4.3 Infinitary Bounds and Sums

One can get even stronger composition principles by considering infinitary bounds and sums. For example, (P.11ξ) can be generalized to a principle to the effect that any non-empty set of (two or more) entities satisfying a suitable condition ψ has an upper bound. Strictly speaking, there is a difficulty in expressing such a principle in a standard first-order language. Some early theories, such as those of Tarski (1929) and Leonard and Goodman (1940), require explicit quantification over sets (see Niebergall 2009a, 2009b; Goodman produced a set-free version of the calculus of individuals in 1951). Others, such as Lewis's (1991), resort to the machinery of plural quantification of Boolos (1984). One can, however, avoid all this and achieve a sufficient degree of generality by relying on an axiom schema where sets are identified by predicates or open formulas. Since an ordinary first-order language has a denumerable supply of open formulas, at most denumerably many sets (in any given domain) can be specified in this way. But for most purposes this limitation is negligible, as normally we are only interested in those sets of objects that we are able to specify. Thus, for most purposes the following axiom schema will do, where ‘φ’ is any formula in the language and ‘ψ’ expresses the condition in question:

(P.14ψ)

General ψ-Bound  (∃wφw ∧ ∀w(φw → ψw)) → ∃z∀w(φw → Pwz).

(The first conjunct in the antecedent is simply to guarantee that ‘φ’ picks out a non-empty set, while in the consequent the variable ‘z’ is assumed not to occur free in ‘ψ’.) The three binary sum axioms corresponding to the schema in (P.12ξ,i) can be strengthened in a similar fashion as follows:

(P.15ψ,i)

General ψ-Sumi  (∃wφw ∧ ∀w(φw → ψw)) → ∃zSizφw,

where

(521)	General Sum1[21]  S1zφw =df ∀v(Pzv ↔ ∀w(φw → Pwv))
(522)	General Sum2  S2zφw =df ∀w(φw → Pwz) ∧ ∀v(Pvz → ∃w(φw ∧ Ovw))
(523)	General Sum3  S3zφw =df ∀v(Ovz ↔ ∃w(φw ∧ Ovw)).

(Here, ‘Sizφw’ may be read: ‘z is a sumi of every w such that φw’ and, again, ‘z’ and ‘v’ are assumed not to occur free in φ; similar restrictions will apply below.) Thus, each (P.15ψ,i) says that if there are some φ-ers, and if every φ-er satisfies condition ψ, then the φ-ers have a sum of the relevant type. It can be checked that each variant of (P.15ψ,i) includes the corresponding finitely principle (P.12ψ,i) as a special case, taking ‘φw’ to be the formula ‘w=x ∨ w=y’ and ‘ψw’ the condition ‘(w=x → ξwy) ∧ (w=y → ξxw)’. And, again, it turns out that in the presence of Strong Supplementation, (P.15ψ,2) and (P.15ψ,3) are equivalent.
One could also consider here a generalized version of the Product principle (P.13ξ), asserting the conditional existence of a maximal common overlapper—a common “nucleus”, in the terminology of Leonard and Goodman (1940)—for any non-empty set of entities satisfying a suitable condition. Adapting from Goodman (1951: 37), such a principle could be stated as follows:

(P.16ψ)

General ψ-Product  (∃wφw ∧ ∀w(φw → ψw)) → ∃zRzφw,

where

(53)	General Product  Rzφw =df ∀v(Pvz ↔ ∀w(φw → Pvw))

