This paper continues the work of Priest and Sylvan in Simplified Semantics for Basic Relevant Logics, a paper on the simplified semantics of relevant logics, such as B⁺ and B. We show that the simplified semantics can also be used for a large number of extensions of the positive base logic B⁺, and then add the dualising '*' operator to model negation. This semantics is then used to give conservative extension results for Boolean negation.

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)

In part I, we presented an algebraic-style of semantics, which we called “content semantics,” for quantified relevant logics based on the weak systemBBQ. We showed soundness and completeness with respect to theunreduced semantics ofBBQ. In part II, we proceed to show soundness and completeness for extensions ofBBQ with respect to this type of semantics. We introducereduced semantics which requires additional postulates for primeness and saturation. We then conclude by showing soundness and completeness forBB d Q and its extentions with respect (...) to this reduced semantics. (shrink)