Appendix: Formal Theories of Parthood

Home , Antisymmetric relation, Process and Reality

Achille C. Varzi

This Appendix gives a brief overview of the main formal theories of parthood, or mereologies, to be found in the literature.1 The focus is on classical theories, so the survey is not meant to be exhaustive. Moreover, it does not cover the many philosophical issues relating to the endorsement of the theories themselves, concerning which the reader is referred to the Selected Bibliography at the end of the volume. In particular, we shall be working under the following simplifying assumptions2: — Absoluteness: Parthood is a two-place relation; it does not hold relative to time, space, spacetime regions, sortals, worlds, or anything else.3 — Monism: There is a single relation of parthood that applies to every entity independently of its ontological category.4 — Precision: Parthood is not a source of vagueness: there is always a fact of the matter as to whether the parthood relation obtains between any given pair of things.5 For deﬁniteness, all theories will be formulated in 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 will be the classical predicate calculus.

1The exposition follows Varzi (2014). For a thorough survey, see Simons (1987). 2The labels and formulations of these assumptions are from Sider (2007). 3For the view that parthood should be a three-place relation relativized to time, see e.g. Thomson (1983). For the view that it should be a four-place relation, see Gilmore (2009). 4For misgivings about Absoluteness and related worries, see e.g. Mellor (2006)andMcDaniel (2009). 5For mereologies that allow for indeterminate or “fuzzy” parthood relations, see e.g. Smith (2005) and Polkowski (2011).

C. Calosi and P. Graziani (eds.), Mereology and the Sciences, Synthese Library 371, 359 DOI 10.1007/978-3-319-05356-1, © Springer International Publishing Switzerland 2014 360 A.C. Varzi

Fig. 1 Basic patterns of mereological relations (shaded cells indicate parthood)

Core Principles

As a minimal requirement on ‘P ’, it is customary to assume that it stands for a partial order—a reflexive, transitive, and antisymmetric relation6: (P.1) Pxx Reflexivity (P.2) .Pxy ^ Pyz/ ! Pxz Transitivity (P.3) .Pxy ^ Pyx/ ! x D y Antisymmetry Together, these three axioms are meant to fix the intended meaning of the parthood predicate. They form the “core” of any standard mereological theory, and the theory that comprises just them is called Ground Mereology,orMforshort.7 A number of additional mereological predicates may then be introduced by definition (Fig. 1):

(1) EQxy Ddf Pxy ^ Pyx equality 8 (2) PPxy Ddf Pxy ^:x D y proper parthood (3) PExy Ddf Pyx ^:x D y proper extension (4) Oxy Ddf 9z.Pzx ^ Pzy) overlap (5) Uxy Ddf 9z.Pxz ^ Pyz/ underlap Fig. 1 Given (P.1)–(P.3), it follows immediately that EQ is an equivalence relation. Moreover, PP and PE are irreflexive, asymmetric, and transitive whereas O and U are reflexive and symmetric, but not transtive. Since the following is a also theorem of M, (6) Pxy $ .PPxy _ x D y/ ‘PP’ could have been used as a primitive instead of ‘P ’. Similarly for ‘PE’. Sometimes ‘P ’ is also defined in terms of ‘O’ via the biconditional

6Unless otherwise specified, all formulas are to be understood as universally closed. 7For a survey of the motivations that may lead to the development of non-standard mereologies in which P is weaker than a partial order, see Varzi (2014: §2.1). 8 In the literature, proper parthood is sometimes defined as asymmetric parthood: PPxy Ddf Pxy ^ :Pyx.GivenAntisymmetry, this definition is equivalent to (2). Without Antisymmetry,however, the two definitions would come apart. (Similarly for ‘PE’.)SeeCotnoir(2010). Appendix: Formal Theories of Parthood 361

Fig. 2 Three unsupplemented models (connecting lines going upwards indicate proper parthood)

(7) Pxy $8z.Ozx ! Ozy/. However, (7) is not provable in M and calls for stronger axioms (specifically, the axioms of theory EM defined below). Since those stronger axioms reflect substantive philosophical theses, ‘P ’and‘PP’(or‘PE’) are the best options to start with. Here we stick to ‘P ’.

Decomposition Principles

M is standardly viewed as embodying the common core of any mereological theory. Yet not every partial order qualiﬁes as parthood, and establishing what further requirements should be added to (P.1)–(P.3) is precisely the question a good mereological theory is meant to answer. One way to extend M is by means of decomposition principles, i.e., principles concerning the part structure of a given whole. Here, one fundamental intuition is that no whole can have a single proper part. There are several ways in which this intuition can be captured, beginning with the following:

(P.4a/ PPxy !9z.PPzy ^:z D x/ (Weak) Company (P.4b/ PPxy !9z.PPzy ^:Pzx/ Strong Company (P.4) PPxy !9z.PPzy ^:Ozx/ (Weak) Supplementation9

(P.4a/ is the literal rendering of the idea in question, but it is too weak: it rules out certain implausible finitary models (Fig. 2, left) but not, for example, models with infinitely descending chains in which the additional parts do not leave any mereological “remainder” (Fig. 2, center). (P.4b/ is stronger, but it still admits of models in which a whole can be decomposed into several proper parts all of which overlap one another (Fig. 2, right). In such cases it is unclear what would be left of the whole upon the removal of any of its proper parts (along with all proper parts thereof). It is only (P.4) that appears to capture the full spirit of the above-mentioned intuition: every proper part must be “supplemented” by another part—a proper part that is completely disjoint (i.e., does not overlap) the first. (P.4) entails both (P.4a/

9In the literature, this principle is sometimes formulated using ‘P ’ in place of ‘PP’inthe consequent. In M the two formulations are equivalent. 362 A.C. Varzi

