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By Alan Ross Anderson;Nuel D. Belnap
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Extra info for Entailment. The logic of relevance and necessity
Sample text
E. set D so that B = A © D has the desired properties, namely D < T A, and B does not support C. We ensure the condition D < T A by a permitting argument. To satisfy the last property we meet the following requirements for all e, Se : $ e total & (M,
The reader will notice that many of these had originated in ordinal recursion theory developed since the early 1960's. The fact that there is a tight link between recursion theory on ordinals and that on weak fragments of P is a fortuitous coinci dence probably not anticipated by the founders of these two subjects. We believe that there is a two-way interaction between them, and expect to see 48 C H O N G AND Y A N G more applications going in either direction. We begin with some definitions and terminologies.
P. e. comeager. Proof. It suffices to define an effective system of partial extension functions / such that for every set A £ BI(C) there is an index e such that fe is dense along A and fe(A\n) % A whenever fe(A\n) is defined. Given a recursive set U such that C = {Ue : e > 0} such a system / is obtained by letting f2e+i(x) — x~i if \x\ £ Ue and / 2 e + i ( x ) | otherwise (e > 0, i < 1). e. nowhere dense via / 2 e or / 2 e + i , respectively. e. meager but not effectively meager. e. meager but not effectively meager.