how to interpret the l-calculus in filters over these models. It is then a wonderful (and wonderfully unintended) coincidence that their insights and ours lead (via Observation 1) to what is essentially the same formal system. Oddly (and from my viewpoint sadly) an analogue of Observation 1 no longer holds when the BCD system is enriched to contain also an analogue ? of the disjunction ? of B+. The result is what Barbanera and Dezani call ?union types? in [ 7]. I shall call this system B?D. And it turns out that, although B+ remains contained in B?D, the converse fails.

I have another idea, to which I shall devote the remainder of this paper.

II. The semantics of B+. I review the semantics of B+.7 A B+-frame8 is a triple K = <0,K,R>, where K is a non-empty set (of ?worlds?, or ?prime theories?), 0 ? K (0 is a ?real world?, or ?preferred theory?),9 and R is a ternary (?accessibility?) relation on K. We define for all x,y ? K,
D?. x ? y =df R0xy
and we set out the following monotonicity postulates:
p1. x ? x
p2. x ? y and Ryzw fi Rxzw
p3. y ? z and Rxzw fi Rxyw
p4. w ? z and Rxyw fi Rxyz
We may think of ? as set inclusion on theories; and Rxyz says that whenever AfiB ? x and A ? y then B ? z; i. e., if a major premiss of fiE is in x and its minor premiss is in y then the conclusion of fiE is in z.10 Note immediately that ? is transitive, by D? and p3. It is also reflexive, by p1.11

Let F be the set of B+ formulas. An interpretation I is a function FxK fi {t,f}, subject to the following conditions:
H. x ? y and I(A,x) = t fi I(A,y) = t (Heredity condition)
TT. I(T,x) = t, for all x ? K
T&. I(A&B,x) = t iff I(A,x)= t and I(B,x)= t
T?. I(A?B,x) = t iff I(A,x)= t or I(B,x)= t
Tfi. I(AfiB,x)= t iff for all y,z ? K (Rxyz fi ( I(A,y) = t fi I(B,z) = t) )
In view of the postulates, we may confine the heredity condition H to sentential variables p, q, r, etc., whence an easy induction via the truth-conditions shows that it holds for all formulas A.

A formula A is verified on an interpretation I just in case I(A,0) = t. And A is B+ valid iff A is verified on all interpretations in all B+ frames. The following is a principal result of [3] (and holds mutatis mutandis for other positive relevant logics, adding additional semantical postulates that correspond combinatorially to further logical axioms):

Soundness and completeness for B+. A is a theorem of B+ iff A is B+ valid.

III. Booleanizing B+. We now show how to add a Boolean negation ? to B+. The method is that used to add ? to R+ in [8]. The result will be a system CB+ (for ?classical B+?). Now that we are in an explicitly Boolean context, we can as in [9] greatly simplify our presentation. We add new definitions D?. A?B =df ?(?A&?B).
DT. T =df p ? ?p (for an arbitrary sentential variable p)

7 Restall in his Ph.D. thesis [10] uses DW+ as his name for B+. He also applies the ?simplified semantics? of [11], which (at least for my money) is just the Boolean semantics under another name. 8 Routley and I have used B+ms (for B+ model structure) previously. But frame seems now the entrenched name for the notion.
9 That there is only one preferred theory 0 produces what [15] calls a ?reduced? semantics. 10 Restall in [10] follows Dunn in writing Ryxz where I have always written Rxyz. Boo! Hiss! 11 These accord with our intended interpretation of ? as ? .