Fig. 3 A weakly supplemented model violating strong supplementation

and (P.4b/ and rules out each of the models in Fig. 2. The extension of M obtained by adding this principle to (P.1)–(P.3) is called Minimal Mereology, or MM.10 There is another, stronger way of expressing the supplementation intuition. It corresponds to the following axiom, which differs from (P.4) in the antecedent: (P.5) :Pyx !9z.Pzy ^:Ozx/ Strong Supplementation In M this principle entails (P.4). The converse, however, does not hold, as shown by the model in Fig. 3. The stronger mereological theory obtained by adding (P.5) to the three core principles of M is called Extensional Mereology,EM. The extensional character of EM may not be manifest in (P.5) itself, but it becomes clearer in view of the following theorem: (8) 9zPPzx ! .8z.PPzx ! PPzy/ ! Pxy/ from which it follows that sameness of mereological composition is both necessary and sufficient for identity: (9) 9zPPzx ! .x D y $8z.PPzx $ PPzy//. Thus, EM is “extensional” precisely insofar as it rules out any model of the sort depicted in Fig. 3, where distinct objects decompose into the same proper parts. There is yet a further way of capturing the supplementation intuition. It corresponds to the following axioms, which differs from (P.5) in the consequent: (P.6) :Pyx !9z8w.Pwz $ .Pwy ^:Owx//. Complementation11 Informally, (P.6) states that whenever an object fails to include another among its parts, there is something that amounts exactly to the difference or relative complement between the first object and the second. Once again, it is easily checked that in M this principle entails (P.5)—thus, a fortiori, (P.4)—whereas the converse does not hold (Fig. 4). It should be noted, however, that (P.6) goes beyond the original supplementation intuition. For while it guarantees that a whole cannot have a single proper part, it also pronounces on the specific mereological makeup of the supplementary part. In particular, it requires the relative complement to exist regardless of its internal structure. If, for example, y is a wine glass and x the stem of the glass, (P.6) entails the existence of something composed exactly of the base and the bowl—a spatially disconnected entity. Whether there exist entities of this

10Strictly speaking, in MM (P.3) is redundant, as it follows from (P.4) along with (P.1) and (P.2). For ease of reference, however, we shall continue to treat (P.3) as an axiom. 11In the literature, (P.6) is also known as the Remainder Principle. Appendix: Formal Theories of Parthood 363

Fig. 4 A strongly supplemented model violating complementation sort, and more generally whether the remainder between a whole and any one of its proper parts adds up to a bona ﬁde entity of its own, is really a question about mereological composition, over and above the conditions on decomposition set by (P.4) and (P.5). Before turning to issues regarding composition, a different sort of decomposition principles is worth mentioning. Let a mereological atom be any entity with no proper parts:

(10) Ax Ddf :9yPPyx. atom Obviously, all the theories considered so far are compatible with the existence of such things. But one may want to demand more than mere compatibility, just as one may want to preclude it. Thus, one may want to require that everything is ultimately composed of atoms, or else that everything is made up of “atomless gunk”12 that divides forever into smaller and smaller parts. These two options are usually formulated as follows: (P.7) 9y.Ay ^ Pyx/ Atomicity (P.8) 9yPPyx Atomlessness These postulates are mutually inconsistent, but taken in isolation they can con- sistently be added to any mereological theory mentioned so far to yield either an atomistic variant or an atomless variant, respectively. Atomistic mereologies admit significant semplifications in the axiomatics. For example, Atomistic EM can be simplified by merging Strong Supplementation (P.5) and Atomicity (P.7) into a single axiom: (P.50) :Pxy !9z.Az ^ Pzx ^:Pzy/ Atomistic Supplementation and the the extensionality thesis (9) can be put more perspicuously as follows: (90) x D y $8z.Az ! .Pzx $ Pzy// This is especially significant if one considers that (P.7) does not quite say that everything is made up of atoms; it merely says that everything has atomic parts, which is consistent with the possibility of infinitely descending chains of decomposition that never bottom out (Fig. 5). Whether stronger versions of (P.7)

12The phrase is from Lewis (1991: 20). 364 A.C. Varzi

Fig. 5 An inﬁnitely descending atomistic model

can be formulated that rule out such dubious patterns is, at the moment, a question that has not been fully explored.13 Concerning atomless mereologies, one may similarly remark that (P.8) is by itself rather weak. For one thing, the unsupplemented model in Fig. 2, middle, qualifies as atomless. To the extent that such models run afoul of the intended notion of a gunky world, this means that (P.8) calls for teories at least as strong as MM, in which case the relevant axiomatization may again be simplified by merging (P.4) and (P.8) into a single axiom: (P.40) Pxy !9z.PPzy ^ .Ozx ! x D y// Atomless Supplementation Moreover, infinite divisibility is loose talk. Given (P.8), gunk may have as few as denumerably many parts; but can it have more? Is there an upper bound on the cardinality on the number of pieces of gunk? Should it be allowed that for every cardinal number there may be more than that many pieces of gunk? (P.8) is silent on these questions, yet these are certainly aspects of atomless mereology that deserve further scrutiny.

Composition Principles

The other main way of extending M is via composition principles,i.e.principles governing the behavior of P in the bottom-up direction: from the parts to the wholes that they compose. We have already seen that the Complementation axiom (P.6) is, in a way, a principle of this sort. Another such principle would be the dual of Atomlessness, to the effect that everything might be “worldless junk”14 that composes forever into greater and greater wholes: (P.9) 9yPPxy Ascent Both (P.6) and (P.9) are consistent with any of the theories considered so far. They are, however, fairly strong principles, which reﬂect speciﬁc views on the overall mereological structure of the universe. More generally, it is customary to consider ways of extending M by means of composition principles that specify the conditions under which one or more things qualify as parts of a larger whole.15

13See Cotnoir (2013) for some work in this direction. 14The phrase is from Schaffer (2010: 64). 15This is a version of the so-called “Special Composition Question”. See van Inwagen (1990: Chap. 2). Appendix: Formal Theories of Parthood 365

Fig. 6 An a-sum that is not a b- or c-sum, and a c-sum that is not an a- or b-sum18