and ‘ψw’ expresses a condition at least as strong as ‘∀x(φx → Owx)’ (again, unless one assumes the Bottom principle (P.10)). This principle includes the finitary version (P.13ξ) as a special case, taking ‘φw’ and ‘ψw’ as above, so the remarks we made in connection with the latter apply here. An additional remark, however, is in order. For there is a sense in which (P.16ψ) might be thought to be redundant in the presence of the infinitary sum principles in (P.15ψ,i). Intuitively, a maximal common overlapper (i.e., a product) of a set of overlapping entities is simply a minimal underlapper (a sum) of their common parts; that is precisely the sense in which a product principle qualifies as a composition principle. Thus, intuitively, each of the infinitary sum principles above should have a substitution instance that yields (P.16ψ) as a theorem, at least when ‘ψw’ is as strong as indicated. However, it turns out that this is not generally the case unless one assumes extensionality. In particular, it is easy to see that (P.15ψ,3) does not generally imply (P.16ψ), for it may not even imply the binary version (P.13ξ). This can be verified by taking ‘ξxy’ and ‘ψw’ to express just the requirement of overlap, i.e., the conditions ‘Oxy’ and ‘∀x(φx → Owx)’, respectively, and considering again the non-extensional model diagrammed in Figure 4. In that model, x and y do not have a product, since neither is part of the other and neither z nor wincludes the other as a part. Thus, (P.13ξ) fails, which is to say that (P.16ψ) fails when ‘φ’ picks out the set {x, y}; yet (P.15ψ,3) holds, for both z and w are things that overlap exactly those things that overlap some common part of the φ-ers, i.e., of x and y.
In the literature, this fact has been neglected until recently (Pontow 2004). It is, nonetheless, of major significance for a full understanding of (the limits of) non-extensional mereologies. As we shall see in the next section, it is also important when it comes to the axiomatic structure of mereology, including the axiomatics of the most classical theor

4.4 Unrestricted Composition

The strongest versions of all these composition principles are obtained by asserting them as axiom schemas holding for every condition ψ, i.e., effectively, by foregoing any reference to ψ altogether. Formally this amounts in each case to dropping the second conjunct of the antecedent, i.e., to asserting the schema expressed by the relevant consequent with the only proviso that there are some φ-ers. In particular, the following schema is the unrestricted version of (P.15ψ,i), to the effect that every specifiable non-empty set of entities has a sumi:

(P.15i)

Unrestricted Sumi  ∃wφw → ∃zSizφw.

For i=3, the extension of EM obtained by adding every instance of this schema has a distinguished pedigree and is known in the literature as General Extensional Mereology, orGEM. It corresponds to the classical systems of Leśniewski and of Leonard and Goodman, modulo the underlying logic and choice of primitives. The same theory can be obtained by extending EM with (P.152) instead, for in the presence of extensionality the two schemas are equivalent. Indeed, it turns out that the latter axiomatization is somewhat redundant: given just Transitivity and Supplementation, Unrestricted Sum2 entails all the other axioms, i.e., GEM is the same theory as (P.2) + (P.4) + (P.152). By contrast, extending EM with (P.151) would result in a weaker theory (Figure 8), though one can still get the full strength of GEM with the help of additional axioms. For example, Hovda (2009) shows that the following will do:

(P.17)

Filtration  (S1zφw ∧ Pxz) → ∃w(φw ∧ Owx).

(in which case, again, Transitivity and Supplementation would suffice, i.e., GEM = (P.2) + (P.4) + (P.151) + (P.17)). For other ways of axiomatizatizing of GEM using (P.151), see e.g. Link (1983) and Landman (1991) (and, again, Hovda 2009). See also Sharvy (1980, 1983), where the extension of M obtained by adding (P.151) is called a “quasi-mereology”.

اقتباس :

Figure 8. A model of EM + (P.152) but not of GEM.

GEM is a powerful theory, and it was meant to be so by its nominalistic forerunners, who were thinking of mereology as a good alternative to set theory. It is also decidable (Tsai 2013a), whereas for example, M, MM, and EM, and many extensions thereof turn out to be undecidable. (For a comprehensive picture of decidability in mereology, see also Tsai 2009, 2011, 2013b.) Just how powerful is GEM? To answer this question, let us focus on the classical formulation based on (P.153) and consider the following generalized sum operator:

(54)	General Sum  σxφx =df ℩zS3zφw.

Then (P.153) and (P.5) can be simplified to a single axiom schema:

(P.18)

Unique Unrestricted Sum3  ∃xφx → ∃z(z=σxφx),

and we can introduce the following definitions:

(55)	Sum  x + y =df σz(Pzx ∨ Pzy)
(56)	Product  x × y =df σz(Pzx ∧ Pzy)
(57)	Difference  x − y =df σz(Pzx ∧ Dzy)
(58)	Complement  ~x =df σzDzx
(59)	Universe  U =df σzPzz.

