5/24/2018 Mereology

1/4

1

Formal Ontology (G9509) Achille C. Varzi

MEREOLOGY (2)

Further Mereological Principles

A. Let Mbe the theory defined by the three basic principles (P.1)(P.3). Mmay be viewed as

embodying the common core of any mereological theory. Not just any partial ordering quali-

fies as a part-whole relation, though, and establishing what further principles should be added

to (P.1)(P.3) is precisely the question a good mereological theory is meant to answer. It is

here that philosophical issues begin to arise.

B. Generally speaking, such further principles may be divided into two main groups:

1. Decomposition Principles (from a whole to its parts): One may consider the idea that

whenever something has a proper part, it has more than onei.e., that there is always

some mereological difference (a remainder) between a whole and its proper parts. This

need not be true in every model for M.

2. Composition Principles (from the parts to the whole): One may consider the idea that

whenever there are some things there exists a whole that consists exactly of those things

i.e., that there is always a mereological sum(or fusion) of two or more parts. Again, this

need not be true in a model for M, and it is a matter of controversy whether the idea should

hold unrestrictedly.

Decomposition Principles

Weak Company

(P.4a) Every proper part is accompaniedbyanother.

PPxy!"z(PPzy#z!x)

Strong Company

(P.4b) No proper part includes all the others.

PPxy!"z(PPzy#Pzx)

Weak Supplementation

(P.4) If an object is a proper part of another, there is a remainder that makes up for the difference.

PPxy!"z(Pzy#Ozx)

Strong Supplementation

(P.5) Unless an object is part of another, there is a remainder that makes up for the difference.

Pyx!"z(Pzy#Ozx)

5/24/2018 Mereology

2/4

2

Discussion

A. The Weak Company Principle (P.4a) is a literal rendering of the idea in question: every proper

part must be accompaniedby another. However, there is an obvious sense in which (P.4a)

only captures the letter of the idea, not the spirit: it rules out models in which an object has a

single proper part but not, for example, a model with an infinitely descending chain in which

the additional proper parts do not leave any remainder:

x

M

B. The Strong Company Principle (P.4b) is stronger: it rules out both models as unacceptable.

However, (P.4b) is still too weak to capture the intended idea. For example, it is satisfied by a

model in which a whole can be decomposed into several proper parts all of which overlap one

another (Figure 2, right), and it may be argued that such models do not do justice to the mean-

ing of proper part: after all, the idea is that the removal of a proper part should leave a re-

mainder, but it is by no means clear what would be left of x once z (along with its parts) is

removed:

x

..

.

..

...

....

...

...

C. The Weak Supplementation Principle (P.4) appears to provide a full formulation of the idea

that nothing can have a single proper part. According to this principle, every proper part must

be supplementedbyanother, disjointpart, and it is this last qualification that captures the

notion of a remainder. Should (P.4), then, be incorporated into M as a further fundamental

principle on the meaning of part?

(a) Some authors (most notably Simons 1987) regard (P.4) as constitutive of the meaning of

part and would accordingly list it with the basic postulates of mereology along with

(P1)(P3).

(b) However, there are theories in the literature violate this principle. For example:

In Brentanos 1933 theory of accidents, a soul is a proper part of a thinking soul even

though there is nothing to make up for the difference. (See Chisholm 1978; Baumgart-

ner and Simons 1994.)

Similarly, in Fines 1982 theory of qua objects, every basic object (John) qualifies as

the only proper part of its incarnations (John quaphilosopher, John qua husband, etc.).

5/24/2018 Mereology

3/4

3

Another example is Whiteheads 1929 theory of extensive connection, where no bound-

ary elements are included in the domain of quantification: on this theory a topologically

closed region includes its open interior as a proper part in spite of there being no bound-

ary elements to distinguish them.

Finally, a counterexample is Thomsons 1998 theory of material constitution (alreadydiscussed in connection with the Antisymmetry postulate, (P.3)), which holds that a ma-

terial object (a statue) and the matter that constitutes it (a lump of clay) are proper parts

of each other although neither has parts disjoint from the other.

In any case, (P.4) is not strong enough to rule out the possibility that two distinct individuals

can be made up of exactly thesameproper parts, as in the following model:

x

D. The Strong Supplementation Principle (P.5) implies all the others and rules out all the models

above, but it is definitely more controversial. It is, in fact, a principle of extensionality, cap-

turing the nominalist dictum No distinction without a difference. This is evident from the

fact that the following is a theorem ofM + (P.5):

(24) "zPPzx!($z(PPzx!PPzy) !Pxy).

from which it follows that no composite objects with the same proper parts can be distinct:

(25) ("zPPzx%"zPPzy)!(x=y &$z(PPzx&PPzy)).