The most basic principles of this sort have the following form, to the effect that for any pair of suitably related entities, i.e., any two entities satisfying a given provision , there is something of which both are part—an underlapper: (P.10) xy ! Uxz -Bound Such principles are quite weak. For example, regardless of how exactly is construed, (P.10) is trivially satisﬁed in any model that includes a universal entity of which everything is part. A stronger sort of requirement is that any pair of suitably related entities have a minimal underlapper, something composed of their parts and nothing else. There are at least three ways of formulating such a requirement, corresponding to three different ways of characterizing the relevant notion of a minimal underlapper, also known as a mereological sum of the two entities in question16: 17 (11a) Sazxy Ddf Pxz ^ Pyz ^8w..Pxw ^ Pyw/ ! Pzw/ a-sum (11b) Sbzxy Ddf Pxz ^ Pyz ^8w.Pwz ! .Oxw _ Oyw// b-sum (11c) Sczxy Ddf 8w.Owz $ .Oxw _ Oyw// c-sum In M these three notions are pairwise distinct (Fig. 6), though they may coincide in the presence of further axioms. For instance, given Strong Supplementation,(11b/ and (11c/ are equivalent (though stronger that (11a//, whereas in the presence of Complementation all three notions coincide so long as there is a universal entity: in that case, each sum of any two things 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.) For each i 2fa; b; cg, we can then extend M by adding a corresponding axiom as follows, where again speciﬁes a suitable binary condition:

(P.11i) xy !9zSizxy -Sumi The non-equivalence of these axioms is immediately veriﬁed by taking to be satisﬁed by all pairs of objects and considering the models in Fig. 6. But the axioms may also differ when is more restrictive. For instance, with expressing overlap,

16The first notion may be found in Eberle (1967)andBostock (1979), the second in Tarski (1935) and Lewis (1991), the third in Simons (1987)andCasati and Varzi (1999). 17 Given Reflexivity and Transitivity, the definiens in (11a) is equivalent to 8w.Pzw $ .Pxw ^ Pyw//. 18The non-extensional model of Fig. 3 also depicts a case in which x and y have a c-sum, in fact two c-sums (themselves), though no a- or b-sum. This runs contrary the intended meaning of ‘sum’, suggesting that (11c/ is best suited to theories at least as strong as EM. See Hovda (2009) for discussion. 366 A.C. Varzi

Fig. 7 A model of -suma violating -sumb and -sumc

the model in Fig. 6, right, still satisfies (P.11c/, but not (P.11a/ or (P.11b/, whereas the model in Fig. 7 satisfies (P.11a/, but not (P.11b/ or (P.11c/.InEM,however,(P.11b/ or (P.11c/ are equivalent, since the corresponding notions of sum coincide. The intuitive force of each (P.11i/ is in fact best appreciated in the context of EM, for in that case the relevant sums must be unique. If we introduce a corresponding binary operator (using ‘š’ for the definite descriptor),

(12i) x Ci y Ddf šzFizxy i-sum then is then easy to see that EM warrants all the “Boolean” properties one might expect. For instance, as long as the arguments satisfy the relevant condition ,19 each operator is idempotent, commutative, and associative:

(13) x D x Ci x (14) x Ci y D y Ci x (15) x Ci .y Ci z/ D .x Ci y/ Ci z and well-behaved with respect to parthood:

(16) Px.x Ci y/ (17) Pxy ! Px.y Ci z/ (18) P.x Ci y/z ! Pxz (19) Pxy $ x Ci y D y

Each (P.11i/ is still fairly weak, for it governs only finitary composition. We get even stronger composition principles by requiring a minimal underlapper to exist for any set of objects satisfying a given condition, including infinite sets (whose sums—or fusions—cannot be generated by means of the binary operators defined above). There is, of course, a technical obstacle to formulating such principles in their full generality without resorting to explicit quantification over sets, since a standard first-order language does not have the resources to specify all sets, but only a denumerable number (in any given domain).20 However, one can achieve a sufficient degree of generality by relying on axiom schemas where the relevant sets are identified through open formulas. Thus, let ‘'’ be any formula in the language,

19If the condition is not satisfied, the sum may not exist, in which case the standard treatment of descriptive terms implies that the corresponding instances of the theorems that follow are false. In classical logic, (13)–(19) should therefore be taken to hold conditionally on the assumption that the relevant variables range over -related entities. 20To overcome this limitation, some early theories such as those of Tarski (1929)andLeonard and Goodman (1940) resort to explicit quantification over sets. Others, such as Lewis (1991), resort to the machinery of plural quantification. Appendix: Formal Theories of Parthood 367 and let ‘ ’ expresses the condition in question. Infinitary variants of the three 21 notions of sum in (11a/–(11c/ can be defined as follows, respectively :

(20a) Faz'w Ddf 8w.'w ! P wz/ ^8v.8w.'w ! Pwv/ ! P zv/ a-fusion (20b) Fbz'w Ddf 8w.'w ! P wz/ ^8v.Pvz !9w.'w ^ Owv// b-fusion (20c) Fcz'w Ddf 8v.Ovz $9w.'w ^ Owv// c-fusion

(‘Fiz'w’ may be read as ‘z is an i-fusion of the '-ers’.) For each such notion, we may then introduce a corresponding principle of inﬁnitary fusion through the following axiom schema, which asserts the existence of an i-fusion .i 2fa; b; cg/ for every non-empty set of objects satisfying :

(P.12i) .9w'w ^8w.'w ! w// !9zFiz'w -Fusioni

It can be checked that each (P.12i/ includes the corresponding ﬁnitary principle (P.11i/ as a special case, taking ‘'w’ to be the formula ‘w D x _ w D y’and ‘ w’ the condition ‘.w D x ! wy/ ^ .w D y ! xw/’. Thus, again, these principles are pairwise distinct in M, though it turns out that in the presence of Strong Supplementation (P.12b/ and (P.12c/ are equivalent. Finally, the strongest versions of all these composition principles are obtained by asserting them as axiom schemas holding for every condition , i.e., effectively, by foregoing any reference to altogether. Formally this amounts in each case to dropping the second conjunct of the antecedent of (P.12i/, i.e., to asserting the schema expressed by the relevant consequent for any non-empty set of objects speciﬁable in the language:

