Pieces of mereology

Abstract

In this paper we will treat mereology as a theory of some structures that are not axiomatizable in an elementary language (one of the axioms will contain the predicate ‘belong’ (‘∈’) and we will use a variable ranging over the power set of the universe of the structure). A mereological structure is an ordered pair M = <M,⊑>, where M is a non-empty set and ⊑ is a binary relation in M, i.e., ⊑ is a subset of M × M. The relation ⊑ is a relation of being a mereological part (instead of ‘<x,y> ∈ ⊑’ we will write ‘x ⊑ y’ which will be read as “x is a part of y”). We formulate an axiomatization of mereological structures, different from Tarski’s axiomatization as presented in [10] (Tarski simplified Leśniewski’s axiomatization from [6]; cf. Remark 4). We prove that these axiomatizations are equivalent (see Theorem 1). Of course, these axiomatizations are definitionally equivalent to the very first axiomatization of mereology from [5], where the relation of being a proper part ⊏ is a primitive one.

Moreover, we will show that Simons’ “Classical Extensional Mereology” from [9] is essentially weaker than Leśniewski’s mereology (cf. Remark 6).