[FOM] Model theory for mereology, without sets?

[David NICOLAS]

Many thanks to Allen Hazen for his response.
Here is a follow-up (which no doubt contains some imprecisions).

A few words about my motivations.
- As I said, mereology was originally conceived by Lesniewski as an 
alternative to set theory.
(Of course, there are many different axiomatic systems of mereology, 
but it's often simpler to just talk of mereology.)
- As noted by Leonard & Goodman (1940), it is not easy to 
characterize the semantics of plural expressions in English in simple 
predicate logic. One way to do so is to use mereological sums, as in 
Leonard & Goodman's "calculus of individuals".
- Link (1983, 1998) has proposed a systematic analysis of sentences 
containing plurals, using an atomic mereology. His mereology is 
equivalent to a complete atomic Boolean algebra minus its zero. 
Moreover, monadic second order logic can be translated into it.
- Friends of plural logic or second order logic (e.g. Schein 1993, 
Oliver & Smiley 2001, Rayo 2002) have argued that mereological sums 
not powerful enough to capture the semantics of plurals. (I am 
currently critically examining their arguments.) Following Boolos, 
they have devised plural logics, i.e. extensions of first-order 
logic, where plural quantification and plural predication are taken 
as primitive. It turns out that monadic second order logic and 
first-order plural logic are inter-translatable. Moreover, when one 
develops a plural logic that has a single type of variables (namely, 
plural variables), the axioms are those of a complete atomic Boolean 
algebra minus its zero (cf. McKay 2006).

This raises several (speculative) questions:
- Is there a strict equivalence between such a plural logic (with 
only plural variables) and Link's atomic mereology?
 From an axiomatic point of view, it seems so. However:
- It is easy to enrich one's metalanguage with plurals. This offers 
means to develop a model theory of plural first order logic, without 
using sets. One takes the notion of ordered pair as primitive. A 
model is then given by some ordered pairs (rather than a set of them).
- But is it so easy to do the same kind of thing for an atomic 
mereology (without using sets not plurals)? A model would be given by 
a mereological sum of ordered pairs. Is this feasible, or is there a 
problem there?
   If this were feasible, could we conclude that plural logic and 
atomic mereology are equivalent? Or in more provocative words, that 
plural logic is atomic mereology in disguise?

David Nicolas
Institut Jean Nicod, CNRS