(P.13i) 9w'w !9zFiz'w Unrestricted Compositioni Once again, the relative strength of these principles varies for each i 2fa; b; cg. In particular, it is noteworthy that adding (P.13b/ to MM would sufﬁce to warrant the equivalence of Weak and Strong Supplementation, (P.4) and (P.5), whereas adding (P.13c/ would not (Fig. 4 would still count as a counteremodel). Given (P.5), however, the two composition principles are equivalent, which means that the theory 22 obtained by adding every instance of (P.13b/ to MM is the same theory obtained by adding every instance of (P.13c/ to EM. This theory is known in the literature as General Extensional Mereology, or GEM. The same theory can be obtained by 23 extending MM with (P.13a/, provided the following axiom is also added :

(P.14) .Faz'w ^ Pxz/ !9w.'w ^ Owx/ Filtration

21 (20a/–(20c/ are to be read on the assumption that the variables ‘z’and‘v’ do not occur free in '. Similar restrictions will apply below. 22Indeed, (P.2) and (P.4) would sufﬁce. 23From Hovda (2009). 368 A.C. Varzi

Classical Mereology

GEM is a powerful theory, and it was meant to be so by its nominalistic forerunners, who were thinking of mereology as a fundamental alternative to set theory.24 Indeed, GEM has such a distinguished pedigree that it has earned the title of Classical Mereology. It is also a decidable theory, whereas for example M, MM, EM, and many extensions thereof are not.25 To see just how powerful GEM is, consider the following operator, where ‘F ’isanyofthe‘Fi’s deﬁned above (which GEM forces to coincide):

(21) x'x Ddf šzFz'x general fusion In terms of this operator—the fusion of all '-ers—GEM can be further simplified, for example by merging (P.5) and (P.13c/ into a single axiom schema: (P.13) 9x'x !9z.z D x'x/ Unique Unrestricted Fusion and we can introduce the following definitions: 26 (22) x C y Ddf z.Pzx _ Pzy/ sum (23) x y Ddf z.Pzx ^ Pzy/ product (24) x y Ddf z.Pzx ^:Ozy/ difference (25) x Ddf z:Ozx complement (26) U Ddf zPzz universe The full strength of GEM can then be appreciated by considering that its models are closed under each of these notions, subject to the satisfiability of the relevant conditions. More exactly: the condition ‘:OzU’ is unsatisfiable, so U cannot have a complement. Likewise products are defined only for overlappers and differences only for pairs that leave a remainder. In all other cases, however, (22)–(26) yield perfectly well-behaved operators. Since such operators are the natural mereological analogues of the familiar set-theoretic operators, with ‘’ 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, modulo the absence of a null entity corresponding to the empty set. Indeed, P is virtually 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. We say ‘virtually’ because, strictly speaking, this is only true of stronger versions of GEM in which infinitary sums are defined using explicit quantification over sets.27 For set-free formulations that, like those considered here, strictly adhere to a standard

24See the classical works of Lésniewski (1927–1931)andLeonard and Goodman (1940). 25For a comprehensive picture of decidability in mereology, see Tasi (2013b). 26 In GEM, this deﬁnition is equivalent to (12i/, for each i 2fa; b; cg. 27As such, the result goes back to Tarski (1935:n.4). Appendix: Formal Theories of Parthood 369

ﬁrst-order language with a denumerable supply of open formulas, the isomorphism does not quite hold. However, this is only a minor limitation, and we can still characterize the exact algebraic strength of GEM as follows: any model of this theory is isomorphic to a Boolean subalgebra of a complete Boolean algebra with the zero element removed (a subalgebra that is not necessarily complete if Zermelo- Frankel set theory with the axiom of Choice is consistent).28 In this connection, two further points are worth stressing. First, the existence of a “null entity” which is part of everything—the analogue of the empty set— is not in principle incompatible with GEM. However, it is easy to see that the only models of GEM with such an entity are trivial one-element models, owing to Weak Supplementation. It is for this reasons that the principles of Unrestricted Composition in (P.13i/ are stated as conditionals warranting the existence of a fusion for any given non-empty set of '-ers. Dropping such a proviso would have disastrous effects, for then the existence of a null entity—the null entity—would be guaranteed by taking ‘'w’ to be the condition ‘8xPwx’. The only way around the disaster would be to revisit the non-basic vocabulary by carefully distinguishing trivial cases of parthood and overlap (involving the ubiquitous null entity) and non-trivial, genuine ones, as in

(27) GPxy Ddf Pxy ^9z:Pxz genuine parthood (28) GOxy Ddf 9z.GPzx ^ GPzy/ genuine overlap and by reformulating all non-core axioms accordingly.29 In this way, one can actually arrive at a variant of GEM that inherits all the strength of a complete Boolean algebra. Nonetheless, the philosophical import of such a theory would remain dubious. Second, note that GEM is fully committed to the existence of U, a “universal entity” of which everything is part. This is not by itself a problem, barring any philosophical concerns about the gerrymendered nature of such an entity. It is, however, not without consequences. In particular, while GEM admits of models in which everything is composed of atoms as well as “gunky” models in which everything divides forever, the necessary existence of U deprives GEM of any “junky” model in which everything composes forever. Thus, while GEM admits of both atomistic and atomless extensions, adding the Ascent principle (P.9) would immediately result in an inconsistent theory.

