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Varieties of Parthood

Technical counterpart to "Parthood-Like Relations" immediately below.

Published in Philosophical Logic: Current Trends in Asia (Springer 2017)

Late draft here.

Parthood‐Like Relations: Closure Principles And Connections To Some Axioms Of Classical Mereology

This paper explores two main claims. The first is that there are general closure principles governing the class of relations that are parthood-like, “part-like,” for short, in a sense to be informally specified. The most distinctive principle suggested is that something that we will call the “overlap-inclusion” of a part-like relation is itself part-like, and our distinctive result is that, given one part-like relation, our principles yield a (possibly distinct) part-like relation that is guaranteed to satisfy the Strong Supplementation axiom of classical mereology. The second main claim is that there is at least one part-like relation that satisfies some axioms that may be described as being “classical mereology with the assumption of anti-symmetry removed.” In particular, if we take set-theoretic membership as a form of parthood, or take the is one of relation from things to pluralities as a form of parthood, we demonstrably get such a relation, provided that our first main claim was correct. A general picture is suggested of how part-like relations among concrete and abstract objects might be structured.

Published in Philosophical Perspectives 2016

Parthood, Pluralities, and Sets (handout)

Handout from a talk I gave at the 2016 California Metaphysics Conference. States some basic ideas about how distinct part-like relations are related to each other; e.g., even if a member of a set is not a "part" of it in the same sense in which a leg might be part of a cat, a relation that includes the leg-to-cat connection and the member-to-set connection might reasonably be regarded as the, or, more likely, a, parthood relation. Similarly, one of some things might be counted as "part of" them; e.g., John is part of John and Paul (where the plural predication is read collectively). And again, some things xx that are among some things yy might be counted as "part of" them; e.g., John and Paul are (collectively) part of (collectively) John, Paul, George, and Ringo. A conjecture was sketchily presented, to the effect that there is a natural stopping point or fixed point for the iterating of logical operations that take us from one part-like relation to another. Some remarks were also made relating these ideas to some similar ideas found in Kit Fine's "Towards a theory of part."

Click here for handout

Logical considerations on composition as identity

Suppose that a certain broom is composed of two parts: a long wooden handle screwed into a brush. It is natural to express the relationship between the broom, on the one hand, and the handle and brush, on the other, with such remarks as: "The broom just is the handle and brush. There is nothing more to the broom than the handle and brush. The handle and the brush, together, are the broom." Composition as identity (CAI) may be taken as a slogan appropriate for, or a very rough formulation of, a general idea that can take rather different specific forms. With regard to the broom, one might (rightly or wrongly) claim that the broom is identical with the handle and the brush. The idea that in general composition is identity has as its most extreme form something like this: whenever something is composed of some things, it is them. But there are more restricted doctrines. It might be thought that for a certain class of things C: whenever something of class C is composed of some things, it is identical with them. And it might be thought that for any things that meet a certain qualification Q, they are identical with something, and indeed compose that thing. Restricted doctrines say that sometimes composition is identity, leaving it open that there are cases of composition or (plural) identity that are not cases in which composition is identity. The goal of this chapter is to formulate, very carefully, some of the logical principles that bear on the tenability of such doctrines. The general moral is that while logic does impose some non-trivial restrictions, it nonetheless leaves some interesting options open.

The final publication is in Composition as Identity Edited by A. J. Cotnoir and Donald L. M. Baxter (Oxford 2014).

Natural mereology and classical mereology

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.

The final publication is in Mereology and the Sciences: Parts and Wholes in the Contemporary Scientific Context, Editors: Claudio Calosi and Pierluigi Graziani.

Tensed mereology

Classical mereology (CM) is usually taken to be formulated in a tenseless language, and is therefore associated with a four-dimensionalist metaphysics. This paper presents three ways one might integrate the core idea of flat plenitude, i.e., that every suitable condition or property has exactly one mereological fusion, with a tensed logical setting. All require a revised notion of mereological fusion. The candidates differ over how they conceive parthood to interact with existence in time, which connects to the distinction between endurance and perdurance. Similar issues arise for the integration of mereology with modality, and much of our discussion applies to this project as well.

The final publication is available at In Journal of Philosophical Logic.

Semantics as information about semantic values

Discussion of Kit Fine's Semantic Relationism. In Philosophy and Phenomenological Research issue as part of symposium on Fine's book.

