A Useful Substructural Logic

Formal systems seem to come in two general kinds: useful and useless. This is painting things starkly, but the point is important. Formal structures can either be used in interesting and important ways, or they can languish unused and irrelevant. Lewis' modal logics are good examples. The systems S4 and S5 are useful in many diierent ways. They map out structures that are relevant to a number of diierent applications. S1, S2 and S3 however, are not so lucky. They are little studied, and used even less. It has become clear that the structures described by S4 and S5 are important in diierent ways, while the structures described by S1 to S3 are not so important. In this paper, we will see another formal system with a number of diierent uses. We will examine a substructural logic which is important in a number of diierent ways. The logic of Peirce monoids, inspired by the logic of relations, is useful in the independent areas of linguistic types and information ow. In what follows I will describe the logic of Peirce monoids in its various guises, sketch out its main properties, and indicate why it is important. As proofs of theorems are readily available elsewhere in the literature, I simply sketch the relevant proofs here, and point the interested reader to where complete proofs can be found. 1 Relation algebras Relation algebras are an interesting generalisation of Boolean algebras. For our purposes we will only need a fragment of the general relation algebras studied by Tarski and others 27]. Take a class of objects and a collection of binary relations on this class. There are a number of ways to form new relations from old: the Boolean con-nectives are some, but there are also others. Pairs of relations can be composed: a and b are related by the composition of and (written) just when for some c, aac and ccb. Composition is associative, but not commutative or idempotent. The identity relation 1' satisses 1' = = 1'. Any relation has a converse, written , which satisses = and a number of other identities. An abstract positive relation algebra is a 6-tuple hR; \; ; 1; ;1'i, such that hR; \; ; 1i is a distributive lattice with top element 1 (the full relation), (composition) is an associative binary operation on R with identity 1' (the identity relation). In addition, composition distributes over …