28See Pontow and Schubert (2006), Theorem 34, for details and proof. 29This strategy is not uncommon in the mathematically oriented literature; see again Pontow and Schubert (2006) for a comprehensive treatment. 370 A.C. Varzi

Summary of GEM

For ease of reference, we conclude by summarizing the main axiomatizations of GEM mentioned above, with some rewriting of bound variables and dropping all redundancies30:

(I) .Pxy ^ Pyz/ ! Pxz Transitivity (P.2) PPxy !9z.PPzy ^:Ozx/ Weak Supplementation (P.4) 9x'x !9zFaz'x Unrestricted Compositiona (P.13a) .Faz'x ^ Pyz/ !9x.'x ^ Oxy/ Filtration (P.14)

(II) .Pxy ^ Pyz/ ! Pxz Transitivity (P.2) PPxy !9z.PPzy ^:Ozx/ Weak Supplementation (P.4) 9x'x !9zFbz'x Unrestricted Compositionb (P.13b)

(III) Pxx Reﬂexivity (P.1) .Pxy ^ Pyz/ ! Pxz Transitivity (P.2) Pxy ^ Pyx/ ! x D y Antisymmetry (P.3) :Pyx !9z.Pzy ^:Ozx/ Strong Supplementation (P.5) 9x'x !9zFcz'x Unrestricted Compositionc (P.13c)

(IV) .Pxy ^ Pyz/ ! Pxz Transitivity (P.3) 9x'x !9z.z D x'x/ Unique Unrestricted Fusion (P.13)

30See also Simons (1987)andHovda (2009) for additional axiom sets. The elegant axiomatization in (IV) is essentially due to Tarski (1929), though the axioms are explicitly given only in the 1956 English translation. Selected Bibliography

Claudio Calosi and Pierluigi Graziani

Aiello, M., Pratt-Hartmann, I., & van Benthem, J. (Eds.). (2007). Handbook of spatial logics. Berlin: Springer. Armstrong, D. (1978). Universals and scientific realism. Cambridge: Cambridge University Press. Bostock, D. (1979). Logic and arithmetic, Vol. 2: Rational and irrational numbers. Oxford: Clarendon. Burkhardt, H., & Dufour, C. A. (1991). Part/whole I: History. In H. Burkhardt & B. Smith (Eds.), Handbook of metaphysics and ontology (pp. 663–673). Munich: Philosophia. Burkhardt, H., Seibt, J., & Imaguire, G. (Eds.). (2014). Handbook of mereology. Munich: Philosophia. Casati, R., & Varzi, A. (1999). Parts and places. Cambridge: MIT. Chisolm, R. M. (1973). Parts as essentials to their wholes. Review of Metaphysics, 26, 581–603. Clarke, B. L. (1981). A calculus of individuals based on connection. Notre Dame Journal of Formal Logic, 22, 204–218. Cotnoir, A. J. (2010). Anti-symmetry and non-extensional mereology. Philosophical Quarterly, 60, 396–405. Cotnoir, A. J. (2013). Beyond atomism. Thought, 2, 67–72. Cruse, D. A. (1979). On the transitivity of the part-whole relation. Journal of Linguistics, 15, 29– 38. Eberle, R. A. (1967). Some complete calculi of individuals. Notre Dame Journal of Formal Logic, 8, 267–278. Eberle, R. A. (1970). Nominalistic systems. Dordrecht: Reidel. Forrest, P. (1996). How innocent is mereology? Analysis, 56, 127–131. Forrest, P. (2002). Nonclassical mereology and its application to sets. Notre Dame Journal of Formal Logic, 43, 79–94. Gilmore, C. (2009). Why parthood might be a four place relation and how it behaves if it is. In L. Hoonefelder, E. Runggaldier, & B. Schick (Eds.), Unity and time in metaphysics (pp. 83– 133). Berlin: De Gruyter. Goodman, N. (1951). The structure of appearance. Cambridge: Harvard University Press. Hempel, C. G. (1953). Reflections on Nelson Goodman’s the structure of appearance. Philosophi- cal Review, 62, 108–116. Hodges, W., & Lewis, D. K. (1968). Finitude and infinitude in the atomic calculus of individuals. Nous, 2, 405–410.

C. Calosi () • P. Graziani Department of Basic Sciences and Foundations, Urbino University, Urbino, Italy e-mail: [email protected]; [email protected]

C. Calosi and P. Graziani (eds.), Mereology and the Sciences, Synthese Library 371, 371 DOI 10.1007/978-3-319-05356-1, © Springer International Publishing Switzerland 2014 372 C. Calosi and P. Graziani

