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| | Composition Principles | |
Let us now consider the second way of extending M mentioned at the beginning of Section 3. Just as we may want to regiment the behavior of P by means of decomposition principles that take us from a whole to its parts, we may look at composition principles that go in the opposite direction—from the parts to the whole. More generally, we may consider the idea that the domain of the theory ought to be closed under mereological operations of various sorts: not only mereological sums, but also products, differences, and more. 4.1 Upper BoundsConditions on composition are many. Beginning with the weakest, one may consider a principle to the effect that any pair of suitably related entities must underlap, i.e., have an upper bound:(P.11ξ) | ξ-Bound ξxy → ∃z(Pxz ∧ Pyz). | Exactly how ‘ξ’ should be construed is, of course, an important question by itself—a version of what van Inwagen (1987, 1990) calls the “Special Composition Question”. A natural choice would be to identify ξ with mereological overlap, the rationale being that such a relation establishes an important tie between what may count as two distinct parts of a larger whole. As we shall see (Section 4.5), with ξ so construed (P.11ξ) is indeed rather uncontroversial. By contrast, the most liberal choice would be to identify ξ with the universal relation, in which case (P.11ξ) would reduce to its consequent and assert the existence of an upper bound for any pair of entities x andy. An axiom of this sort was used, for instance, in Whitehead's (1919, 1920) mereology of events.[17] In any case, and regardless of any specific choice, it is apparent that (P.11ξ) does not express a strong condition on composition, as the consequent is trivially satisfied in any domain that includes a universal entity of which everything is part, or any entity sufficiently large to include both x and y as parts regardless of how they are related. 4.2 SumsA stronger condition would be to require that any pair of suitably related entities must have aminimal underlapper—something composed exactly of their parts and nothing else. This requirement is sometimes stated by saying that any suitable pair must have a mereological “sum”, or “fusion”,[18] though it is not immediately obvious how this requirement should be formulated. Consider the following definitions:(391) | Sum1[19] S1zxy =df ∀w(Pzw ↔ (Pxw ∧ Pyw)) | (392) | Sum2 S2zxy =df Pxz ∧ Pyz ∧ ∀w(Pwz → (Owx ∨ Owy)) | (393) | Sum3 S3zxy =df ∀w(Ozw ↔ (Owx ∨ Owy)) | (‘Sizxy’ may be read: ‘z is a sumi of x and y’. The first notion is found e.g. in Eberle 1967, Bostock 1979, and van Benthem 1983; the second in Tarski 1935 and Lewis 1991; the third in Needham 1981, Simons 1987, and Casati and Varzi 1999.) Then, for each i ∈ {1, 2, 3}, one could extend M by adding a corresponding axiom as follows, where again ξ specifies a suitable binary condition:(P.12ξ,i) | ξ-Sumi ξxy → ∃zSizxy. | In a way, (P.12ξ,1) would seem the obvious choice, corresponding to the idea that a sum of two objects is just a minimal upper bound of those objects relative to P (a partial ordering). However, this condition may be regarded as too weak to capture the intended notion of a mereological sum. For example, with ξ construed as overlap, (P.12ξ,1) is satisfied by the model of Figure 7, left: here z is a minimal upper bound of x and y, yet z hardly qualifies as a sum “made up” of x and y, since its parts include also a third, disjoint item w. Indeed, it is a simple fact about partial orderings that among finite models (P.12ξ,1) is equivalent to (P.11ξ), hence just as weak.By contrast, (P.12ξ,2) corresponds to a notion of sum that may seem too strong. In a way, it says—literally—that any pair of suitably ξ-related entities x and y compose something, in the sense already discussed in connection with (35): they have an upper bound all parts of which overlap either x or y. Thus, it rules out the model on the left of Figure 7, precisely because w is disjoint from both x and y. However, it also rules out the model on the right, which depicts a situation in which z may be viewed as an entity truly made up of x and y insofar as it is ultimately composed of atoms to be found either in x or in y. Of course, such a situation violates the Strong Supplementation principle (P.5), but that's precisely the sense in which (P.12ξ,2) may seem too strong: an anti-extensionalist might want to have a notion of sum that does not presuppose Strong Supplementation.The formulation in (P.12ξ,3) is the natural compromise. Informally, it says that for any pair of suitably ξ-related entities x and y there is something that overlaps exactly those things that overlap either x or y. This is strong enough to rule out the model on the left, but weak enough to be compatible with the model on the right. Note, however, that if the Strong Supplementation axiom (P.5) holds, then (P.12ξ,3) is equivalent to (P.12ξ,2). Moreover, it turns out that if the stronger Complementation axiom (P.6) holds, then all of these principles are trivially satisfied in any domain in which there is a universal entity: in that case, regardless of ξ, the sum of any two entities is just the complement of the difference between the complement of one minus the other. (Such is the strength of (P.6), a genuine cross between decomposition and composition principles.) - اقتباس :
- Figure 7. A sum1 that is not a sum3, and a sum3 that is not a sum2.