(The analogue for P is already provable in M, since P is reflexive and antisymmetric.) This

goes far beyond the intuition that lies behind the basic supplementation principle (P.4). Doesit go too far?

(a) On the one hand, it is sometimes argued that sameness of parts is not sufficient for iden-

tity, as some entities may differ exclusively with respect to the arrangementof their parts.

Familiar examples include the following:

Two sentences made up of the same wordsJohn loves Mary and Mary loves

Johnwould be a case in point (Hempel 1953: 110; Rescher 1955: 10).

Likewise, the identity of a bunch of flowers may depend crucially on the arrangements

of the individual flowers (Eberle 1970: 2.10).

A second familiar sort of example comes from the literature on material constitution,where the principle of mereological extensionality is sometimes taken to contradict the

possibility that an object may be distinct from the matter constituting it, though both may

be ultimately made up of the same basic parts:

A cat can survive the annihilation of its tail. But the amount of feline tissue consisting

of the cats tail and the rest of the cats body cannot survive the annihilation of the tail.

Thus, a cat and the corresponding amount of feline tissue have different (tensed or mo-

dal) properties. Thus they are distinct by the Principle of the Indiscernibility of Identi-

5/24/2018 Mereology

4/4

4

cals. (Wiggins 1968; see also Doepke 1982, Lowe 1989, Johnston 1992, Baker 1997,

Merav 2003, and Sanford 2003, inter alia, for similar or related arguments.)

(b) On the other hand, it is sometimes argued that sameness of parts is not necessaryfor iden-

tity, as some entities may survive mereological change. Example:

If a cat survives the annihilation of its tail, then the tailed cat (before the accident) and

the tailless cat (after the accident) are numerically the same in spite of their having dif-

ferent proper parts (Wiggins 1980).

E. Some nomenclature:

1. The extension of M obtained by adding (P.4) is calledMinimal Mereology, MMfor short.

It can be checked that in the presence of (P.1) and (P.2), the Antisymmetry postulate (P.3)

is derivable from (P.4) and is therefore redundant in MM.

2. The extension of M obtained by adding (P.5) is called Extensional Mereology, EM for

short. It can be checked that, given M, (P.4) follows from (P.5), though not vice versa,

hence EMproperly includes MM.

References

Baker L. R., 1997, Why Constitution Is Not Identity,Journal of Philosophy94: 599-621.

Baumgartner W. and Simons P. M., 1994, Brentanos Mereology,Axiomathes 5: 55-76.

Brentano F., 1933, Kategorienlehre, ed. A. Kastil, Hamburg: Meiner (Eng. trans. by R. M. Chisholm

and N. Guterman: The Theory of Categories, The Hague: Nijhoff, 1981).

Chisholm R. M., 1978, Brentanos Conception of Substance and Accident, in R. M. Chisholm and R.

Haller (eds.),Die Philosophie Brentanos, Amsterdam: Rodopi, pp. 197-210.

Doepke F. C., 1982, Spatially Coinciding Objects, Ratio24: 45-60. Eberle R. A., 1970,Nominalistic Systems, Dordrecht: Reidel.

Fine, K., 1982,Acts, Events, and Things,in W. Leinfellner, E. Kraemer e J. Schank (eds.),Language

and Ontology. Proceedings of the 6th International Wittgenstein Symposium , Hlder-Pichler-

Tempsky, Vienna, pp. 97-105.

Hempel C. G., 1953, Reflections on Nelson Goodmans The Structure of Appearance,

Philosophical Review 62: 108-116.

Johnston M., 1992, Constitution Is Not Identity',Mind101: 89-105.

Lowe E. J., 1989, Kinds of Being: A Study of Individuation, Identity and the Logic of Sortal Terms,

Oxford: Blackwell.

Meirav A., 2003, Wholes, Sums and Unities, Dordrecht: Kluwer.

Rescher N., 1955, Axioms for the Part Relation,Philosophical Studies6: 8-11.

Sanford D., 2003, Fusion Confusion,Analysis 63: 1-4.

Simons P. M., 1987, Parts. A Study in Ontology, Oxford: Clarendon.

Thomson J. J ., 1998, The Statue and the Clay, Nos 32: 149-173.

Whitehead A. N., 1929,Process and Reality. An Essay in Cosmology, New York: Macmillan.

Wiggins D., 1968, On Being in the Same Place at the Same Time,Philosophical Review 77: 90-95.

Wiggins D., 1980, Sameness and Substance, Oxford: Blackwell.