Non-symmetric Relations

Let us say that a relation r is symmetric iff whenever x bears r to y, y bears r to x; otherwise, r is non-symmetric. In this paper, I will argue for the thesis that necessarily, there are no non-symmetric relations. What is this predicate ‘. . . bears . . . to . . . ’ in terms of which ‘symmetric’ was defined? According to one important theory of relations, this predicate is primitive, in the same sense in which the two-place predicate ‘instantiates’ (‘has’, ‘exemplifies’) is primitive according to some theories of properties. When x bears r to y, this is not so in virtue of any more basic truths involving x, r and y; there is no interesting answer to the question ‘What is it for x to bear r to y?’ But it is not uncontroversial that ‘bears’ is primitive in this sense. Various analyses of this predicate have been proposed. For example, a believer in states of affairs might analyse ‘x bears r to y’ as ‘there is a state of affairs s such that s relates x to y, and r is the universal component of s.’ The predicates ‘. . . relates . . . to . . . ’ and ‘. . . is a universal Russell (1903, p. 25), by contrast, rather misleadingly defines ‘not-symmetric’ to mean: not symmetric, and also not asymmetric—where an asymmetric relation is one for which there are no x and y such that x bears it to y and y bears it to x.