Provides a comprehensive overview and introduction to the Routledge Handbook of Propositions.

This paper argues that the theory of structured propositions is not undermined by the Russell-Myhill paradox. I develop a theory of structured propositions in which the Russell-Myhill paradox doesn't arise: the theory does not involve ramification or compromises to the underlying logic, but rather rejects common assumptions, encoded in the notation of the $\lambda$-calculus, about what properties and relations can be built. I argue that the structuralist had independent reasons to reject these underlying assumptions. The theory is given both a (...) diagrammatic representation, and a logical representation in a novel language. In the latter half of the paper I turn to some technical questions concerning the treatment of quantification, and demonstrate various equivalences between the diagrammatic and logical representations, and a fragment of the $\lambda$-calculus. (shrink)

On a highly influential way to think of modality, that I call ‘relationalism’, the modality of a state is explained by its being composed of properties, and these properties being related by a higher-order and primitively modal relation. Examples of relationalism are the Dretske-Tooley-Armstrong account of natural necessity, many dispositional essentialist views, and Wang’s incompatibility primitivism. I argue that relationalism faces four difficulties: that the selection between modal relations is arbitrary, that the modal relation cannot belong to any logical order, (...) that to explain how the modal relation can relate properties of different adicities additional ideological complexity has to be introduced, and that not all modal constraints are relational. From the discussion, I will extract desiderata for a successor theory of modality. (shrink)

Relationism holds that objects entirely depend on relations or that they must be eliminated in favour of the latter. In this article, I raise a problem for relationism. I argue that relationism cannot account for the order in which non-symmetrical relations apply to their relata. In Section 1, I introduce some concepts in the ontology of relations and define relationism. In Section 2, I present the Problem of Order for non-symmetrical relations, after distinguishing it from the Problem of Differential Application. (...) I also examine four main existing strategies to solve it. In Section 3, I develop my argument. The first step consists in arguing that—among those strategies—relationism can only accept directionalism. The second step consists in arguing that directionalism is affected by a serious problem: the Problem of Converses. I also show that relationists who embrace directionalism cannot solve this problem. In Section 4, I introduce and rebut several strategies on behalf of relationists to cope with my argument. In Section 5, I briefly draw some conclusions. (shrink)

This paper develops a novel theory of abstraction—what we call collective abstraction. The theory solves a notorious problem for noneliminative structuralism. The noneliminative structuralist holds that in addition to various isomorphic systems there is a pure structure that can be abstracted from each of these systems; but existing accounts of abstraction fail for nonrigid systems like the complex numbers. The problem with the existing accounts is that they attempt to define a unique abstraction operation. The theory of collective abstraction instead (...) simultaneously defines a collection of distinct abstraction operations, each of which maps a system to its corresponding pure structure. The theory is precisely formulated in an essentialist language. This allows us to throw new light on the question to what extent structuralists are committed to symmetric dependence. Finally, we apply the theory of collective abstraction to solve a problem about converse relations. (shrink)

‘Instantiation-directed slot theorists’ believe that properties/relations have slots which are filled by their instances/relata e.g., where Abigail is taller than Bronia, there are two slots in the relation Taller Than such that Abigail fills the first slot and Bronia fills the second. This crude statement of the theory runs into ‘The Problem of Filling’, whereby a natural understanding of the relation between slots, filling, and instantiation leads to absurd results. This paper examines a variety of solutions to that problem, one (...) of which is an extension of slot theory that adds an additional category of entities, ‘slotites’. (shrink)

There are two ways to characterize symmetric relations. One is intensional: necessarily, _Rxy_ iff _Ryx_. In some discussions of relations, however, what is important is whether or not a relation gives rise to the same completion of a given type (fact, state of affairs, or proposition) for each of its possible applications to some fixed relata. Kit Fine calls relations that do ‘strictly symmetric’. Is there is a difference between the notions of necessary and strict symmetry that would prevent them (...) from being used interchangeably in such discussions? I show that there is. While the notions coincide assuming an intensional account of relations and their completions, according to which relations/completions are identical if they are necessarily coinstantiated/equivalent, they come apart assuming a hyperintensional account, which individuates relations and completions more finely on the basis of relations’ real definitions. I establish this by identifying two definable relations, each of which is necessarily symmetric but nonetheless results in distinct facts when it applies to the same objects in opposite orders. In each case, I argue that these facts are distinct because they have different grounds. (shrink)

Maureen Donnelly’s (2016) relative positionalism correctly handles any fixed arity relation with any symmetry such a relation can have, yielding the intuitively correct way(s) in which that relation can apply. And it supplies an explanation of what is going on in the world that makes this the case. But it has at least one potential shortcoming — one that its opponents are likely to seize upon: it can only handle relations with fixed arities. It is unable to handle relations with (...) variable arities. I argue that, all else being equal, relative positionalism ought nonetheless to be preferred to its closest competitors — at least to the extent that the explanation it supplies of relational application is plausible — even though those competitors can handle variable arity relations in addition to fixed arity relations. (shrink)

2020 update of the entry "Properties".

Maureen Donnelly has recently argued that directionalism, the view that relations have a direction, applying to their relata in an order, is unable to properly treat certain symmetric relations. She alleges that it must count the application of such a relation to an appropriate number of objects in a given order as distinct from its application to those objects in any other ordering of them. I reply by showing how the directionalist can link the application conditions of any fixed arity (...) relation, no matter its arity or symmetry, and its converse(s) in such a way that directionalism will yield the correct ways in which it can apply. I thus establish that directionalism possesses the same advantage Donnelly's own account of relations, relative positionalism, has over traditional positionalist accounts of relations, which do not properly treat symmetric relations. I then note some advantages that directionalism has over its closest competitors. This includes Donnelly's relative positionalism, since directionalism is not, like relative positionalism, committed to the involvement of relative properties in every irreducibly relational claim. I close by conceding that, as Donnelly notes, directionalism is committed to the primitive relation of order-sensitive relational application. But I don't find this notion as mysterious as Donnelly does. I conclude that, even if one construes this feature of directionalism as a drawback, the two views are at worst at a draw, other things being equal, since this drawback is mitigated by the advantage directionalism has over relative positionalism. (shrink)