Abstract
Two apparently opposing philosophical views on mereological issues are contrasted, and a device is suggesting for assessing the extent to which their difference is terminological, rather than substantial. According to the view associated with Classical Mereology, the part-whole relation is guaranteed, a priori, to have a particular and neat algebraic structure. According to a more naturalistic view, the global structure of the part-whole relation turns instead on how things in nature happen to be organized. The main technical idea of the paper is that there is a non-arbitrary way to transform any relation into one that obeys the axioms of Classical Mereology. The main philosophical idea is that if we apply the transformation to the ontology and structure the naturalistic philosopher believes in, the resulting Classical relational structure may be ontologically acceptable to the naturalistic philosopher, and the relation that relates the objects in the structure might be sufficiently “natural” that the naturalistic philosopher ought to acknowledge that it deserves the label “part-whole relation.” Thus, in the most dramatic case, the naturalistic philosopher might hold a view that turns out to be merely terminologically different from that of the Classical Mereologist. Sets, or pluralities, of the original naturalistic objects will typically be “added,” by the transformation, to the original ontology to which it is applied. This raises questions, only briefly addressed here, about the place of such entities in the study of part-whole relations.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
If we reject a sharp distinction between a priori and not, in favor a graduated distinction, then we may substitute “very close to as a priori as it gets” for “a priori” here.
- 2.
Lewis (1991, p. 75).
- 3.
“Sparse,” because not every arbitrary collection corresponds to a universal.
- 4.
See below for a formally exact statement of this constraint.
- 5.
One can get an especially clear view of the effect of Φt by considering how it can be built up from iterated application of a simpler transformation. Define \(\varPhi _{t_{0}}\) so that
$$\displaystyle{\varPhi _{t_{0}}(\langle X,\mathbf{R}\rangle ) =\langle X,\mathbf{R} \cup \{\langle x,z\rangle (\in X \times X): \exists y\,(x\mathbf{R}y \wedge y\mathbf{R}z)\}\rangle.}$$\(\varPhi _{t_{0}}(\mathfrak{B})\) is a first approximation of \(\varPhi _{t}(\mathfrak{B})\); a second approximation is \(\varPhi _{t_{0}}(\varPhi _{t_{0}}(\mathfrak{B}))\). One can show that \(\varPhi _{t}(\mathfrak{B})\) is the “limit” of the approximations. More precisely: let \(\mathfrak{B}^{0}\) be \(\mathfrak{B} =\langle X,\mathbf{R}\rangle\) and let \(\mathfrak{B}^{i+1}\) be \(\varPhi _{t_{0}}(\mathfrak{B}^{i})\). Let Ri be the relation in \(\mathfrak{B}^{i}\). Then Rt, the relation of \(\varPhi _{t}(\mathfrak{B})\), is the relation
$$\displaystyle{\{\langle x,y\rangle (\in X \times X): \exists i \in \mathbb{N}\ \langle x,y\rangle \in \mathbf{R}^{i}\}.}$$ - 6.
See Sect. 4 of Hovda (2009); the five conditions here correspond to the five axioms in the last of the five axiom-sets given there.
- 7.
Given any structure \(\langle X,\mathbf{R}\rangle\), let \(A = \mathcal{P}(X)\). Given any x ∈ X, let \([x]\ =\ \{ y \in X: x\mathbf{R}y \wedge y\mathbf{R}x\}\). Let B be \(\{e \in A: \exists x \in X \wedge e = [x]\}\). Let S be the relation on B defined as follows: for any e and f in B,
-
eSf if and only if (∃z ∈ e)(∃w ∈ f) zRw.
Then define \(\varPhi _{a}(\langle X,\mathbf{R}\rangle ) =\langle B,\mathbf{S}\rangle\). In general, this transformation is much more natural when combined with prior application of Φr and Φt; the composite Φa ∘Φr ∘Φt transforms any relational structure into a partial ordering.
-
- 8.
For a fully general definition, we need some way to tell apart the members of a cluster that are of lower rank from the others; in our application, these were members of D rather than of E∘. Assuming that our set theory provides a natural way to rank everything in the universe (as does Zermelo-Fraenkel set theory with ur-elements, choice, and foundation) a general transformation Φρ on arbitrary \(\langle X,\mathbf{R}\rangle\) may be defined by first applying Φt, then, taking a cluster to be a maximal set of members of X that bear Rt to one another, for each cluster, choosing its single lowest ranked member, if there is one, and the union of all its lowest-ranked sets, otherwise. Φρ is then defined by taking the “chosen” items as carrier set and taking the “inherited” relation.
- 9.
- 10.
See Hovda (2013) for a discussion of the interaction of formalistic mereology with time and tense. The main idea pursued in Hovda (2013) is to re-conceive formalistic mereology while taking tense (or metaphysical modality) seriously, and allowing objects (including fusions) to change their parts. To wed, in a natural way, the approach in Hovda (2013) with the idea in this paper would seem to require a set theory in which sets can change their members. Such a set theory should be buildable by modifying untensed set theory in something like the manner that Hovda (2013) modifies untensed Classical Mereology to yield a tensed mereology.
Here is the barest sketch of how this would go. Naïve Set Theory consists of the axiom of Extensionality (sets x and y are the same iff x and y have the same members) together with the Naïve Comprehension scheme for set existence. The scheme is this (for any predicate ϕ(x) in which x occurs free and y does not, an instance of the scheme is): there exists at least one set y such that: for all x, x ∈ y iff ϕ(x). Let Tensed Naïve Set Theory be Tensed Extensionality (sets x and y are the same iff it is always the case that x and y have the same members) together with a tensed correlate of Comprehension: there exists at least one set y such that it is always the case that for all x (x ∈ y iff ϕ(x)). An instance of this scheme thus implies that there is a set y such that at every time, for every x, x ∈ y at that time iff x is (at that time) a dog. This set would have no members when there are no dogs, and its membership would wax and wane with the existence of dogs. Of course the tensed scheme inherits the inconsistency of the Naïve Comprehension scheme; to find a reasonable, consistent tensed set theory, one would modify ZFC with ur-elements, or the like.
References
Fine, K. (1999). Things and their parts. Midwest Studies in Philosophy, 23, 61–74.
Fine, K. (2010). Towards a theory of part. Journal of Philosophy, 107(11), 559–589.
Hovda, P. (2013). Tensed mereology. Journal of Philosophical Logic, 42(2), 241–283. doi:10.1007/s10992-011-9220-4.
Hovda, P. (2009). What is classical mereology? Journal of Philosophical Logic, 38(1), 55–82. http://dx.doi.org/10.1007/s10992-008-9092-4.
Koslicki, K. (2008). The structure of objects. Oxford/New York: Oxford University Press.
Lewis, D. (1983). New work for a theory of universals. Australasian Journal of Philosophical, 61, 343–377.
Lewis, D. (1991). Parts of classes. Oxford/Cambridge: Blackwell.
Sider, T. (2011). Writing the book of the world. Oxford/New York: Oxford University Press.
Van Inwagen, P. (1990). Material beings. Ithaca: Cornell University Press.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Hovda, P. (2014). Natural Mereology and Classical Mereology. In: Calosi, C., Graziani, P. (eds) Mereology and the Sciences. Synthese Library, vol 371. Springer, Cham. https://doi.org/10.1007/978-3-319-05356-1_7
Download citation
DOI: https://doi.org/10.1007/978-3-319-05356-1_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-05355-4
Online ISBN: 978-3-319-05356-1
eBook Packages: Humanities, Social Sciences and LawPhilosophy and Religion (R0)