A → (B → C).pdfVIP

An Algebraic Approach to prove γ-admissibility of Relevant Modal Logics Takahiro Seki tseki@adm.niigata-u.ac.jp University Evaluation Center, Niigata University The rule γ , sometimes called material detachment, is given by A B A B The admissibility of γ is regarded as one of the most important problems in the area of relevant logics. The γ-admissibility was proved first by algebraic method in [3]. After Routley-Meyer semantics and metavaluational techniques were established, the method of proving the γ-admissibility has been preferred using them. (See [4] and [2], for example.) Recently, algebraic study of logics has been increased greatly. When the paper [3] was written, algebraic technique was not developed enough in the area of relevant logics. This paper discusses an application of the technique for proving the γ-admissibility in [3] to modern algebraic models for weaker relevant modal logics. Later, we use to denote implication in the metalanguage. 1 Relevant Modal Logics The language of relevant modal logics consists of (i) propositional variables; (ii) logical connectives , , , and ; (iii) modal operators and ; and (iv) constant t. Formulas are defined in the usual way, and are denoted by capital letters A, B, C , etc. When necessary, we use subscripts for capital letters. Prop and Wff will denote the set of all propositional variables and of formulas, respectively. The relevant modal logic L is defined as follows. (a) Axioms (A1) A A (A9) (A A) A (A2) A B A (A10) A A (A3) A B B (A