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Notes toMereology

1. The term was coined in1927 by Leśniewski (see below), probably as a variant of the term‘merology’ originally used to indicate the field ofanatomy concerned with body fluids and elementary tissues; see Simons(1997: n. 4). In some literature, ‘mereology’ is also usedwith reference to work in General Systems Theory (not covered by thisentry) devoted to the study of system decomposition, as inMesarovićet al. (1970). See also Winther (2011).

2. For detailed treatments ofHusserl's formal theory of parts and wholes, see Sokolowski (1968),Smith and Mulligan (1982), Simons (1982), Null (1983), Blecksmith andNull (1991), Fine (1995b), and Casari (2000, 2007). On Brentano'stheory, see Baumgartner and Simons (1993) and Baumgartner (2013).

3. A detailed exposition ofLeśniewski's mereology may be found in Simons (1987:64–81). For comprehensive surveys, see also Luschei (1965), Clay(1981), Miéville (1984), Gessler (2005), and Urbaniak (2013:ch. 5), as well as the entry onLeśniewski. A simplified versionof Leśniewski's mereology was presented in Tarski (1937).

4. An early version of theCalculus was developed in ch. 4 of Leonard's (1930) dissertation(written under Whitehead's supervision) and was presented under thesame title by Leonard and Goodman at the 1936 meeting of theAssociation for Symbolic Logic in Cambridge, MA. For a comprehensiveaccount, see Eberle (1970) and Breitkopf (1978). For an historicalreconstruction, see Cohnitz and Rossberg (2006: ch. 4) and Rossberg(2009). On Whitehead's precursory work, see Simons (1991b).

5. Actually, the original Calculus of Individuals had variables for classes; a class-free, purely nominalistic version of the system appeared later in Goodman (1951). On the link between mereology and nominalism, see Eberle 1970. For a recent characterization, see Meyers (2014).

6. In the literature, theparthood relation is often represented by non-alphabetic symbolssupporting infix notation, such as ‘≤’ or‘<’ (the notation used in Leonard and Goodman 1940 andfavored by Simons 1987) and the like. The same applies to thenon-primitive mereological relations defined in the remainder of thisentry. For instance, proper parthood is sometimes symbolized as‘≪’ (but also, again, as ‘<’),overlap as ‘○’, disjointness as‘⥍’, ‘≀’,‘†’, ‘|’, or ‘)(’,etc. This has resulted in a plethora of different notations that isboth overwhelming and confusing, as no canonical choice has beensettled upon. Here we stick to the safer practice of sacrificing allinfix notation in favor of ordinary, mnemonic upper-case letters(single or compound), with the only exception of the identitypredicate.

7. This choice of logic isnot without consequences. Particularly when it comes to themereological operators of sum, product, etc. defined in Sections4.2–4.4, a free logic would arguably provide a more adequatebackground (see Eberle 1970, Simons 1991), as would a logic equippedwith the apparatus of plural quantification (Lewis 1991). However,here we shall go along with the simplifications afforded by theassumption of a classic logical background (with descriptive termstreatedà la Russell).

8. The labels follow Varzi (1996) and Casati and Varzi (1999, ch. 3).

9. That this is just a matterof “choosing a suitable primitive” is confirmed by thefact that in 1920 Leśniewski himself provided an alternativeaxiomatization of Mereology based on ‘P’ (see1927–1931, ch. VII). It should be noted, however, thatthroughout his writings Leśniewski used ‘part’(część) in the sense in which we are using‘proper part’ (PP). His word for our use of‘part’ (P) was ‘ingredient’(ingredyens), apparently following a suggestion by LucjanZarzecki. See Leśniewski (1916, §2).

10. The version of theCalculus of Individuals presented in Leonard’s (1930) dissertationused yet another primitive, corresponding to the sum operator definedin (40_i) below (withi=3). The idea ofusing ‘D’ as a primitive may also be found inLeśniewski, who provided an axiomatization of Mereology based onthis predicate—in his terminology: ‘exterior’(zewnętrzny)—already in 1921; see(1927–1931, ch. X).

11. Up to the Winter 2012edition. References to this entry in the literature may be affected bythe current revision.

12. Thanks to PaoloMaffezioli for helpful comments on this point.

13. In the literature,(P.4) is sometimes called ‘Weak Supplementation’, incontrast to the Strong Supplementation principle (P.5) discussed inSection 3.2. This is also Simons's terminology. (To be precise,Simons's formulation of (P.4) uses ‘PP’ also in theconsequent, as in (P.4′) below, but in a standard setting thedifference is immaterial owing to the reflexivity and transitivity ofP.)