Note that (55) and (56) yield the binary operators defined in (403) and (49) as special cases. Moreover, in GEM the General ψ-Product principle (P.16ψ) is also derivable as a theorem, with ‘ψ’ as weak as the requirement of mutual overlap, and we can introduce a corresponding functor as follows:

(60)	General Product  πxφx =df σz∀x(φx → Pzx).

The full strength of the theory can then be appreciated by considering that its models are closed under each of these functors, modulo the satisfiability of the relevant conditions. To be explicit: the condition ‘DzU’ is unsatisfiable, so U cannot have a complement. Likewise products are defined only for overlappers and differences only for pairs that leave a remainder. Otherwise, however, (55)–(60) yield perfectly well-behaved functors. Since such functors are the natural mereological analogues of the familiar set-theoretic operators, with ‘σ’ in place of set abstraction, it follows that the parthood relation axiomatized by GEM has essentially the same properties as the inclusion relation in standard set theory. More precisely, it is isomorphic to the inclusion relation restricted to the set of all non-empty subsets of a given set, which is to say a complete Boolean algebra with the zero element removed—a result that can be traced back to Tarski (1935: n. 4) and first proved in Grzegorczyk (1955: §4).[22]
There are other equivalent formulations of GEM that are noteworthy. For instance, it is a theorem of every extensional mereology that parthood amounts to inclusion of overlappers:

(61)	Pxy ↔ ∀z(Ozx → Ozy).

This means that in an extensional mereology ‘O’ could be used as a primitive and ‘P’ defined accordingly, as in Goodman (1951), and it can be checked that the theory defined by postulating (61) together with the Unrestricted Sum principle (P.153) and the Antisymmetry axiom (P.3) is equivalent to GEM (Eberle 1967). Another elegant axiomatization of GEM, due to an earlier work of Tarski (1929),[23] is obtained by taking just the Transitivity axiom (P.2) together with the Sum2-analogue of the Unique Unrestricted Sum axiom (P.18). By contrast, it bears emphasis that the result of adding (P.153) to MM is not equivalent to GEM, contrary to the “standard” characterization given by Simons (1987: 37) and inherited by much literature that followed, including Casati and Varzi (1999) and the first edition of this entry.[24] This follows immediately from Pontow's (2004) counterexample mentioned at the end of Section 4.3, since the non-extensional model in Figure 4 satisfies (P.153), and was first noted in Pietruszczak (2000, n. 12). More generally, in Section 4.2 we have mentioned that in the presence of the binary Product postulate (P.13ξ), with ξ construed as overlap, the Strong Supplementation axiom (P.5) follows from the weaker Supplementation axiom (P.4). However, the model shows that the postulate is not implied by (P.153) any more than it is implied by its restricted variants (P.15ψ,3). Apart from its relevance to the proper characterization of GEM, this result is worth stressing also philosophically, for it means that (P.153) is by itself too weak to generate a sum out of any specifiable set of objects. In other words, fully unrestricted composition calls for extensionality, on pain of giving up both supplementation principles. The anti-extensionalist should therefore keep that in mind. (On the other hand, a friend of extensionality may welcome this result as an argument in favor of adopting (P.152) instead of (P.153), for we have already noted that such a way of sanctioning unrestricted composition turns out to be enough, in MM, to entail Strong Supplementation along with the existence of all products and, with them, of all sums; see Varzi 2009, with discussion in Rea 2010 and Cotnoir 2016 . In this sense, the standard way of characterizing composition given in (35), on which (P.152) is based, is not as neutral as it might seem. On this and related matters, indicating that the axiomatic path to “classical extensional mereology” is no straightforward business, see also Hovda 2009 and Gruszczyński and Pietruszczak 2014.)
Would we get a full Boolean algebra by supplementing GEM with the Bottom axiom (P.10), i.e., by positing the mereological equivalent of the empty set? One immediate way to answer this question is in the affirmative, but only in a trivial sense: we have already seen in Section 3.4 that, under the axioms of MM, (P.10) only admits of degenerate one-element models. Such is the might of the null item. On the other hand, suppose we rely on the “non-trivial” notions of genuine parthood and genuine overlap defined in (37)–(38). And suppose we introduce a corresponding family of “non-trivial” operators for sum, product, etc. Then it can be shown that the theory obtained from GEM by adding (P.10) and replacing (P.5) and (P.153) with the following non-trivial variants:

(P.5G)	Genuine Strong Supplementation  ¬Pyx → ∃z(GPzy ∧ ¬GOzx)
(P.153G)	Genuine Unrestricted Sum3  ∃wφw → ∃z∀v(GOzv ↔ ∃w(φw ∧ GOwv))

is indeed a full Boolean algebra under the new operators (Pontow and Schubert 2006). This shows that, mathematically, mereology does indeed have all the resources to stand as a robust and yet nominalistically acceptable alternative to set theory, the real source of difference being the attitude towards the nature of singletons (as already emphasized by Leśniewski 1916 and eventually clarified in Lewis 1991). As already mentioned, however, from a philosophical perspective the Bottom axiom is by no means a favorite option. The null item would have to exist “nowhere and nowhen” (as Geach 1949: 522 put it), or perhaps “everywhere and everywhen” (as in Efird and Stoneham 2005), and that is hard to swallow. One may try to justify the gulp in various ways, perhaps by construing the null item as a non-existing individual (Bunge 1966), as a Meinongian object lacking all nuclear properties (Giraud 2013), as an Heideggerian nothing that nothings (Priest 2014a and 2014b: §6.13), or as the ultimate incarnation of divine omnipresent simplicity (Hudson 2006b, 2009). But few philosophers would be willing to go ahead and swallow for the sole purpose of neatening up the algebra.
Finally, it is worth recalling that the assumption of atomism generally allows for significant simplifications in the axiomatics of mereology. For instance, we have already seen that AEM can be simplified by subsuming (P.5) and (P.7) under a single Atomistic Supplementation principle, (P.5′). Likewise, it is easy to see that GEM is compatible with the assumption of Atomicity (just consider the one-element model), and the resulting theory has some attractive features. In particular, it turns out that AGEM can be simplified by replacing any of the Unrestricted Sum postulates in (P.15i) with the more perspicuous

(P.15i′)

Atomistic Sumi  ∃wφw → ∃zSiz(Av ∧ ∃w(φw ∧ Pvw)),

which asserts, for any non-empty set of entities, the existence of a sumi composed exactly of all the atoms that compose those entities. Indeed, GEM also provides the resources to overcome the limits of the Atomicity axiom (P.7) discussed in Section 3.4. For, on the one hand, the infinitely descending chain depicted in Figure 6 is not a model of AGEM, since it is missing all sorts of sums. On the other, in GEM one can actually strenghten (P.7) in such a way as to require explicitly that everything be made entirely atoms, as in

(P.7′)

Strong Atomicity  ∃yAy ∧ PxσyAy.