Review of Kathrin Koslicki's The Structure of Objects in Notre Dame Philosophical Reviews

What is Classical Mereology? (draft)

Classical mereology is a formal theory of the part-whole relation, essentially involving a notion of mereological fusion, or sum. The theorems are very close to those for complete Boolean algebras. There are various different definitions of fusion in the literature, and various axiomatizations for classical mereology. In the context of a correct axiomatization, the definitions of fusion are provably equivalent, but they are not inter-changeable in the axioms themselves. We examine the relations between the main definitions of fusion and correct some technical errors in prominent discussions of the axiomatization of mereology. We show the equivalence of four different ways to axiomatize classical mereology, using three different notions of fusion. We also clarify the connection between classical mereology and complete Boolean algebra by giving two "neutral" axiom sets which can be supplemented by one or the other of two simple axioms to yield the full theories; one of these uses a notion of "strong complement" that helps explicate the connections between the theories.
This paper is in Journal of Philosophical Logic; the official publication is available here

Quantifying Weak Emergence

In Minds and Machines.
The concept of weak emergence is a refinement or specification of the intuitive, general notion of emergence. Basically, a fact about a system is said to be weakly emergent if its holding both (i) is derivable from the fundamental laws of the system together with some set of basic (non-emergent) facts about it, and yet (ii) is only derivable in a particular manner, called "simulation." This essay analyzes the application of this notion Conway's Game of Life, and concludes that a modification of the notion would provide a better refinement of the general notion of emergence. It is proposed that emergence be taken as a matter of degree, defined in terms of the amount of simulation required to derive a fact.

How to be an Atomist

A manuscript detailing one branch of thought about identity and plurals: the thesis that talk of things that are not (mereological) atoms can be systemetically and adequately interpreted without talking about anything but mereological atoms and their properties.

Two defenses of composition as identity
Approximate ms of a talk I gave at the 2005 Western Canadian Philosophical Association. (A longer version was given at Reed.)

Two defenses as postscript file (recommended)
Handout from talk as postscript file (recommended)
Two defenses as PDF file (figures may not render properly)
Handout from talk as PDF file (figures may not render properly)

Metaphysical indeterminacy

(with Mahrad Almotahari)

A report of work done together with my student Mahrad Almotahari, funded by a Ruby Grant, administered by Reed College, from the summer of 2004. Argues that the vagueness connected with identity and existence is, at least in some cases, metaphysical indeterminacy. Presents a novel response to Gareth Evans' argument against indeterminacy of identity, on which the determinacy of 'a=a' is denied.

ABSTRACT: We motivate and consider the ramifications of the thesis that there is worldly (or metaphysical) indeterminacy, not mere semantic indeterminacy (or vagueness) about the boundaries and identities of objects. We give general considerations about what objects are that we think help to show that it is plausible that their lack of sharp boundaries is a feature of them, not just of the language used to describe them. Then we consider the difficulties raised by the fact that it is then natural to suppose further that there can be indeterminacy about the identities of objects. In particular, we focus on the argument of Gareth Evans that purports to show (as we interpret it) that indeterminacy of identity statements could not be due to worldly indeterminacy, on pain of logical incoherence. We criticize the response to this argument of Terence Parsons, a leading advocate of worldly indeterminacy of identity. We offer our own approach, on which the threat of EvansÂ' argument can be effectively defused.

Vagueness and the world

A talk I gave at various institutions in 2002. Argues that for certain kinds of cases, vagueness should be seen as a metaphysical phenomenon, and that in these cases, supervaluational treatments of vagueness (like David Lewis') are not correct.

The Nature and Logic of Vagueness

My dissertation, UCLA, 2001, chaired by David Kaplan.

ABSTRACT: The dissertation considers both metaphysical and logical issues related to the vagueness of natural language. The principle metaphysical claim is that the vagueness of language is, at least in some cases, a direct result of indeterminacy in the subject matter of the language, rather than any sort of flaw of the language. A limited defense of this claim is given, as well as criticism of alternative views. A number of logical issues are addressed. First, the relationship between the notion of determinacy and the idea of an unsharp line is considered, and it is suggested that the relationship is not as simple as it may at first seem, and that the idea of the unsharp line may be irreducible. Next, it is urged that there is a methodological fork in the road for the systematic treatment of vague language, including formal semantics. On one path, we accept certain intuitively puzzling propositions, exemplified by Â"This is red or it is not the case that this is red, though it is indeterminate which.Â" On the other path, we reject classical principles of reasoning in our own reasoning both in and about vague language. Some limited arguments are given for taking the former path, and a formal system relevant to this path is motivated and discussed. Finally, both the metaphysical and logical work of the prior parts of the dissertation are brought to bear on the subject of the indeterminacy of identity.