Core Principles

With these provisos, and barring for the moment the complications arising from the consideration of intensional factors (such as time and modalities), we may proceed to review some core mereological notions and principles. Ideally, we may distinguish here between (a) those principles that are simply meant to fix the intended meaning of the relational predicate ‘part’, and (b) a variety of additional, more substantive principles that go beyond the obvious and aim at greater sophistication and descriptive power. Exactly where the boundary between (a) and (b) should be drawn, however, or even whether a boundary of this sort can be drawn at all, is by itself a matter of controversy.

2.1 Parthood as a Partial Ordering

The usual starting point is this: regardless of how one feels about matters of ontology, if ‘part’ stands for the general relation exemplified by (1)–(8) above, and perhaps also (12)–(15), then it stands for a partial ordering—a reflexive, transitive, antisymmetric relation:

(16)	Everything is part of itself.
(17)	Any part of any part of a thing is itself part of that thing.
(18)	Two distinct things cannot be part of each other.

As it turns out, most theories put forward in the literature accept (16)–(18). Some misgivings are nonetheless worth mentioning that may, and occasionally have been, raised against these principles.
Concerning reflexivity (16), two sorts of worry may be distinguished. The first is that many legitimate senses of ‘part’ just fly in the face of saying that a whole is part of itself. For instance, Rescher (1955) famously objected to Leonard and Goodman's theory on these grounds, citing the biologists' use of ‘part’ for the functional subunits of an organism as a case in point: no organism is a functional subunit of itself. This is a legitimate worry, but it appears to be of little import. Taking reflexivity (and antisymmetry) as constitutive of the meaning of ‘part’ simply amounts to regarding identity as a limit (improper) case of parthood. A stronger relation, whereby nothing counts as part of itself, can obviously be defined in terms of the weaker one, hence there is no loss of generality (see Section 2.2 below). Vice versa, one could frame a mereological theory by taking proper parthood as a primitive instead. As already Lejewski (1957) noted, this is merely a question of choosing a suitable primitive, so nothing substantive follows from it. (Of course, if one thinks that there are or might be objects that are not self-identical, for instance because of the loss of individuality in the quantum realm, or for whatever other reasons, then such objects would not be part of themselves either, yielding genuine counterexamples to (16). Here, however, we stick to a notion of identity that obeys traditional wisdom, which is to say a notion whereby identity is an equivalence relation subject to Leibniz's law.) The second sort of worry is more serious, for it constitutes a genuine challenge to the idea that (16) expresses a principle that is somehow constitutive of the meaning of ‘part’, as opposed to a substantive metaphysical thesis about parthood. Following Kearns (2011), consider for instance a scenario in which an enduring wall, W, is shrunk down to the size of a brick and eventually brought back in time so as to be used to build (along with other bricks) the original W. Or suppose wall W is bilocated to my left and my right, and I shrink it to the size of a brick on the left and then use it to replace a brick from W on the right. In such cases, one might think that W is part of itself in a sense in which ordinary walls are not, hence that either parthood is not reflexive or proper parthood is not irreflexive. For another example (also by Kearns), if shapes are construed as abstract universals, then self-similar shapes such as fractals may very well be said to contain themselves as parts in a sense in which other shapes do not. Whether such scenarios are indeed possible is by itself a controversial issue, as it depends on a number of background metaphysical questions concerning persistence through time, location in space, and the nature of shapes. But precisely insofar as the scenarios are not obviously impossible, the generality and metaphysical neutrality of (16) may be questioned. (Note that those scenarios also provide reasons to question the generality of many other claims that underlie the way we ordinarily talk, such as the claim that nothing can be larger than itself, ornext to itself, or qualitatively different from itself. Such claims might be even more entrenched in common sense than the claim that proper parthood is irreflexive, and parthood reflexive; yet this is hardly a reason to hang on to them at every cost. It simply shows that our ordinary talk does not take into account situations that are—admittedly—extraordinary.)
