# [FOM] Model theory for mereology, without sets?

David NICOLAS dnicolas at gmx.net
Sun May 13 06:15:42 EDT 2007

```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

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