On the Distinction between Relational and Functional Type Theory

It is commonly believed that it makes no difference whether one starts with relational types or functional types in formulating type theory, since one can either start with relations as primitive and represent functions as relations or start with functions as primitive and represent relations as functions. It is also commonly believed that the formula-based logic of relational type theory is equivalent to the term-based logic of functional type theory. However, in this paper, the authors argue that there are systems with logics that can be properly characterized in relational type theory, but not in functional type theory. We investigate an important difference between relational type theories (RTTs) and functional type theories (FTTs). It is often thought that for each RTT, there is an FTT which is a mere variant (or vice versa), since relations and functions are interdefinable. It is often concluded, therefore, that one can equally well start with relational types or functional types as the basis for the syntax and ontology of type theory. In what follows, we describe a system, namely, object theory (e.g., Zalta 1983, 1988) containing wffs that are not terms and that cannot be converted to terms by embedding them in λ-expressions. A formulabased logic for this system can be easily constructed in an RTT. We then consider whether the logic of this system can be reformulated as an FTT. Paul Oppenheimer and Edward Zalta 2 We encounter what appear to be insoluble problems in so doing. This suggests that RTT and FTT are not, after all, mere variants and that RTT has certain capabilities not shared by FTT. Thus, we take our efforts to show an important difference between these two type-theoretic environments. The difference lies primarily in the contrast between formula logics and term logics, i.e., in the difference between logics whose fundamental axioms and rules apply to formulas and logics whose fundamental axioms and rules apply to terms. Indeed, the distinction between formulas and terms seems to have collapsed in the original development of FTT and we see the difference between RTT and FTT described below as due in part to this decision not to distinguish formulas and terms in FTT. 1. Sketches of RTT and FTT In this section, we sketch two simple (as opposed to ramified) type theories, one relational and one functional. Type symbols will be used to categorize the domains of quantification as well as the classes of expressions that range over those domains.