Hoffman, J., & Rosenkrantz, G. (1999). Mereology. In R. Audi (Ed.), The Cambridge dictionary of philosophy (2nd ed., pp. 557–558). Cambridge: Cambridge University Press. Hovda, P. (2009). What is classical mereology. Journal of Philosophical Logic, 38(1), 55–82. Iris, M. A., Litowitz, B. E., & Evens, M. (1988). Problems of the part-whole relation. In M. Evens (Ed.), Relations models of the lexicon (pp. 261–288). Cambridge: Cambridge University Press. Johansson, I. (2004). On the transitivity of parthood relations. In H. Hochberg & K. Mulligan (Eds.), Relations and predicates (pp. 161–181). Lancaster/Frankfurt: Ontos. Johansson, I. (2006). Formal mereology and ordinary language: Reply to Varzi. Applied Ontology, 1(2), 157–161. Leonard, H. S., & Goodman, N. (1940). The calculus of individuals and its uses. Journal of Symbolic Logic, 5, 45–55. Lesniewski,´ S. (1927–1931). O podstawach matematyki. Przegla¸d Filozoﬁczny, 30, 164–206; 31, 261–291; 32, 60–101; 33, 77–105; 34, 142–170. Eng. trans. by Barnett, D. I. (1992). On the foundations of mathematics. In S. J. Surma et al. (Eds.), S. Le«sniewski, collected works (Vol. 1, pp. 174–382). Dordrecht: Kluwer. Lewis, D. K. (1991). Parts of classes. Oxford: Blackwell. McDaniel, K. (2009). Structure-making. Australasian Journal of Philosophy, 87, 251–274. Mellor, D. H. (2006). Wholes and parts: The limits of composition. South African Journal of Philosophy, 25, 138–145. Moltmann, F. (1997). Parts and wholes in semantics. Oxford: Oxford University Press. Niebergall, K.-G. (2011). Mereology. In L. Horsten & R. Pettigrew (Eds.), The continuum companion to philosophical logic (pp. 271–298). New York: Continuum. Polkowski, L. (2011). Approximate reasoning by parts. An introduction to rough mereology. Berlin: Springer. Polkowski, L., & Skowron, A. (1994). Rough mereology. In Z. W. Ras & M. Zemankova (Eds.), Proceedings of the 8th international symposium on methodologies for intelligent systems (ISMIS 94), Charlotte (pp. 85–94). Berlin: Springer. Pontow, C. (2004). A note on the axiomatics of theories in parthood. Data & Knowledge Engineering, 50, 195–213. Pontow, C. & Schubert, R. (2006). A mathematical analysis of theories of parthood. Data & Knowledge Engineering, 59, 107–138. Rescher, N. (1955). Axioms for the part relation. Philosophical Studies, 6, 8–11. Schaffer, J. (2010). Monism: The priority of the whole. Philosophical Review, 119, 31–76. Sider, T. (2007). Parthood. Philosophical Review, 116, 51–91. Simons, P. (1987). Parts: A study in ontology. Oxford: Clarendon. Simons, P. (1991a). Free part-whole theory. In K. Lambert (Ed.), Philosophical applications of free logic (pp. 285–306). Oxford: Oxford University Press. Simons, P. (1991b). Part/whole II: Mereology since 1900. In H. Burkhardt & B. Smith (Eds.), Handbook of metaphysics and ontology (pp. 209–210). Munich: Philosophia. Simons, P. (2003). The universe. Ratio, 16, 237–250. Sällström, P. (Ed.). (1983–1986). Parts & wholes: An inventory of present thinking about parts and wholes (Vol. 4). Stockholm: Forskningsrdsnmnden. Smith, B. (1982). Annotated bibliography of writings on part-whole relations since Brentano. In B. Smith (Ed.), Parts and moments. Studies in logic and formal ontology (pp. 481–552). Munich: Philosophia. Smith, N. J. J. (2005). A plea for things that are not quite all there. Journal of Philosophy, 102, 381–421. Tarski, A. (1929). Les fondements de la géométrie des corps. Ksiega Pamiatkowa Pierwszkego Polskiego Zjazdu Matematycznego. Suppl. to Annales de la Société Polonaise de Mathéma- tique, 7, 29–33; Eng. trans. by Woodger, J. H. (1956). Foundations of the geometry of solids. In A. Tarski (Ed.), Logics, semantics, metamathematics. Papers from 1923 to 1938 (pp. 24–29). Oxford: Clarendon. Tarski, A. (1935). Zur Grundlegung der Booleschen Algebra. I. Fundamenta Mathematicae, 24, 177–198; Eng. trans. by Woodger, J. H. (1956). On the foundations of the Boolean algebra, In Selected Bibliography 373

A. Tarski (Ed.), Logics, semantics, metamathematics, papers from 1923 to 1938 (pp. 320–341). Oxford: Clarendon. Thomson, J. J. (1983). Parthood and identity across time. Journal of Philosophy, 80, 201–220. Tsai H.-c. (2009). Decidability of mereological theories. Logic and Logical Philosophy, 18, 45–63. Tsai H.-c. (2011). More on the decidability of mereological theories. Logic and Logical Philoso- phy, 20, 251–265. Tsai H.-c. (2013a). Decidability of general extensional mereology. Studia Logica, 101(3), 619–636. Tsai H.-c. (2013b). A comprehensive picture of the decidability of mereological theories. Studia Logica, 101(5), 987–1012. Uzquiano, G. (2006) The price of universality. Philosophical Studies, 129, 137–169. van Inwagen, P. (1990). Material beings. Ithaca: Cornell University Press. Varzi, A.C. (2000). Mereological commitments. Dialectica, 54, 283–305. Varzi, A.C. (2006). A note on transitivity of parthood. Applied Ontology, 1, 141–146. Varzi, A.C. (2008). The extensionality of parthood and composition. The Philosophical Quarterly, 58, 108–133. Varzi, A.C. (2009b) Universalism entails extensionalism. Analysis, 69(4), 599–604. Varzi, A. C. (2014). Mereology. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy (Spring 2014 ed.). http://plato.stanford.edu/archives/spr2014/entries/mereology. Whitehead, A. N. (1929). Process and reality: An essay in cosmology. New York: McMillian. Winston, M., Chafﬁn, R., & Herrmann, D. (1987). A taxonomy of part-whole relations. Cognitive Science, 11, 417–444. Yi, B. U. (1999). Is mereology ontologically innocent? Philosophical Studies, 93, 141–160. Index

A Class, 55, 94, 107, 109, 114–120, 123, 125, Accuracy, 192, 269, 270, 272–279 130–134, 136–139, 142, 157, 164, Act of assembling, 248 221–223, 237–240, 242, 243, 256, Adjacency space, 297–312, 314–316 257, 271, 278, 297, 298, 301, 314, Affordances, 162, 190, 199, 201, 203, 204, 326, 349–353, 356 209, 210 C-mereology, 205–210 Allogeneic, 183, 184, 186 Coincidence in time, 256, 257 Anti-Symmetry, 24, 62, 63, 102, 106, 141, 147, Collective, 27, 177, 178, 181, 207, 218 152–154, 156 Complement, 92, 94, 124, 149–153, 163, Artifact, 29, 171, 217, 247–258 189, 193, 199, 202, 209, 223, 232, Ascent; 364, 369 233, 253, 256, 297, 303, 305, 306, A-theory of time, 6 308–311, 326, 362–365, 368 Atom, 13, 16, 78, 150, 156, 168, 177, 192, Complementation, 223, 305, 309, 362, 364, 193, 196, 206, 207, 209, 249, 297, 365 299–301, 306, 363 Component, 2, 25, 26, 44, 59–61, 70–72, 75, Atomicity, 363 79, 82, 157, 165, 168, 174–184, 186, Atomic part, 214, 300, 301, 306, 307, 330, 192, 196, 197, 200, 206, 209, 219, 331, 333, 340, 341, 363 220, 238, 256, 280, 295, 307, 308, Atomlessness, 363, 364 310, 354 Autogeneic, 183 Composition, 2, 10, 12, 24, 27, 28, 47, 48, 50, 53–82, 95, 102, 178, 196, 199, 200, 205–207, 209, 227, 252, 333, 351, B 361–367, 369, 370 Betweenness relation, 245, 246, 264, 265 Compound, 46, 78, 80, 177–179, 181, 182, Bona-ﬁde part, 257, 363 184, 193–195, 198–200, 203, 204, Boolean algebra, 101, 107, 233, 234, 303, 306, 208, 210 368, 369 Connection, 87, 91, 95, 101, 102, 105, 106, Boundary, 170, 178, 179, 183–185, 226, 229, 108–110, 115, 116, 121, 129, 141, 232, 233, 235, 236, 238–242, 244, 165, 172, 173, 183, 214, 219, 220, 254–257, 259, 302, 315 223–227, 235, 239–242, 255–258, B-theory of time, 6 282, 294–299, 306, 324, 326–328, 330–333, 338–339, 343–345, 356, 369 C Connection ingredient, 219, 222, 224, C-boundary, 240–242 226–228, 231, 237, 239, C-closure, 240 241, 282 C-interior, 239–240 Container, 172, 174–180, 182, 325