(See Shiver 2015.) It should be noted, however, that such advantages come at a cost. For regardless of the number of atoms one begins with, the axioms of AGEM impose a fixed relationship between that number, κ, and the overall number of things, which is going to be 2κ–1. As Simons (1987: 17) pointed out, this means that the possible cardinality of an AGEM-model is restricted. There are models with 1, 3, 7, 15, 2ℵ[size=9]0, and many more cardinalities, but no models with, say, cardinality 2, 4, 6, or ℵ0. Obviously, this is not a consequence of (P.15i) alone but also of the other axioms of GEM (the unsupplemented pattern in Figure 2, left, satisfies (P.15i) for each i and has 2 elements, and can be expanded at will to get models of any finite cardinality, or indefinitely to get a model with ℵ0 elements, as in Figure 2, center; see also Figure 8 for a supplemented non-filtrated model of (P.151) with 4 elements and Figure 7, right, for a supplemented non-extensional model of (P.153) with 6 elements). Still, it is a fact that in the presence of such axioms each (P.15i) rules out a large number of possibilities. In particular, every finite model of AGEM—hence of GEM—is bound to involve massive violations of what Comesaña (2008) calls “primitive cardinality”, namely, the intuive thesis to the effect that, for any integer n, there could be exactly n things. And since the size of any atomistic domain can always be reached from below by taking powers, it also follows that AGEM cannot have infinite models of strongly inaccessible cardinality. Such is, as Uzquiano (2006) calls it, the “price of universality” in the context of Atomicity.[/size]
What about ÃGEM, the result of adding the Atomlessness axiom (P.8)? Obviously the above limitation does not apply, and the Tarski model mentioned in Section 3.4 will suffice to establish consistency. However, note that every GEM model—hence every ÃGEM model—is necessarily bound at the top, owing to the existence of the universal entity U. This is not by itself problematic: while the existence of U is the dual the Bottom axiom, a top jumbo of which everything is part has none of the formal and philosophical oddities of a bottom atom that is part of everything (though see Section 4.5 for qualifications). Yet a philosopher who believes in infinite divisibility, or at least in its possibility, might feel the same about infinite composability. Just as everything could be made of atomless gunk that divides forever into smaller and smaller parts, everything might be mereological “junk”—as Schaffer (2010: 64) calls it—that composes forever into greater and greater wholes. (One philosopher who held such a view is, again, Whitehead, whose mereology of events includes both the Atomlessness principle and its upward dual, i.e.:

(P.19)

Ascent  ∃yPPxy.

See Whitehead 1919: 101; 1920: 76). GEM is compatible with the former possibility, andÃGEM makes it into a universal necessity. But neither has room for the latter. Indeed, the possibility of junk might be attractive also from an atomist perspective. After all, already Theophilus thought that even though everything is composed of monads, “there is never an infinite whole in the world, though there are always wholes greater than others ad infinitum” (Leibniz,New Essays, I-xiii-21). Is this a serious limitation of GEM? More generally, is this a serious limitation of any theory in which the existence of U is a theorem—effectively, any theory endorsing at least the unrestricted version of (P.14ψ)? (In the absence of Antisymmetry, one may want to consider this question by understanding the predicate ‘PP’ in (P.19) in terms of the stronger definition given in (20′); see above, ad (P.8′).) Some authors have argued that it is (Bohn 2009a, 2009b, 2010), given that junk is at least conceivable (see also Tallant 2013) and admits of plausible cosmological and mathematical models (Morganti 2009, Mormann 2014). Others have argued that it isn't, because junk is metaphysically impossible (Schaffer 2010, Watson 2010). Others still are openly dismissive about the question (Simons 1987: 83). One may also take the issue to be symptomatic of the sorts of trouble that affect any theory that involves quantification over absolutely everything, as the Unrestricted Sum principles in (P.15i) obviously do (see Spencer 2012, though his remarks focus on mereological theories formulated in terms of plural quantification). One way or the other, from a formal perspective the incompatibility with Ascent may be viewed as an unpleasant consequence of (P.15i), and a reason to go for weaker theories. In particular, it may be viewed as a reason to endorse only finitary sums, which is to say only instances of (P.12ξ,i), or perhaps its unrestricted version:

(P.12i)

Finitary Unrestricted Sumi  ∃zSizxy.

(See Contessa 2012 and Bohn 2012: 216 for explicit suggestions in this spirit.) This would be consistent with the existence of junky worlds as it is consistent with the existence of gunky worlds. Yet it should be noted that even this move has its costs. For example, it turns out that in a world that is both gunky and junky (what Bohn calls “hunk”) (P.12i) is in tension with the Complementation principle (P.6) for each i (Cotnoir 2014). Moreover, while (P.12i) is compatible with junky worlds, i.e., models that fully satisfy the Ascent axiom (P.19), it is in tension with the possibility of worlds containing junky structures along with other, disjoint elements (Giberman 2015).