The?-admissibility is one of the most important problems in the realm of relevant logics. To prove the 7-admissibility, either the method of normal models or the method using metavaluations may be employed. The?-admissibility of a wide class of relevant modal logics has been discussed in Part I based on a former method, but the?-admissibility based on metavaluations has not hitherto been fully considered. Sahlqvist axioms are well known as a means of expressing generalized forms of formulas with modal operators. This (...) paper shows that? is admissible for relevant modal logics with restricted Sahlqvist axioms in terms of the method using metavaluations. (shrink)

In this paper, we investigate neighbourhood semantics for modal extensions of relevant logics. In particular, we combine the neighbourhood interpretation of the relevant implication (and related connectives) with a neighbourhood interpretation of modal operators. We prove completeness for a range of systems and investigate the relations between neighbourhood models and relational models, setting out a range of augmentation conditions for the various relations and operations.

In relevant logics, necessary truths need not imply each other. In justification logic, necessary truths need not all be justified by the same reason. There is an affinity to these two approaches that suggests their pairing will provide good logics for tracking reasons in a fine-grained way. In this paper, I will show how to extend relevant logics with some of the basic operators of justification logic in order to track justifications or reasons. I will define and study three kinds (...) of frames for these logics. For the first kind of frame, I show soundness and highlight a difficulty in proving completeness. This motivates two alternative kinds of frames, with respect to which completeness results are obtained. Axioms to strengthen the justification logic portions of these logics are considered. I close by developing an analogy between the dot operator of justification logic and theory fusion in relevant logics. (shrink)

The admissibility of Ackermann’s rule γ is one of the most important problems in relevant logic. While the γ-admissibility of normal modal logics based on the relevant logic R has been previously discussed, the case for weaker relevant modal logics has not yet been considered. The method of normal models has often been used to prove the γ-admissibility. This paper discusses which relevant modal logics admit γ from the viewpoint of the method of normal models.