Algebraic study of two deductive systems of relevance logic
The paper studies from an algebraic point of view both the classical relevant logic \({\mathbf R}\) and a weaker version \({\mathbf W} {\mathbf R}\), which arises from \({\mathbf R}\) by dropping the Identity Law \(p \to p\). \({\mathbf W} {\mathbf R}\) is given by its consequence operation \(\vdash_{{\mathbf W} {\mathbf R}}\) defined as \[ \Gamma \vdash_{{\mathbf W} {\mathbf R}} \varphi \Leftrightarrow \exists \varphi_1, \ldots, \varphi_n \in \Gamma:\;\vdash_{\mathbf R} (\varphi_1 \wedge \ldots \wedge \varphi_n) \to \varphi. \] It is provided with a classical style Gentzen calculus. Both \({\mathbf W} {\mathbf R}\) and \({\mathbf R}\) are investigated by means of abstract logic, i.e., abstract algebras with closure operators. One of the most interesting results is that on any algebra of suitable type there is an isomorphism between the lattice of all \({\mathbf W} {\mathbf R}\)-logics on it and the lattice of all congruences of the algebra whose quotient is an \({\mathbf R}\)-algebra.
scientific article; zbMATH DE number 769230