ABSTRACT
Given these three conditions, it follows that A 1\ (B v C) f-vn (A 1\ B) v (AI\ C).1 To live without distribution, we need to modify one of our three conditions. Which should it be?
In answering this question, you need to consider how the completeness result for a frame semantics is demonstrated. In the previous chapter, we proved completeness by considering the canonical frame, made up of prime theories. In the class of prime theories, the conditions (1), (2) and (3) are respected. A f-B if and only if in every prime theory x, if A E x then B E x. Similarly, A 1\ B E x iff A E x and B E x, and A v B E x iff A E x or B E x. If prime theories respect all three conditions, the natural weakening is to consider the class of theories. In this case, (1) and (2) are respected, but (3) fails. We can have a theory including A v B without the theory containing A or containing B. A natural way to construct models for logics without distribution is to consider models in which condition (3) fails. It is clear that we need it to fail in the leftto-right direction: that is, we require points x at which x If-A v B while x l,fl A 1 Belnap's "Life in the Undistributed Middle" [25] makes this point rather forcefully and examines the case for living without distribution.