Similar considerations apply to the transitivity principle, (17). On the one hand, several authors have observed that many legitimate senses of ‘part’ are non-transitive, fostering the study of mereologies in which (17) may fail (Pietruszczak 2014). Examples would include: (i) a biological subunit of a cell is not a part of the organ(ism) of which that cell is a part; (ii) a handle can be part of a door and the door of a house, though a handle is never part of a house; (iii) my fingers are part of me and I am part of the team, yet my fingers are not part of the team. (See again Rescher 1955 along with Cruse 1979 and Winston et al. 1987, respectively; for other examples see Iris et al. 1988, Moltmann 1997, Hossack 2000, Johnston 2002, 2005, Johansson 2004, 2006, and Fiorini et al. 2014). Arguably, however, such misgivings stem again from the ambiguity of the English word ‘part’. What counts as a biological subunit of a cell may not count as a subunit, i.e., a distinguished part of the organ, but that is not to say that it is not part of the organ at all. Similarly, if there is a sense of ‘part’ in which a handle is not part of the house to which it belongs, or my fingers not part of my team, it is a restricted sense: the handle is not afunctional part of the house, though it is a functional part of the door and the door a functional part of the house; my fingers are not directly part of the team, though they are directly part of me and I am directly part of the team. (Concerning this last case, Uzquiano 2004: 136–137, Schmitt 2003: 34, and Effingham 2010b: 255 actually read (iii) as a reductio of the very idea that the group-membership relation is a genuine case of parthood, as mentioned above ad (11).) It is obvious that if the interpretation of ‘part’ is narrowed by additional conditions, e.g., by requiring that parts make a functional or direct contribution to the whole, then transitivity may fail. In general, if x is a φ-part of y and y is a φ-part of z, x need not be a φ-part of z: the predicate modifier ‘φ’ may not distribute over parthood. But that shows the non-transitivity of ‘φ-part’, not of ‘part’, and within a sufficiently general framework this can easily be expressed with the help of explicit predicate modifiers (Varzi 2006a; Vieu 2006; Garbacz 2007). On the other hand, there is again a genuine worry that, regardless of any ambiguity concerning the intended interpretation of ‘part’, (17) expresses a substantive metaphysical thesis and cannot, therefore, be taken for granted. For example, it turns out that time-travel and multi-location scenarios such as those mentioned in relation to (16) may also result in violations of the transitivity of both parthood (Effingham 2010a) and proper parthood (Gilmore 2009; Kleinschmidt 2011). And the same could be said of cases that involve no such exotica. For instance, Gilmore (2014) brings attention to the popular theory of structured propositions originated with Russell (1903). Already Frege (1976: 79) pointed out that if the constituents of a proposition are construed mereologically as (proper) parts, then we have a problem: assuming that Mount Etna is literally part of the proposition that Etna is higher than Vesuvius, each individual piece of solidified lava that is part of Etna would also be part of that proposition, which is absurd. The worse for Russell's theory of structured propositions, said Frege. The worse, one could reply, for the transitivity of parthood (short of claiming that the argument involves yet another equivocation on ‘part of’).
Concerning the antisymmetry postulate (18), the picture is even more complex. For one thing, some authors maintain that the relationship between an object and the stuff it is made of provides a perfectly ordinary counterexample of the antisymmetry of parthood: according to Thomson (1998), for example, a statue and the clay that constitutes it are part of each other, yet distinct. This is not a popular view: as already mentioned, most contemporary authors would either deny that material constitution is a relation of parthood or else treat it as improper parthood, i.e., identity, which is trivially antisymmetric (and symmetric). Moreover, those who regard constitution as a genuine case of proper parthood tend to follow Aristotle's hylomorphic conception and deny that the relation also holds in the opposite direction: the clay is part of the statue but not vice versa (see e.g. Haslanger 1994, Koslicki 2008). Still, insofar as Thomson's view is a legitimate option, it represents a challenge to the