Zeitschr. zyxwvutsrqpo f. math. Logik zyxwvuts und zyxwvutsrqp Grandlagen d. Math. Bd. 18, zyxwvutsrqpo S. zyxwvutsrq 117-182 (1972) zyxwvutsrqpo VARIABLE BINDING TERM OPERATORS by JOHN CORCORAN, WILLIAMHATCHER, and JOHN HERRING in Buffalo, New York and Quebec (Canada) Introduction The study of variable binding term operators (vbtos) in logic dates from ‘the beginning of the modern period. Perhaps because the RUSSELL device of contextual definition was used to establish eliminability-in-principle of the common vbtos, full deductive and semantic treatment of them was not regarded as important. However, the fact that symbols of a certain kind are eliminable-in-principle does not by itself imply that the best framework for logic is achieved by renouncing use of such symbols. Adoption of function symbols as primitives is now widespread though, of course, they are eliminable-in-principle in a logic with identity. Moreover, the expedient of introducing vbtos by contextual definition leads quickly to elaborate technical maneuvers involving numerous metatheorems in order to justify ther use in a natural and easy way (cf. QUINE [6], pp. 133, 140ff, for example). The system presented in this paper provides the basis for a standard handling of vbtos. Given the semantics of variable binding term operators, it is seen that deductive completeness can be achieved by addition of a single axiom scheme, here called the zy truth set principle (see below). The whole situation is analogous to the relationship between the predicate logics with and without identity. Once a semantics for logic with identity is given, deductive completeness is achieved by addition of schemes to the deductive system of logic without identity. The proof of completeness of logic wit,h identity can be obtained from completeness of logic without identity by showing that every model of the theory got by taking the added schemes as proper axioms in logic without identity is equivalent to some interpretation in the new Semantics (cf. HEN= [3], pp. zyxwv 64-65). Similar observations hold for soundness. The soundness and completeness proofs for logic with identity and vbtos given below exploit the analogy. Of course, in the case both of logic with identity and of logic with identity and vbtos, the well-known HENKIN-HASENJAEGER methods can be used to construct direct completeness proofs not based on completeness results for the simpler systems. This type of completeness proof does not indicate, however, the exact interrelation between the original system and its extension. In case this exact interrelation is of interest, a proof based on the completeness of the original system is to be preferred. 1. The logics 2K and 2(X + V) Let LK be a first order language with identity where K indicates the set of non- logical constants possibly including individual constants and function symbols. Let AK be some deductive system which is sound and complete with respect to some 12 Ztschr. f. math. Logik