C. Calosi and P. Graziani (eds.), Mereology and the Sciences, Synthese Library 371, 375 DOI 10.1007/978-3-319-05356-1, © Springer International Publishing Switzerland 2014 376 Index

Containment, 162, 166, 168–171, 176–178, Function, 54, 96, 105, 111, 112, 119, 164, 184–186, 218, 227, 228, 259 170–173, 176, 179, 181–184, 186, Contains, 2, 9, 16, 41, 44, 58, 64, 72, 92, 94, 191, 206, 229, 259, 260, 268, 269, 96, 106, 107, 109, 110, 113, 116, 280–282, 302, 306, 308, 314, 334, 161, 162, 165–180, 182–186, 194, 336, 337, 340, 346, 347, 349, 352 214, 218, 227, 228, 246, 254, 256, Fundamentality, 53 264–266, 291, 295, 303, 311, 325, Fuzzy equivalence, 112 326, 333, 352, 355 Context dependence, 192 Continuity, 2, 85–102, 178, 219, 259 G Corner slice, 35–50 General Extensional Mereology (GEM), 64, Coverage, 269, 270, 272–275, 277, 279 81, 300, 367–370 General relativity, 2, 9–12 Genetic identity, 167, 170–173 ; D Genuine Overlap 369 ; Decision algorithm, 170, 171, 220, 268, 269 Genuine Parthood 369 Decision rule, 170, 268–270, 273, 279, 283, Grain, 10, 31, 174–179, 181, 182, 199 284 Granular agent, 281 Dent, 253, 254 Granular logic, 282–284 Dependence, 12, 79, 82, 161, 162, 192, 197, Granular reﬂection, 270, 273–275 198, 201, 210, 218, 284 Granule of knowledge, 220, 268–279 Design, 191–193, 213, 217, 219, 247–252, 255, 259, 281, 301, 305, 306, 325, 328, 330, 333, 345–347, 353–355 H Dichotomy Paradox; 84 Has-component, 165, 177, 178, 181 Difference, 24, 30, 67, 88, 128, 142, 143, 145, Has-ﬁat-part, 178, 181 173, 186, 199, 201, 204, 205, 208, Has-grain, 177, 178, 181 232, 241, 247, 250, 256, 298, 305, Hausdorff distance, 115, 312, 313 307, 308, 362, 365, 368 Histological image, 316, 317 Digital topology, 314–315 Hole, 183, 253 Discrete closure, 307–309, 314, 315, 317 Discrete interior, 306–309, 313, 318–320 Dualist substantivalism, 9, 29, 31, 33 I Immaterial object, 168, 174–177, 180 Indeterminacy, 166–170 Individuals, 60, 74, 75, 101, 125, 166, E 169–172, 181, 196, 200, 206, 207, Endurance, 12, 14, 35–40, 43, 44, 46, 47, 49 210, 218, 219, 221, 222, 297, 299, Entanglement, 72–75, 80, 82, 193, 195 316, 326, 328, 355 Equidistance relation, 245 Information system, 213, 230–231, 255, 268, Extension, 2, 53, 63–65, 67, 70, 73, 76, 78, 280, 282–284 79, 81, 82, 105, 112, 113, 116, 159, Ingredient, 9, 11, 199, 219, 221–224, 226–228, 169, 207, 211, 214, 222–224, 228, 231, 235–237, 239, 241, 244, 247, 249–251, 256, 258, 282, 283, 296, 248, 253, 256, 257, 282 300, 301, 306, 356, 360–363, 365, 367, 369 Extensionality of artifacts, 249, 250 J Joins predicate, 253

F Fiat part (vs. bonaﬁde part), 181 L Fills predicate, 253, 254 Location, 7, 17–19, 21, 23, 24, 33, 35, 36, Fits predicate, 253 44–46, 48, 144, 145, 158, 168, 169, Formation (of robots), 265, 266 185, 187, 263, 275, 313, 328, 345 Index 377

