Part/sum based theories

One of the best known criticisms of set-based theories is a metaphysical one. Its most famous articulation comes from Link (1998). Imagine that the following sentence is true under the collective construal:

(183) Link’s daughters made a mess in the living room.

For (183) to be understood, says the set-based theory, it needs to be paraphrased as (184):

(184) The set of Link’s daughters made a mess in the living room.

But note that (184) entails (185):

(185) A set made a mess in the living room.

Herein lies a basic problem. For after all, a set is an abstract entity. How could it do something as concrete as make a mess in a room?

Link’s complaint, in other words, is that sets are too abstract an object to serve as a mediator between a plurality and a predicate3_{. Instead, he argues for a theory}

based on the notion of sums (also known as a mereological theory). The intuition behind sums is that of standard part/whole relations. Take, for instance, a book. A book is an entity; it is normal to talk of it as a singular, and to conceive of it as a “thing”. But a book also has parts: a cover, and pages. If I open a book and point to a page, I can talk of the page as an entity on its own. The book and its page may have different properties; the book, for instance, may be heavy, even though the page 3_{Some researchers agree that (185) is entailed by (183) but do not accept that this is a problem.}

See Landman (1989) for an example of a response to Link which argues thatLink’s daughters is not concrete in the same sense that each of the daughters is.

is light. The book and page, then, are examples of how two things in the world can be conceived of as entities, yet stand in a part/whole relation to each other.

The sum-based theory of plurality extends this view to all elements. If we can talk of an object αand another objectβ, we can talk of their sumαtβ. αandβ are parts ofαtβ, butαtβis itself a thing in the world. It is possible to attribute things to it. For example, Andrea and Amy are entities, and therefore AndreatAmy is also an entity4. This entity may have properties that neither Andrea or Amy have; for example, the property of meeting. Thus, instead of plural entities referring to sets, they are taken to refer to sums.

It is worth noting, however, that the metaphysical assumptions of the sum theory are not entirely innocuous. Oliver and Smiley (2001) point out, for instance, that if the sum relation is taken to be a purely nominalistic relation, defined only in terms of spatio-temportal coincidence, unwelcome results arise. For instance, for any particular point in time, the sum of Russell and Whitehead coincides spatially with the sum of the molecules that comprise Russell and Whitehead. But that means that the following inference holds5_:

(186) a. Russell and Whitehead are logicians. ⇒

b. _logicians(RusseltW hitehead) ⇒

c. _logicians(t{x:x is a molecule & x≤ RusselltW hitehead}) ⇒

d. The molecules that comprised Russell and Whitehead are logicians.

But this inference is clearly invalid. Oliver and Smiley argue that this means that a notion of sum that is not purely spatiotemporal must be adopted; but going down this road may lead right back to the type of statements over abstractions it was designed to avoid. A sum-theorist might reply that the problem is with the treatment of the 4_{Perhaps the first application of the sum operator to discrete individuals such as humans in this}

manner can be found in Massey (1976).

predicate _logicians; perhaps it does not distribute down to the level of molecules. But such a solution requires a mechanism that will let predicates tell apart different types of parts, which ends up reintroducing the same problem.

Beyond the metaphysical point, there is an additional attraction to the sum-based theories in that there is no type-distinction between parts and their sums. While most sum-based theories make a sortal distinction between atoms (which have no smaller parts) and non-atoms (which do), both are of type e. Set-based theories, however, face an option; they can assign singulars type e and plurals type he, ti, which means that predicates such asdance, which can either apply to singulars or plurals, need to be construed as ambiguous or otherwise type-shifted. Otherwise, individuals are also taken to be singleton sets, which seems like an unintuitive complexity for sentences not involving plurals6_{. Sum-based theories do not need to posit such an ambiguity.}

Generally, sum-based theories are quite similar to set-based theories in their im- plementation; most results formulated in one type of theory are easily carried over to the other. Because of this, there are those who deny that there is a substantial dif- ference between the two, including most critics of both views, and also some of those who adopt them, such as Schwarzschild (1996) and Landman (2000). Nonetheless, sum-based theories are probably the most common today. In addition to the highly influential work by Link, other sum-based theories include Hoeksema (1988), Krifka (1990), Moltmann (1997), Winter (2002) and Landman (2000), which will form the basis of my own formal framework.