What is Classical Mereology?

Classical mereology is a formal theory of the part-whole relation, essentially involving a notion of mereological fusion, or sum. There are various different definitions of fusion in the literature, and various axiomatizations for classical mereology. Though the equivalence of the definitions of fusion is provable from axiom sets, the definitions are not logically equivalent, and, hence, are not interchangeable when laying down the axioms. We examine the relations between the main definitions of fusion and correct some technical errors in prominent discussions of the axiomatization of mereology. We show the equivalence of four different ways to axiomatize classical mereology, using three different notions of fusion. We also clarify the connection between classical mereology and complete Boolean algebra by giving two “neutral” axiom sets which can be supplemented by one or the other of two simple axioms to yield the full theories; one of these uses a notion of “strong complement” that helps explicate the connections between the theories. Note: This paper is a draft of “What is Classical Mereology?” in Journal of Philosophical Logic, c © Springer 2008. The original publication is available at www.springerlink.com, or at http://dx.doi.org/10.1007/s10992-008-9092-4 The idea of amereological fusion ormereological sum has become a commonplace in philosophical literature. Those who use the notion casually may do so without giving an exact definition. Some very rough explanation like “the fusion of some things is what you get when you put them together” is enough for some purposes. Something more substantial and What is classical mereology? DRAFT 2 precise is often wanted, however, for there are appeals to such principles as If every proper part of x is part of y, and every proper part of y is part of x, then x = y which are supposed to follow from some axioms in which the notion of fusion plays a central role. When definitions of fusion are given, they are not always the same. In fact, there are many slightly different definitions, of which two are quite common in the literature. These two definitions are often run together, but they are logically distinct. It is true that once we have “the correct” axioms in place, or any equivalent set of axioms, then the two definitions can be shown to be equivalent (i.e., their equivalence logically follows from those axioms). But when actually giving an axiomatization intended to yield “the correct” theory, which we will call classical mereology, the difference matters. There is fairly universal agreement on what the theorems of classical mereology ought to be—not on whether they are true, but on what they are. (Roughly, they are the same as the theorems derived from the axioms for complete Boolean algebra, except without a zero element.) The difference between the definitions of fusion makes for a difference in how one can get those theorems. It turns out that Peter Simons’ system SC in Parts does not suffice to get the desired theorems. Casati and Varzi’s definition of system GEM in Parts and Places suffers from an unintended ambiguity; on one disambiguation, we do get the desired theorems, on the other, we do not. These mistakes, first addressed in the literature by Carsten Potow in [10], are fairly easily fixed, however, once they are noticed: we will see that, as in [10], one way is to replace a weak “supplementation” axiom with a stronger one; we also show that another way is to replace the “weaker” definition of fusion with the other one.1 Wewill also consider an alternative axiom set that does not directly use either common definition of fusion; rather it splits a fusion existence axiom into two parts and uses the notion of minimal upper bound in place of fusion, gaining, perhaps, in intuitive appeal what it loses in brevity. Using a related axiom set, we will give a very clear picture of the close connection 1I wish to express my gratitude to Pontow for very useful comments on an earlier draft of this paper. What is classical mereology? DRAFT 3 between mereologies and complete Boolean algebras. The connection was known to Tarski (see [15] and [16]) and has been given a recent treatment in [11]. The treatment given here differs from others in that it crucially uses the concept of a “strong complement” in the axiomatizations, which sheds an alternative light on the roles of the “supplementation” axioms of mereology and the complement and “distribution” axioms of Boolean algebra. Along the way, we will correct an axiomatization found in the work of Fred Landman and in the work of Manfred Krifka that uses the notion of minimal upper bound. We presuppose no substantial knowledge