The intuitive idea behind these principles is in fact best appreciated in the presence of (P.5), hence extensionality, for in that case the relevant sums must be unique. Thus, consider the following definition, where i ∈ {1, 2, 3} and ‘℩’ is the definite descriptor):In the context of EM, each (P.12ξ,i) would then imply that the corresponding sum operator has all the “Boolean” properties one might expect (Breitkopf 1978). For example, as long as the arguments satisfy the relevant condition ξ,[20] each +i is idempotent, commutative, and associative,(41) | x = x +i x | (42) | x +i y = y +i x | (43) | x +i (y +i z) = (x +i y) +i z, | and well-behaved with respect to parthood:(44) | Px(x +i y) | (45) | Pxy → Px(y +i z) | (46) | P(x +i y)z → Pxz | (47) | Pxy ↔ x +i y = y. | (Note that (47) would warrant defining ‘P’ in terms of ‘+i’, treated as a primitive. For i=3, this was actually the option endorsed in Leonard 1930: 187ff.)Indeed, here there is room for further developments. For example, just as the principles in (P.12ξ,i) assert the existence of a minimal underlapper for any pair of suitably related entities, one may at this point want to assert the existence of a maximal overlapper, i.e., not a “sum” but a “product” of those entities. In the present context, such an additional claim can be expressed by the following principle:(P.13ξ) | ξ-Product ξxy → ∃zRzxy, | where(48) | Product Rzxy =df ∀w(Pwz ↔ (Pwx ∧ Pwy)), | and ‘ξ’ is at least as strong as ‘O’ (unless one assumes the Bottom principle (P.10)). In EM one could then introduce the corresponding binary operator,and it turns out that, again, such an operator would have the properties one might expect. For example, as long as the arguments satisfy the relevant condition ξ, × is idempotent, commutative, and associative, and it interacts with each +i in conformity with the usual distribution laws:(50) | x +i (y × z) = (x +i y) × (x +i z) | (51) | x × (y +i z) = (x × y) +i (x × z). | Now, obviously (P.13ξ) does not qualify as a composition principle in the main sense that we have been considering here, i.e., as a principle that yields a whole out of suitably ξ-related parts. Still, in a derivative sense it does. It asserts the existence of a whole composed of parts that are sharedby suitably related entities. Be that as it may, it should be noted that such an additional principle is not innocuous unless ‘ξ’ expresses a condition stronger than mere overlap. For instance, we have said that overlap may be a natural option if one is unwilling to countenance arbitrary scattered sums. It would not, however, be enough to avoid embracing scattered products. Think of two C-shaped objects overlapping at both extremities; their sum would be a one-piece O-shaped object, but their product would consist of two disjoint, separate parts (Bostock 1979: 125). Moreover, and independently, if ξ were just overlap, then (P.13ξ) would be unacceptable for anyone unwilling to embrace mereological extensionality. For it turns out that the Strong Supplementation principle (P.5) would then be derivable from the weaker Supplementation principle (P.4) using only the partial ordering axioms for ‘P’ (in fact, using only Reflexivity and Transitivity; see Simons 1987: 30f). In other words, unless ‘ξ’ expresses a condition stronger than overlap, MM cum (P.13ξ) would automatically include EM. This is perhaps even more remarkable, for on first thought the existence of products would seem to have nothing to do with matters of decomposition, let alone a decomposition principle that is committed to extensionality. On second thought, however, mereological extensionality is really a double-barreled thesis: it says that two wholes cannot be decomposed into the same proper parts but also, by the same token, that two wholes cannot be composed out of the same proper parts. So it is not entirely surprising that as long as proper parthood is well behaved, as per (P.4), extensionality might pop up like this in the presence of substantive composition principles. (It is, however, noteworthy that it already pops up as soon as (P.4) is combined with a seemingly innocent thesis such as the existence of products, so the anti-extensionalist should keep that in mind.) | |
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