The Theory of Relations, Complex Terms, and a Connection Between λ and Calculi

This paper motivates and introduces a new method of interpreting complex relation terms in a second-order quantified modal language. The new method of interpreting these terms establishes an interesting connection between λ and calculi, and the resulting semantics provides a precise understanding of the theory of relations. In addition to motivating the new method generally, several research problems in connection with previous, algebraic methods for interpreting complex relation terms are discussed and solved. Relations are not sets and predication is not set membership. To assert that John loves Mary or that 1 < 2 is to assert that John bears a certain (two-place) relation, loves, to Mary and that 1 bears a certain (two-place) relation, less than, to 2. It is not to assert that 〈John,Mary〉 or 〈1,2〉 is an element of some set. Similarly, to assert that John is happy or that 2 is prime is to assert that John or 2 has (or exemplifies or instantiates) a certain property (i.e., one-place relation), namely, being happy or being ∗I’d like to thank Allen Hazen, Bernard Linsky, Christopher Menzel, Seyed Mousavian, Uri Nodelman, Paul Oppenheimer, and Susanne Riehemann for the insightful comments and observations they made about this paper. I’d like to thank an anonymous referee of this journal for suggesting that I be more explicit about the motivation for the ideas in this paper. I’d also like to thank the members of the logic group at the University of Alberta/Edmonton for reading and discussing the first draft of the paper. Edward N. Zalta 2 prime. It is not to assert that John or 2 is an element of some set. Yet atomic predications of the form Fx1 . . .xn (e.g., Rxy or P x) in predicate calculi are standardly modeled and interpreted as claims solely about set membership: n-place predicates of the predicate calculus are standardly interpreted as denoting or signifying sets of n-tuples and the n-place predicates of the modal predicate calculus are standardly interpreted as denoting or signifying functions that map each possible world to a set of n-tuples. Although this standard interpretation allows us to investigate the metatheoretical properties of these calculi in set-theoretic terms, such an interpretation is nevertheless philosophically incorrect. In a philosophically proper interpretation, predicates denote or signify relations, not sets or functions from worlds to sets, and if we want to use set theory to represent or model the truth conditions of exemplification claims, relations should play some role in those truth conditions. In Section 1, I rehearse the prima facie case for this last claim and thereby provide general motivation for developing an intensional interpretation of the modal predicate calculus in which the predicates denote relations. Although such intensional interpretations have been proposed before, they give rise to a number of research problems. These are described in Section 2 and the discussion there motivates specific features of the system presented in the main sections of the paper, namely, Sections 3 – 5. This system achieves the research goals implicitly defined by the discussion in Sections 1 and 2, and one of its distinguishing features is that an -calculus in the metalanguage is used to interpret the λ-calculus in the object language. The system not only provides a better conception of the predications expressed by primitive atomic formulas of the (modal) predicate calculus, but also provides us with a formalism for asserting a precise theory of relations conceived as genuine entities in their own right and not some other thing. 1 General Motivation To focus our attention, consider a second-order modal language with definite descriptions (i.e., complex individual terms, interpreted rigidly for simplicity) and λ-expressions interpreted relationally rather than functionally (thereby construing them as complex predicates or n-place relation terms). In the traditional interpretation of this language, relations are assumed to be functions from possible worlds to sets of indi3 Relations, Complex Terms, and λ and Calculi viduals and so the latter are assigned as the semantic values of the simple predicates and λ-expressions. Though this traditional interpretation suffices for the study of the metatheoretical properties of this language, it fails to offer a philosophically proper understanding of the language as a whole, for the following reasons: • The interpretation represents relations and predication purely in terms of sets and set membership, and so doesn’t acknowledge the fact that relations aren’t sets and that predication involves those relations. • The interpretation turns the second-order comprehension principle into a comprehension principle for sets or functions rather than for relations. • The interpretation doesn’t allow us to assert that there are necessarily equivalent but distinct relations. • The interpretation allows one to mistakenly suppose that the truth value of a sentence changes from world-to-world because the meaning of the sentence changes from world-to-world. This would be a misconception of modal language. • Relations and predication have 0-place cases (a 0-place relation is a proposition and the 0-place case of predication is just truth). But sets and set membership, and functional application, don’t have a 0-place case. We discuss these in turn. 1.1 Relations are not Sets The reason that relations are not sets and predication is neither set membership nor functional application is that a relation characterizes its arguments, whereas a set merely collects its members and a function merely correlates its arguments and values. There is a important difference between characterization, collection, and correlation.1 When one asserts: 1In the following remarks, one might substitute the notions of classify and classification for collect and collection. However, in some literatures, classification is based on one or more shared common characertistics and so already presupposes the notion of characterization. The notions of collect and collection don’t carry this presupposition and so better capture the essential difference between a set and its members. Edward N. Zalta 4