of mereology or Boolean algebra, and the technical arguments are intended to be accessible to nonspecialists interested in a fairly self-contained, careful treatment. The paper is almost purely technical in nature; we do not address the question of whether classical mereology is a plausible theory. Part One: Definitions of fusion We begin with an explication of the devices needed for a formal language in which classical mereology might be expressed. Suppose we have a first-order (or higher) language2 that includes a special 2-place predicate ≤, meant to represent “is part of” or “is a part of”.3 Thus ∀x(Cat(x) → ∃y(Tail(y) ∧ y ≤ x)) says that every cat has a tail as part. For any terms s 2Strictly speaking, when we get to axioms and theorems, we will be interested in not a single axiomatic theory of mereology, but rather any system that results from introducing a new relation symbol ≤ into a system by (augmenting its language and) adding certain axioms and axiom-schemes. For our model examples, below, we assume we are working in pure unrestricted mereology: pure, meaning ≤ is the only non-logical expression in the language; unrestricted meaning that the quantifiers of the mereology axioms are unrestricted. For most of our purposes, we may assume unrestrictedness (the uniform imposition of explicit restriction being a routine matter) and what other expressions there are in the language will not matter. For informal examples, we will often assume our language contains predicates like ‘is a cat’ and ‘is a dog’. The availability of set-theory or higher-order logical devices in the language will be addressed below. 3Sharvy suggests in [13] (cf. [12]) that “is part of” and “is a part of” have rather different meanings, but classical mereology treats a single relation. What is classical mereology? DRAFT 4 and t, pick a variable v not free in s or in t and stipulate:4 s ◦ t abbreviates ∃v(v ≤ s ∧ v ≤ t) s o t abbreviates ¬ s ◦ t s t abbreviates s ≤ t ∧ ¬ s = t ‘101 ◦ 102’ can be paraphrased as ‘Rooms 101 and 102 have a common part’ or ‘Rooms 101 and 102 overlap.’ ‘101 o 102’ says that they do not overlap, or are disjoint, and ‘102 101’ says that Room 102 is a proper part of room 101; it is a part, but is not the whole.5 One could, instead, take ◦ or o or as primitive, and define ≤ and the others in terms of the primitive, but this substantially affects the axiomatization, as we will see later in this paper. It seems most natural to take ≤ as primitive. Schematic fusion-definitions We now look at the two common definitions of fusion. According to the first, roughly put, a fusion of the F’s is a thing x such that for every thing y, y overlaps x iff y overlaps one of the F’s. We will first look at a way of formalizing this that uses an open sentence Fx in place of the notion of “the F’s.” Wewill use the expression φ(y) to stand for anywff (well-formed formula) whose free variables may or may not include y, and so on for any variable. For any variable x, any wff φ(x), and any term t distinct from the variable x, find a variable y that does not occur free in φ(x) or in t, and stipulate that SCHEMATIC TYPE-1 FUSION Fu1(t, [x | φ(x)]) abbreviates ∀y(y ◦ t ↔ ∃x(φ(x) ∧ y ◦ x)) (read “t is a fusion of the first type, of the condition φ(x)” or, perhaps, “t fuses the φ’s”). For example, 4We will use lower-case italic letters (s, t, x, etc.) as meta-language variables meant to stand for terms and variables of the object language; the object language will be in sans-serif font (x ≤ y, etc.). We will be a little loose with use/mention. 5One might complain about the fact that in formal mereology, everything is treated as part of itself. The usual reply is that this is a mere formal convenience, eliminable in principle. What is classical mereology? DRAFT 5 Fu1(a, [x | ∃z(Cat(z) ∧ Loves(x, z))]) abbreviates ∀y(y ◦ a ↔ ∃x(∃z(Cat(z) ∧ Loves(x, z)) ∧ y ◦ x)) and says that a fuses (in the first sense) the things that love a cat. Note that we do not, with our notation, take for granted that there is at most one fusion of cat-lovers. Roughly put, “fusing” is a relation between a thing (the fusion) and a condition, or between a thing (the fusion) and some things (that get “fused”). But since we here assume only a first-order language, no such “relation” can be explicitly mentioned; its logical type would be beyond the type associated with first-order relation symbols. Note, for example, that though we know how to say a fuses the cats, it is not immediately evident how we might say that a fuses some cats: we want something like ∃ψ (∃x ψ(x) ∧ ∀x(ψ(x) → Cat(x)) ∧ ∀y(y ◦ a ↔ ∃x(ψ(x) ∧ y ◦ x))) but, of course, this is nonsense, unless ‘ψ’ here is being used as a secondorder