Locational endurantism, 14, 16, 17, 22–27, 33 124, 126, 130, 149, 151, 155, 156, Lorentz contraction, 41 169, 175, 177–179, 213, 221, 222, 224–225, 227, 233, 235, 237, 238, 258, 293–297, 300, 312, 316, 319, M 320, 327, 328, 333, 341, 342, 356, Many-slice constitution view (MSCV), 14–33 360, 361, 365, 368, 369 Material object, 1, 2, 5–14, 16–29, 31–33, 36, 39, 46, 96, 142, 143, 145, 146, 165, 168, 169, 174–176, 179, 180, 209 P Mathematical morphology, 214, 313–314, 316 Part-for, 172, 181, 182 Mereological fallacy, 200, 210 Parthood, 2, 12–14, 16–21, 23–25, 27, 30–33, Mereological structure, 2, 134–135, 138, 141, 35, 36, 40, 53–82, 101, 102, 141, 214, 364 158, 161–186, 213, 214, 248, Mereological sum relation, 25, 63, 64, 68, 72, 293–297, 299, 306, 359–370 73, 78, 79, 81, 102, 123–139, 141, Parts, 1, 5, 36, 53, 85, 105, 123, 141, 163, 189, 174, 184, 365 217, 293, 323 Mereological universalism, 185 Path (in space-time), 9 Mereology rough, 220, 227 Path constitution view (PCV), 11–16, 23–25, Mereoposet, 137–139 30–33 Metrical geometry, 2, 39, 41, 46, 85–102, Perceptron, 279, 280 105–121, 136, 191–193, 219, Perdurance, 14, 35–38, 40, 44, 46, 49 232–239, 244, 245, 261 Planning, 217, 219, 220, 255, 259–267 Michalski index, 270, 274 Plurality, 10, 12, 27, 28, 31, 32, 142 Minkowski space-time, 39–46, 48, 49 Point-free geometry, 105, 106, 110–115, 118 Missing value, 275–277 Potential ﬁeld, 259–262 Modalities of action, 203, 204 (Priority) Monism, 82 Mode of access, 197, 199, 200, 202, 204, 210 (Priority) Pluralism, 82 Momentary thing, 257 Product, 58–60, 67, 70, 80, 81, 111, 161, 200, Monadic, 170, 196, 197 201, 210, 229, 237, 240, 247, 248, Monist substantivalism, 22, 23, 25 252, 253, 270, 283, 304, 368 Motion, 2, 41, 42, 85–102, 196, 255, 263 Proper extension, 360 MSCV. See Many-slice constitution view Proper Part, 1, 62, 63, 65–69, 74, 75, 78, 124, (MSCV) 125, 138, 165, 178, 258, 293, 297, Multilocation, 5, 19, 20, 25–29 300, 333, 340–342, 360–362 Multi-valued logic, 101, 102, 105–121 Proximal distance, 312

N Q Natural kinds, 142–144, 146 Quantum Field Theory (QFT), 2, 3, 9, 11 Neuron (McCulloch-Pitts), 279, 280 Quantum state, 76, 77 Nihilism, 27, 69 Quasi-topological operator, 306–310 Non-tangential, 109, 110, 226, 239, 253, 256, 295, 303, 306–308 R Realism, 50, 76, 172, 181, 182, 264 O Reﬂexivity, 14, 18, 19, 24, 32, 62, 63, 65, 91, Ontology, 9, 12, 35, 44, 46, 75, 78, 81, 88, 143, 102, 106, 112, 113, 123, 146, 148, 163, 164, 169, 186, 207, 218, 221, 165, 245, 341, 360, 365, 370 255, 354 Region, 2, 5, 36, 77, 86, 105, 136, 144, 168, Organicism, 72, 73 196, 231, 294, 359 Origin, 108, 170–173, 176, 248, 313, 336 Regions-as-pluralities multilocationism Overlap, 5, 10–19, 22, 25–27, 32, 37, 40, (RPM), 25–33 62–64, 67, 68, 79, 93, 109, 116, Regular open/closed set, 233–235, 295 378 Index

Relational, 70, 71, 91, 145–148, 153, 157, Tensor product, 59, 60, 67, 70 195–197, 204, 206, 210, 211, 266, Time-slice, 36, 41, 44, 46, 257 304, 306 Topology, 101, 103, 169, 199, 219, 220, Relationism, 6, 7 224, 226, 232, 237, 238, 243, 308, Relativity, 1, 2, 9–12, 35, 36, 39, 41–43, 46–48, 314–315 50, 200–203 Transitivity, 18, 19, 62, 63, 65, 66, 102, 106, Restricted composition, 2, 10, 12, 27, 28, 53, 112, 113, 123, 141, 146, 153, 162, 64, 68, 69, 73, 74, 78, 81, 102, 207, 164, 165, 181, 182, 206, 207, 209, 367, 369, 370 230, 231, 245, 271, 300, 341, 360, ROM model, 255 365, 370 Rough inclusion, 227–231, 242, 245, 246, 261, t-slice, 36, 38–41, 44–46, 49, 256, 257 263, 268, 270, 271, 273, 275–278, 280–284 RPM. See Regions-as-pluralities U multilocationism (RPM) Universalism (unrestricted composition), 184, 185 S Universe, 82, 101, 141, 154, 214, 223, 224, Separative partial order, 133–134 237, 241, 248, 249, 251, 256, 271, Sets, 1, 7, 36, 55, 87, 106, 123, 142, 164, 189, 272, 277, 283, 297, 299, 303, 305, 218, 293, 333 310, 311, 315, 364, 368 Shape, 6, 13, 17, 25, 29, 36, 41, 43, 46, 48, 71, 164, 166, 199, 209, 252–258, 263, 266, 313 V Similarity, 112–114, 120, 143, 171, 208, Vagueness, 47, 48, 50, 69, 70, 166–170, 214, 270–271, 277, 278, 313 359 S-Mereology, 205–210 Spatial part (s-part), 36, 37, 39 Special composition question, 68, 69, 72, 102, 364 W Strong Company, 361 Weak Company, 361 Strong Supplementation Principle (SSP), Weak supplementation principle (WSP), 63, 63, 64, 66–68, 76–78, 133, 134, 66, 67, 129–131, 134, 137 136–139, 300 Whitehead, 2, 85–97, 101, 105–108, 110, 113, Subsets, 56, 60, 67, 90, 96, 106, 107, 113, 116, 219, 220, 223 115, 119, 135, 145, 146, 148, 152, Whole-for, 182 154, 156, 157, 169, 197, 206–209, Wholes, 53, 78, 79, 165, 179, 189–201, 222–223, 232, 283, 295–300, 303, 203–211, 293, 339, 340, 353, 313, 355, 368 364 Super strong supplementation principle, 136 WSP. See Weak supplementation principle Supersubstantivalism, 5, 7–11 (WSP) Supertask, 87–90, 95–97 Supremum relation, 126 X Xenogeneic, 171, 181–183, 186 T Tangential ingredient, 226, 236–237, 239, 256 Temporal part (t-part), 1, 5, 6, 10, 12–14, 16, 17, 23, 27, 30–33, 36–41, 43–46, Z 48, 50, 158 Zeno’s Paradox, 87