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 Composition Principles

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التوقيع : رئيس ومنسق القسم الفكري

عدد الرسائل : 1500

الموقع : center d enfer
تاريخ التسجيل : 26/10/2009
وســــــــــام النشــــــــــــــاط : 6

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مُساهمةComposition Principles

Composition Principles Ouo_0010

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 Bounds

Conditions 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 Sums

A 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.)
Composition Principles Figure7_2013
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):
(40i)x +i y =df ℩zSizxy.
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,
(49)x × y =df ℩zRzxy,
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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رد: Composition Principles
مُساهمة الجمعة مارس 11, 2016 11:49 am من طرف free men

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 ∧ ∀ww → ψw)) → ∃zww → 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 ∧ ∀ww → ψw)) → ∃zSizφw,
where
(521)General Sum1[21] 
S1zφw =df ∀v(Pzv ↔ ∀ww → Pwv))
(522)General Sum2 
S2zφw =df ∀ww → Pwz) ∧ ∀v(Pvz → ∃ww ∧ Ovw))
(523)General Sum3 
S3zφw =df ∀v(Ovz ↔ ∃ww ∧ 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 ∧ ∀ww → ψw)) → ∃zRzφw,
where
(53)General Product 
Rzφw =df ∀v(Pvz ↔ ∀ww → Pvw))
and ‘ψw’ expresses a condition at least as strong as ‘∀xx → 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 ‘∀xx → 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
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رد: Composition Principles
مُساهمة الجمعة مارس 11, 2016 11:50 am من طرف free men

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) → ∃ww ∧ 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”.
Composition Principles Figure8_2013
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, MMM, 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(zxφx),
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 σzxx → 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 → ∃zv(GOzv ↔ ∃ww ∧ 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 ∧ ∃ww ∧ 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).
 

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