Abelard's Theory of Relations: Reductionism and the Aristotelian Tradition
Brower, Jeffrey E., The Review of Metaphysics
Due to the influence of Bertrand Russell and Gottlob Frege, twentieth-century philosophers have devoted a great deal of attention to questions concerning the logic and metaphysics of relations. However, systematic philosophical interest in relations does not originate in the twentieth century, or even in the modern era. On the contrary, it originates in antiquity, dating back at least to Aristotle's short treatise, the Categories.(1) In the Categories, Aristotle identifies relations (or relatives, ta pros ti) as one of the ten irreducible kinds of being, and devotes an entire chapter--the seventh chapter of his treatise--to analyzing their nature and ontological status.
Aristotle's discussion in Categories 7 provides the starting point for a long and rich tradition of thinking about relations, one which stems from antiquity, runs through the Middle Ages, and eventually makes its way into the early modern period. What is distinctive about this tradition is the commitment of its adherents to the view that relations are in some sense reducible to the monadic properties of related things. Twentieth-century philosophers typically assume that, if Simmias is taller than Socrates, this is to be explained by an entity to which both Simmias and Socrates are somehow jointly attached (namely, the dyadic or two-place property being-taller-than). By contrast, Aristotle and his followers assume that Simmias's being taller than Socrates is to be explained by a pair of monadic properties, one of which inheres hi Simmias and points him toward (pros) Socrates, and another of which inheres in Socrates and points him toward Simmias.
Despite the prominence of the Aristotelian tradition in the history of philosophy, and despite the stature of the philosophers whose support it claims, reductionism about relations is now widely rejected on the basis of broadly logical considerations. Bertrand Russell, for example, has argued that the meaning of relational propositions is unanalyzable;(2) and C. I. Lewis and C. H. Langford have constructed a formal proof which allegedly shows that dyadic relations cannot, without contradiction, be reduced to monadic predicates or concepts.(3) Due to the influence of these sorts of considerations, there is widespread conviction that advances in twentieth-century logic have discredited theories that are in any way reductive of relations. In this paper, I challenge the reigning consensus by examining one theory representative of the Aristotelian tradition. On the basis of this examination, I argue that reductive theories are capable of far more subtlety and sophistication than contemporary philosophers have recognized, and that when properly understood, they can be defended against all the standard logical objections.
In what follows I focus on the work of Peter Abelard (1079-1142), an influential medieval logician who developed his theory of relations in the course of commenting on Categories 7.(4) Like other Aristotelians, Abelard accepts the view that relations are reducible to the monadic properties of related things. On his theory, however, the relation between Simmias and Socrates is not to be explained by a set of peculiar monadic properties--say, being-taller-than-Socrates and being-shorter-than-Simmias. Rather it is to be explained by a pair of ordinary heights--say, being-six-feet-tall in the case of Simmias and being-five-feet-ten in the case of Socrates. Indeed, according to Abelard, the relation between Simmias and Socrates is nothing over and above the possession by these individuals of their respective heights.
Although Abelard commits himself to a form of reductionism about relations, we shall see that his theory is perfectly compatible with the advances made by twentieth-century logicians. Abelard is careful to distinguish questions about ontology from questions about logic, and to commit himself to reducing relations only at the level of ontology. Thus, he argues that Simmias's being taller than Socrates is nothing but Simmias, Socrates, and their respective heights. …