Positive Fragments of Relevance Logic and Algebras of Binary Relations

Abstract

We prove that algebras of binary relations whose similarity type includes intersection, union, and one of the residuals of relation composition form a nonfinitely axiomatizable quasivariety and that the equational theory is not finitely based. We apply this result to the problem of the completeness of the positive fragment of relevance logic with respect to binary relations.

Lower Semilattice-Ordered Residuated Semigroups and Substructural Logics

“…2 The same non-finite axiomatizability results hold for R cd (· , +, ;, \) and R(· , +, ;, \, /), see [HM11] and [Mi11]. In contrast, we will show finite axiomatizability for state semantics over R cd (· , ;, \), and Corollary 6.4 can be interpreted as a completeness result for relevance logic of the language consisting of conjunction, composition (or fusion) and implication.…”

“…For instance, [HM11] shows that finite axiomatization of state validities for R cd (· , +, \) is not possible, hence establishing incompleteness of the positive fragment of relevance logic.…”

Residuated Kleene Algebras

“…We define the class of relational action lattices, RAL, as that subclass of AL whose elements can be represented on binary relations. The equational theory of RAL turns out to be nonfinitely axiomatizable; see [7]. Here we give an alternative, simpler proof of this fact, and show that strong, finite quasi-axiomatization is impossible in this case, too; see Theorem 4.1.…”

“…The construction in [7] is rather involved, since we needed dense algebras (where x ≤ x;x is valid) so that they can be applied to relevance logic. A simpler construction is available from [11] that has been used to show nonfinite axiomatizability of the equational theories of some residuated algebras in [12].…”

Relevant logic and relation algebras

“…Rather surprisingly, Maddux was able to show in [8] that another well-known relevant logic, namely RM, is complete with respect to idempotent, commutative, representable relation algebras. For several other results in similar vein, the reader is referred to Bimbó et al [5] and Hirsch and Mikulás [4]. The results are mostly negative, and they all focus on representable relation algebras.…”