putative generality of (18). Second, one may wonder about the possibility of unordinary cases of symmetric parthood relationships. Sanford (1993: 222) refers to Borges's Aleph as a case in point: “I saw the earth in the Aleph and in the earth the Aleph once more and the earth in the Aleph …”. In this case, a plausible reply is simply that fiction delivers no guidance to conceptual investigations: conceivability may well be a guide to possibility, but literary fantasy is by itself no evidence of conceivability (van Inwagen 1993: 229). Perhaps the same could be said of Fazang's Jeweled Net of Indra, in which each jewel has every other jewel as part (Jones 2012). However, other cases seem harder to dismiss. Surely the Scholastics were not merely engaging in literary fiction when arguing that each person of the Trinity is a proper part of God, and yet also identical with God (see e.g. Abelard, Theologia christiana, bk. III). And arguably time travel is at least conceivable, in which case again (18) could fail: if time-traveling wall W ends up being one of the bricks that compose (say) its own bottom half, H, then we have a conceivable scenario in which W is part of H and H is part of W while W ≠ H (Kleinschmidt 2011). Third, it may be argued that antisymmetry is also at odds with theories that have been found acceptable on quite independent grounds. Consider again the theory of structured propositions. If A is the proposition that the universe exists—where the universe is something of which everything is part—and if A is true, then on such a theory the universe would be a proper part of A; and since A would in turn be a proper part of U, antisymmetry would be forfeit (Tillman and Fowler 2012). Likewise, if A is the proposition that B is true, and B the proposition that A is contingent, then again A and B would be part of each other even though A ≠ B(Cotnoir 2013b). Finally, and more generally, it may be observed that the possibility of mereological loops is to be taken seriously for the same sort of reasons that led to the development of non-well-founded set theory, i.e., set theory tolerating cases of self-membership and, more generally, of membership circularities (Aczel 1988; Barwise and Moss 1996). This is especially significant in view of the possibility of reformulating set theory itself in mereological terms—a possibility that is extensively worked out in the works of Bunt (1985) and especially Lewis (1991, 1993b) (see also Burgess 2015 and Hamkins and Kikuchi forthcoming). For all these reasons, the antisymmetry postulate (18) can hardly be regarded as constitutive of the basic meaning of ‘part’, and some authors have begun to engage in the systematic study of “non-well-founded mereologies” in which (18) may fail (Cotnoir 2010; Cotnoir and Bacon 2012; Obojska 2013).
In the following we aim at a critical survey of mereology as standardly understood, so we shall mainly confine ourselves to theories that do in fact accept the antisymmetry postulate along with both reflexivity and transitivity. However, the above considerations should not be dismissed. On the contrary, they are crucially relevant in assessing the scope of mereology and the degree to which its standard formulations and extensions betray intuitions that may be found too narrow, false, or otherwise problematic. Indeed, they are crucially relevant also in assessing the ideal desideratum mentioned at the beginning of this section—the desideratum of a neat demarcation between core principles that are simply meant to fix the intended meaning of ‘part’ and principles that reflect more substantive theses concerning the parthood relation. Classical mereology takes the former to include the threefold claim that ‘part’ stands for a reflexive, transitive and antisymmetric relation, but this is not to say that “anyone who seriously disagrees with them had failed to understand the word” (Simons 1987: 11), just as departure from the basic principles of classical logic need not amount to a “change of subject” (Quine 1970: 81). And just as the existence of widespread and diversified disagreement concerning the laws of logic may lead one to conclude that “for all we know, the only inference left in the intersection of (unrestricted) alllogics might be the identity inference: From A to infer A” (Beall and Restall 2006: 92), so one might take the above considerations and the corresponding development of non-classical mereologies to indicate that there may be “no reason to assume that any useful core mereology […] functions as a common basis for all plausible metaphysical theories” (Donnelly 2011: 246)

Core Principles

2.1 Parthood as a Partial Ordering

Core Principles :: تعاليق

Core Principles