14. Strictly speaking, onemay draw a distinction between the matter of which an object is made(feline tissue, clay) and the specific portion or lump of matter thatactually constitutes the object; see Chappel (1973), Laycock (1975),Grandy (1975), Burge (1977), and H. Cartwright (1979) for some classicworks on the distinction and the entry ontheMetaphysics of MassExpressions for a comprehensive survey. We shall ignore thedetails here, but see Zimmerman (1995), Koslicki (1999), Barnett(2004), Kleinschmidt (2007), Donnelly and Bittner (2008), Laycock(2011), and Tanksley (2010) for some recent discussion relevant to thecases at issue.

15. (P.6) is sometimescalled the ‘Remainder Principle’. See e.g. Simons (1987:88).

16. On this point, I amthankful to Anthony Shiver and Aaron Cotnoir for helpfuldiscussion.

17. In Whitehead (1920: 76)the axiom is initially assumed to hold only for finite events, but therestriction is dropped in “reading over the proofs”(pp. 197–198).

18. Both terms go back toLeśniewski and are used interchangeably in most currentliterature. Some authors, however, use ‘sum’ specificallyfor the notion corresponding to (39₂) (or its infinitaryextension (52₂) discussed below) and ‘fusion’for the notion corresponding to (39₃) (or(52₃)). See e.g. Gruszczyński and Pietruszczak (2010)and Gruszczyński (2013).

19. Given Reflexivity andTransitivity, the definiens in (39₁) is equivalent to

This is how the notion of a sum₁ is sometimes defined inthe literature, for ease of comparison with sum₂. Fordefinitions that do not presuppose Transitivity, see Pietruszczak(2014).

20. If the condition is notsatisfied, the sum may not exist, in which case the standard treatmentof descriptive terms that we are assuming implies that thecorresponding instances of the principles that follow arefalse. Strictly speaking, (41)–(47) should therefore be inconditional form. For instance, (41) should read

On this understanding, we use the non-conditional form forperspicuity. Ditto for (50)–(51) below.

21. Again, givenReflexivity and Transitivity, the definiens in (52₁) isequivalent to

22. Actually, Tarski'sresult refers to a stronger version ofGEM in whichinfinitary sums are characterized using explicit quantification oversets, rather than schematic formulas. This is of course a relevantdifference, in view of Cantor's (1891) theorem. For set-freeformulations which, like those considered here, strictly adhere to astandard first-order language with a denumerable supply of openformulas, the correct way of summarizing the algebraic strength ofGEM is this: Any model of this theory is isomorphicto a Boolean subalgebra of a complete Boolean algebra with the zeroelement removed—a subalgebra that is not necessarily complete ifZermelo-Frankel set theory with Choice is consistent. SeePontow and Schubert (2006), Theorem 34, for details and proof.

23. Actually, the axiomsappear in the 1956 English translation but not the French 1929original. For details, see Betti (2013).

24. June 2003 edition. Theinaccuracy was corrected in the Spring 2009 edition.

25. Some authors use‘universalism’ for the narrower thesis that everycollection of distinct material objects compose a further materialobject (see e.g. Effingham 2011b). In the general sense represented by(P.15), universalism is also known as ‘conjunctivism’ (VanCleve 1986, Chisholm 1987), ‘collectivism’ (Hoffman andRosenkrantz 1999), or ‘maximalism’ (Simons 2006); otherauthors speak more generally of ‘unrestricted composition’(Lewis 1986), or the ‘general fusion principle’ (Casatiand Varzi 1999), or simply the ‘fusion principle’ (Heller1990). In the presence ofU, the principle is also closelyrelated to the ‘doctrine of arbitrary undetached parts’attacked by van Inwagen (1981), though, again, the latter doctrine islimited to thematerial content of occupiable regions ofspace.

26. Of course, whichmereological principles should hold remains controversial, and maydepend on the specificde dicto account one endorses. Seee.g. Donnelly (2014: 56ff) on the status of the UnrestrictedSum₂ principle.

27. Strictly speaking,Parsons relies on the notion of a sum₁, hence unrestrictedminimal upper bounds, but the argument applies also under the otherconstruals of that notion examined in Section 4.2.

28. Strictly speaking,Smith works with a notion of (concrete) part that involves a doubleworld-time index, but for the simple principles under discussion hereboth indices can be omitted as irrelevant.