A major question for the relevant logics has been, “Under what conditions is Ackermann's ruleγ from -A ∨B andA to inferB, admissible for one of these logics?” For a large number of logics and theories, the question has led to an affirmative answer to theγ problem itself, so that such an answer has almost come to be expected for relevant logics worth taking seriously. We exhibit here, however, another large and interesting class of logics-roughly, the Boolean extensions of theW — (...) free relevant logics (and, precisely, the well-behaved subsystems of the 4-valued logicBN4) — for which γ fails. (shrink)

This paper presents completeness and conservative extension results for the boolean extensions of the relevant logic T of Ticket Entailment, and for the contractionless relevant logics TW and RW. Some surprising results are shown for adding the sentential constant t to these boolean relevant logics; specifically, the boolean extensions with t are conservative of the boolean extensions without t, but not of the original logics with t. The special treatment required for the semantic normality of T is also shown along (...) the way. (shrink)

This paper is a study of four subscripted Gentzen systems G u R +, G u T +, G u RW + and G u TW +. [16] shows that the first three are equivalent to the semilattice relevant logics u R +, u T + and u RW + and conjectures that G u TW + is, equivalent to u TW +. Here we prove Cut Theorems for these systems, and then show that modus ponens is admissible — which (...) is not so trivial as one normally expects. Finally, we give decision procedures for the contractionless systems, G u TW + and G u RW +. (shrink)

[10] offers two (cut-free) subscripted Gentzen systems, G 2 T + and G 2 R +, which are claimed to be equivalent in an appropriate sense to the positive relevant logics T + and R +, respectively. In this paper we show that that claim is false. We also show that the argument in [10] for the further claim that cut and/or modus ponens is admissible in two other subscripted Gentzen systems, G 1 T + and G 1 R +, (...) is unsound. (shrink)

Although the system T of ticket entailment is obviously related to its cousins E and R , it is motivated along quite distinctive lines in Anderson and Belnap [1975]. It would seem, accordingly, that T is more nearly akin to the system P W studied in Martin [1978] than to E and R. The result presented here, however, at least suggests the contrary.