A Useful Substructure! 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 different ways. They map out structures that are relevant to a number of different applications. SI, 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 different ways, while the structures described by SI to S3 are not so important. In this paper, we will see another formal system with a number of different uses. We will examine a substructural logic which is important in a number of different ways. The logic of Peirce monoids, inspired by the logic of relations, is useful in the independent areas of linguistic types and information flow. 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