natural axioms for classical mereology

We present a new axiomatization of classical mereology in which the three components of the theory—ordering, composition, and decomposition principles—are neatly separated.The equivalence of our axiom systemwith other, more familiar systems is established by purely deductivemethods, along with additional results on the relative strengths of the composition and decomposition axioms of each theory. As a formal theory of parts and wholes, mereology should tell us three sorts of thing. It should say: (i) what sort of relation parthood is; (ii) what sorts of condition govern mereological composition, i.e., intuitively, what it takes to form a whole by “adding things” together; (iii) what sorts of condition govern mereological decomposition, i.e., intuitively, what happens when we “subtract things” from a given whole. Classical mereology—the theory stemming from the work of Leśniewski [3] and of Leonard and Goodman [2]—provides a clear answer to each of these questions. It answers (i) by taking parthood to be a partial order (i.e., a reflexive, transitive, antisymmetric relation); it answers (ii) by taking composition to be unrestricted (so that adding any number of things together, no matter how disparate and gerrymandered those thingsmight be, always yields a further thing); and its answers (iii) by taking decomposition to be fully subtractive (so that the mereological difference between any two things, when it exists, always leaves an exact remainder). Interestingly, however, none of the extant axiomatizations of classical mereology does the job explicitly. All are explicit about (ii), and all are (more or less) explicit about (i), modulo redundancies. When it comes to (iii), however, the answer usually comes as a theorem. That is, the axioms do not quite address the question directly but natural axioms for classical mereology 2 rather tell us something else, typically in the form of a supplementation principle, from which the answer follows. There is, of course, nothing wrong with this way of proceeding. Yet it would be more natural and theoretically more elegant if in each case the answer could be captured directly (and non-redundantly) by means of suitable axiom(s). In this note we offer a new axiomatization of classical mereology that does just that. In section 1, we introduce our axiom system and explain how the axioms give clear and natural answers to questions (i), (ii), and (iii). In section 2, we explore more deeply the connection between composition and decomposition, providing a formal argument to the effect that they are ‘two-sides of the same coin’.This serves to illustrate the connections between (ii) and (iii) displayed by our axiomatization. In Section 3 we prove the equivalence of our system with other, more familiar axiom systems by purely deductive methods. We conclude in Section 4 with some additional remarks on the relative strengths of the composition and decomposition axioms of each theory vis à vis our initial questions. 1 the axiom system cm We assume a standard first-order language with identity supplied with a distinguished binary predicate constant, P , to be interpreted as the parthood relation. The underlying logic is the classical predicate calculus. To introduce our axiom system for classical mereology, CM, we begin with some definitions. (D.1) PPxy ∶≡ Pxy ∧ x ≠ y Proper Parthood (D.2) Oxy ∶≡ ∃z(Pzx ∧ Pzy) Overlap (D.3) Fφx ∶≡ ∀z(φ → Pzx) ∧ ∀y(∀z(φ → Pzy) → Pxy) Fusion D.1 is a standard definition to the effect that a proper part is any part distinct from the whole.1 D.2 simply states that things overlap whenever they have at least one part in common. As for D.3, from the standpoint of mereology this is a slightly unusual definition of the fusion predicate, which is meant to capture the notion of something being composed of a specified collection of things.2 It is, however, straightforward and intuitive from the perspective of lattice theory and algebra. Where φ is any open formula with just z free, Fφx says that x is a minimal upper bound, relative to P , of the objects satisfyingφ: the first conjunct states that x is an upper bound of the φs, while the second states any (other) upper bound includes x as a part. 1The usual alternate definition, PPxy ∶≡ Pxy ∧ ¬Pyx, is equivalent in this system. 2For other definitions, see e.g. [1]. natural axioms for classical mereology 3