or plural variable; we will consider this possibility in more detail momentarily. Further, the expression “the fusion of cat-lovers” has to be justified by showing that our axioms entail that if some things are fused by z and also by w, then z = w. Yet, since we are using schemes in a standard first-order setting, we have another kind of uniqueness for free. If every φ is a ψ, and vice-versa, then anything that fuses the φ’s fuses the ψ’s: ∀x(φ(x) ↔ ψ(x)) → ∀z (Fu1(z, [x | φ(x)]) ↔ Fu1(z, [x |ψ(x)])) For the second notion of fusion: for any φ(x), t, x, as above (in the following we will often suppress qualifications like these), find y as above and stipulate TYPE-2 FUSION Fu2(t, [x | φ(x)]) abbreviates ∀x(φ(x) → x ≤ t) ∧ ∀y(y ≤ t → ∃x(φ(x) ∧ y ◦ x)) (“t is a fusion of the second type, of φ(x)”). Roughly the second notion of fusion is the one used by Alfred Tarski in What is classical mereology? DRAFT 6 [16] and David Lewis in [7]. The former notion is used by Simons [14], (see his SD9 on p. 37) and Casati and Varzi [2], p. 46. Casati and Varzi seem to assume that the difference does not matter in their reference to Tarski’s system on p. 47. Schematic vs. non-schematic We had to say “roughly” in connection with Tarski and Lewis because their definitions are non-schematic. It is possible, and sometimes desirable6, to use sets, second-order quantification, plural quantification, or some other auxilliary device in place of the schematic [x | φ(x)] that we used, to give definitions of fusion to similar effect. E.g., if we were helping ourselves to set theory, then we would define Type-2 fusion like this: SET-THEORETIC TYPE-2 FUSION Fu2(t, s) abbreviates ∀x(x ∈ s → x ≤ t) ∧ ∀y(y ≤ t → ∃x(x ∈ s ∧ y ◦ x)) (Tarski gives an obviously equivalent definition of what is called ‘sum’, in the translation, in [16].) Going the plural route, Lewis would replace ‘x ∈ s’ with ‘x is one of Xs’; one could also aim to get the intended effect using monadic second-order variables. In the case of sets, it is common and natural to take the quantifiers in the mereology axioms (formulated in a language that contains both ≤ and ∈) to be restricted to a set (and thus to give a single axiom of fusion-existence instead of an axiom scheme). To see the expressive power of the use of auxilliaries, note that it is easy to say that a is a set-theoretic type-1 fusion of a set of cats: ∃ψ (Set(ψ) ∧ ∃x x ∈ ψ ∧ ∀x(x ∈ ψ → Cat(x)) ∧ ∀y(y ◦ a ↔ ∃x(x ∈ ψ ∧ y ◦ x))) with ‘ψ’ just another first-order variable. Using auxilliaries, we get an “explicit” definition of the fusion relationship, as in something of the form “for all x and y, x fuses y just in case. . . ” or of the form “for any x and any Ys, x fuses Ys just in case. . . ” In the case 6And sometimes not desirable. E.g., the nominalist might wish to to avoid commitment to sets in defining fusions; also, one may wish to consider what happens when unrestricted fusion axiom-schemes are added to something else, like an already given first-order theory, e.g., a modal formal language, or set theory. Cf. Uzquiano’s discussion of the difficulties of combining set theory and mereology, in [17]. What is classical mereology? DRAFT 7 of set theory, the fusion relationship acquires the logical type of a standard relation between objects: fuser and fused are both objects (things in the range of the first-order quantifiers). With plural logic, the logical type is of a relation between an object and some objects. As we noted, with a schematic definition of fusion, no such relation is even hinted at (except perhaps in our abbreviatory conventions), and no “explicit” definition is possible: there is nothing to put in the blank in “for all x and all , x fuses just in case. . . ”7 Thus we have a second kind of ambiguity in the notion of mereological fusion, among the purely schematic and alternative non-schematic versions. Fortunately, most of the issues we discuss arise in parallel for all of these alternatives, so, for our purposes, it usually does not matter which is chosen. Informally, we will ignore the differences among the schematic, set-theoretic, and plural versions, when the differences do not matter. Formally, we will finesse the issue by adopting the notation Fu2(t, φx) in place of the schematic Fu2(t, [x | φ(x)]) or the set theoretic Fu2(t, φ) (where φ is taken as a first-order variable whose range includes sets). Officially, φx is an abbreviation to be unabbreviated differently according to whether one wants to proceed schematically or by sets, or by plural variables, etc. Similarly for Fu1. For example, Fu1(t, φx) is always partially unabbreviated as ∀y(y ◦ t ↔ ∃x(φx ∧ y ◦ x)), but the occurrence of ‘φx’ in this will be (partially) unabbreviated as ‘φ(x)’ on a schematic treatment, and (completely) unabbreviated as ‘x ∈ φ’ on a set-theoretic treatment (with φ a first-order variable), and as ‘x is one of the φs’ on a plural variable treatment (with φs a plural variable), and so forth. 