In this paper we study the termination behavior of Russell’s description elimination rewrite system. We discuss certain claims made by Kripke (2005) in his paper concerning the possible nontermination of elimination of descriptions.

In the ordinary way of representing relations, the order of the relata plays a structural role, but in the states themselves such an order often does not seem to be intrinsically present. An alternative way to represent relations makes use of positions for the arguments. This is no problem for the love relation, but for relations like the adjacency relation and cyclic relations, different assignments of objects to the positions can give exactly the same states. This is a puzzling situation. (...) The question is what is the internal structure of relations? Is the use of positions still justified, and if so, what is their ontological status? In this paper mathematical models for relations are developed that provide more insight into the structure of relations “out there” in the real world. (shrink)

We study the logical structure of relations, and in particular the notion of occurrences of objects in a state. We start with formulating a number of principles for occurrences and defining corresponding mathematical models. These models are analyzed to get more insight in the formal properties of occurrences. In particular, we prove uniqueness results that tell us more about the possible logical structures relations might have.

Argument-places play an important role in our dealing with relations. However, that does not mean that argument-places should be taken as primitive entities. It is possible to give an account of relations in which argument-places play no role. But if argument-places are not basic, then what can we say about their identity? Can they, for example, be reconstructed in set theory with appropriate urelements? In this article, we show that for some relations, argument-places cannot be modeled in a neutral way (...) in V[A], the cumulative hierarchy with basic ingredients of the relation as urelements. We argue that a natural way to conceive of argument-places is to identify them with abstract, structureless points of a derivative structure exemplified by positional frames. In case the relation has symmetry, these points may be indiscernible. (shrink)

How we talk about relations has a great influence on how we think about relations. By saying that Spain defeated the Netherlands we obviously say something entirely different from saying that the Netherlands defeated Spain. This makes many of us think that in the underlying relation itself one of the relata comes first and the other comes second. However, there are good reasons to view the order as a representational artifact. In this paper I present a new logic that allows (...) us to think and talk about relations in a coordinate-free manner. The logic is minimalistic in the sense that it has no predicates except equality, but its expressive power is the same as that of first-order predicate logic. (shrink)

A theory of relations is presented that provides a detailed account of the logical structure of relational complexes. The theory draws a sharp distinction between relational complexes and relational states. A salient difference is that relational complexes belong to exactly one relation, whereas relational states may be shared by different relations. Relational complexes are conceived as structured perspectives on states ‘out there’ in reality. It is argued that only relational complexes have occurrences of objects, and that different complexes of the (...) same relation may correspond to the same state. (shrink)

A new logic is presented without predicates—except equality. Yet its expressive power is the same as that of predicate logic, and relations can faithfully be represented in it. In this logic we also develop an alternative for set theory. There is a need for such a new approach, since we do not live in a world of sets and predicates, but rather in a world of things with relations between them.