the presence of extensionality (16), the entity which is the sum of all entities satisfying predicate F has its uniqueness guaranteed. In this case, we can define a general sum as:

(28) σxF(x) =_def ιz∀y((y • z) ↔ ∃w(F(w) ∧ (y • w))

The product of all members of a set G of overlapping objects can be defined as follows: Let W be set of all those things that are part of every member of set G, i.e.,

(29) ∀x W(x) ↔ ∀y (G(y) → (x ≤ y))

The unrestricted product of all members of set G can, hence, be defined as the sum of all members of W, that is, by replacing F for W in the formula schema (27). The result of adding schema (27) to CMM or CEM is a theory named GEM (General Extensional Mereology) or Classical Extensional Mereology.

As demonstrated, for instance, in (Varzi, 2003), all closure operations (19-22) can be defined via choice of suitable predicates F to be substituted in (27). This gives the full strength of GEM, which has the algebraic structure of a quasi-boolean algebra (boolean algebra with a zero element removed)³¹.

Figure 5.5 below represents schematically the logical space of these different mereological theories.

31 In the case of Bunge’s Assembly theory, it is as expressive as a complete Boolean lattice.

Ground Mereology (Partial Order)

Minimal Mereology (+ Weak Supplementation)

Extensional Mereology (+ Strong Supplementation,

Extensionality principle ) Closure Extensional Mereology =

Closure Minimal Mereology (+ Closure Operations, uniqueness condition)

General Extensional Mereology (+ Unrestricted Fusion)

A final way to extend the theories in figure 5.5 is by considering the issue of Atomism. A mereological atom is an entity that has no proper parts:

(30) At(x) =_def ¬∃y (y < x)

In an atomistic mereology everything has atomic parts, i.e., (31) ∀x ∃y At(y) ∧ (y ≤ x)

Conversely, in an atomless mereology the following axiom holds:

(32) ¬∃x At(x)

Formulas (31) and (32) are clearly incompatible, but taken in isolation they can be added to any of the theories depicted in figure 5.5. In other words, the question of atomism is ortogonal and compatible to any of the mereologies discussed so far. Adding (31) to a theory X yields its atomistic version AX, whereas adding (32) yields its corresponding atomless version AX.

Finally, it is important to emphasize two important features of atomistic mereologies:

1. Atomistic mereologies (and only them) admit finite models, i.e., decomposition of parts must eventually come to an end;

Figure 5-5 Relations between different mereological theories.

The arrows go from the weaker to the stronger theory

PROBLEMS WITH MEREOLOGY AS A THEORY OF CONCEPTUAL PARTS 151

2. When atoms are considered, important simplifications can be made to many of the axioms discussed so far (Varzi, 2003). For instance, the weak supplementation principle (15) can be replaced by

(33) ∀x,y (y < x) → ∃z At(z) ∧ (z < x) ∧ ¬(z < y)

5.2 Problems with Mereology as a Theory of