Relevant logic and relation algebras.
Tomasz Kowalski · Transactions of the Association for Computational Linguistics · 2013
In 2007, Maddux observed (see [7]) that relevant logics have a natural interpretation in the language of relation algebras, with respect to which certain standard relevant logics are sound. He asked whether this interpretation was complete as well. In 2008, Mikulas proved in [10] that the relevant logic R was not complete with respect to square-increasing, commutative, representable relation algebras, answering thereby the original Maddux’ question in the negative. 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 Bimbo et al. [5] and Hirsch and Mikulas [4]. The results are mostly negative, and they all focus on representable relation algebras. In this note, I broaden the perspective a little and show that the answer to Maddux’ question is in the positive, if one does not require representability. I was encouraged to do so by a surprisingly wide attention one little result of mine has received. This result (see [6]), motivated by Meyer’s study [9] of the logic B, deals with weakly-associative relation algebras. Since these are obviously non-representable, relinquishing representability seemed natural.