7It is worth noting that even if we use auxilliaries to define fusion, schemes will still be invoked when the auxilliary theory is axiomatized (as in the Separation scheme of set theory, or the Comprehension schemes of plural and second-order logic) and the resulting notions of fusion will thus logically link back to these schemes. Basically, utilitzing set theory, our schematic ‘[x | φ(x)]’ will be linked to ‘{x : φ(x)}’; utilizing plural quantification, with ‘Xs’ a plural variable, it gets linked to ‘Xs such that x is one of them if and only if φ(x)’. What is classical mereology? DRAFT 8 Minimal Upper Bounds Now, it is easy enough to say ‘z is a fusion of all lovers of cats’, but if we are required to spell out (in English) the defined notion in terms of the partwhole relation, we are left with quite a mouthful; without a lot of training, it is far from easy to understand just what is being said. There is a perhaps more intuitive notion that, in conjunction with the right axioms, is basically equivalent: the notion of a minimal upper bound. It is rather intuitive that if z is the fusion of all cats, then, whatever else it is, it has every cat as a part. That is to say, it is an “upper bound” on the cats: ∀x(Cat(x) → x ≤ z) But it is not just any upper bound. According to classical mereology, there is some object which is the fusion of all objects, call it the universe, and of which everything is a part. Thus, every cat is part of the universe, so the universe is an upper bound on the cats. But the fusion of cats should be something smaller than the universe; no dogs should be part of it, for example. What’s special about z, the fusion of the cats, is that it is aminimal upper bound (mub), a part of any upper bound on the cats: ∀w((Cat(x) → x ≤ w) → z ≤ w) For a compact notation for mubs, stipulate MIN UPPER BOUND Mub(t, φx) abbreviates ∀x(φx → x ≤ t) ∧ ∀w(∀x(φx → x ≤ w) → t ≤ w) We use the term minimal instead of least so as not to build uniqueness into our very definition. The axiom of Anti-symmetry (see below) is enough, however, to guarantee that any mub of φx is identical with every mub of φx, so with Anti-symmetry in place, minimal amounts to (uniquely) least. (The terms supremum and join are sometimes used for formally the same notion.) We will eventually see that in classical mereology, for any φx, if, and only if, there is an x with φx, there is exactly one type-1 fusion of φx, exactly one type-2 fusion, and exactly one minimal upper bound, and they are all the same thing. Hence, once the right axioms are in place, one could use the notion of least upper bound (supremum, join) in place of fusion.8 8Acknowledgment is due to Tony Martin for directing my attention to the notion of least upper bound in connection with the notion of fusion; see footnote 16. What is classical mereology? DRAFT 9 The use of the notion of least upper bound in place of type-1 or type-2 fusion can be found in the formal linguistic literature in connection with the semantics of mass nouns and plurals, e.g., in Krifka [3], Landman [4] and [5], and Link [8]. Landman and Krifka intend to capture classical mereology with their axiomatizations, but they do not quite succeed, as we note below when we show how to use mubs in place of fusions to axiomatize classical mereology. Richard Sharvy uses the notion of least upper bound as his central fusion-like concept, but favors a notion of quasimereology, which is weaker than classical mereology; see p. 234 of [13] and footnote 8 of [12]. Part Two: Axiomatizations short of classical mereology Adopting nomenclature from Casati and Varzi9, let us have the systemM (Ground Mereology) be the set of axioms Reflexivity ∀x x ≤ x Anti-symmetry ∀x∀y((x ≤ y ∧ y ≤ x) → x = y) Transitivity ∀x∀y∀z(x ≤ y ∧ y ≤ z → x ≤ z) These say ≤ is (in mathematician’s parlance) a partial ordering. We will show shortly that the two notions of fusion are not equivalent, in the sense that we cannot derive, using first-order logic alone ∀z(Fu1(z, φx) ↔ Fu2(z, φx)) without further assumptions. In the presence of Transitivity, however, the right-to-left direction can be derived. The second type of fusion thus may be said to be the stronger notion of fusion. Fusion existence axioms Consider now the system GM1 that results from adding to M instances of a scheme (or, if one is using auxiliaries, a single axiom) asserting the existence of type-1 fusions. For any wff φx, if the variable z is not free in φx then