Models of Data

Abstract

To nearly all the members of this Congress, the logical notion of a model of a theory is too familiar to need detailed review here. Roughly speaking, a model of a theory may be defined as a possible realization in which all valid sentences of the theory are satisfied, and a possible realization of the theory is an entity of the appropriate set-theoretical structure. For instance, we may characterize a possible realization of the mathematical theory of groups as an ordered couple whose first member is a nonempty set and whose second member is a binary operation on this set. A possible realization of the theory of groups is a model of the theory if the axioms of the theory are satisfied in the realization, for in this case (as well as in many others), the valid sentences of the theory are defined as those sentences which are logical consequences of the axioms. To provide complete mathematical flexibility I shall speak of theories axiomatized within general set theory by defining an appropriate set-theoretical predicate (e.g., ‘is a group’) rather than of theories axiomatized directly within first-order logic as a formal language. For the purposes of this paper, this difference is not critical. In the set-theoretical case, it is convenient sometimes to speak of the appropriate predicate’s being satisfied by a possible realization. But whichever sense of formalization is used, essentially the same logical notion of model applies.1