Models for structures

Let I = 〈S, ⇒〉 be an implication structure. Let T be the set of all strongly closed subsets of S – that is, those subsets U of S for which whenever A1, …, An ⇒ B, and all the Ai's belong to U, then B belongs to U. In this section, when we speak of theories, it will be these strong theories that are intended. Let R be any binary relation on the set T of all theories of the structure I. Under these conditions we shall say that F = 〈T, R〉 is a frame for the implication structure I.

We now wish to introduce the notion of a model of (or for) the structure I. Essentially this is given by specifying a function f, the model function, that associates to each A in S a collection f(A) of theories of I, that is, some subset of T, and satisfies certain additional conditions.

We do not always have connectives available in an implication structure. Consequently, the usual way of providing conditions for a model are not generally available. The use of “∨,” for example, in indicating disjunction, was a convenience for indicating the elements in the structure that are assigned by the disjunction operator (if it assigns an element at all). Similar remarks hold for our use of the other connective signs. Thus, the usual method of exploiting the syntax to define the notion of a model by saying what the model assigns to “A ∨ B,” “A ⇒ B,” “A & B,” “¬A,” and so forth, is not available generally.