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. S1, S2 and S3 however, are not so lucky. They are (...) little studied, and used even less. It has become dear that the structures described by S4 and S5 are important in different 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 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, and point the interested reader to where complete proofs can be found. (shrink)

In "The Logical Structure of Linguistic Commitment I" (The Journal of Philosophical Logic 23 (1994), 369-400), we sketch a linguistic theory (inspired by Brandom's Making it Explicit) which includes an "expressivist" account of the implication connective, →: the role of → is to "make explicit" the inferential proprieties among possible commitments which proprieties determine, in part, the significances of sentences. This motivates reading (A → B) as "commitment to A is, in part, commitment to B". Our project is to study (...) the logic of →. LSLC I approximates (A → B) as "anyone committed to A is committed to B", ignoring issues of whether A is relevant to B. The present paper includes considerations of relevance, motivating systems of relevant commitment entailment related to the systems of commitment entailment of LSLC I. We also consider the relevance logics that result from a commitment reading of Fine's semantics for relevance logics, a reading that Fine suggests. (shrink)

What is the logic of entailment? The latter half of the twentieth century has seen, for even the simplest languages, a proliferation of distinct formal entailment systems, each having those willing to defend its status as the answer. Among those defenders, and among the most adamant and mutually critical, are the champions of strict implication and relevance logic. To an outsider, this debate must seem singularly odd. Here we have a group of philosophers who cannot agree on the validity of (...) a simple, easily comprehensible inference scheme. They seem to think that they are not talking past one another---that there is some reality each seeks to describe---for they continue to write articles declaring that the opposing view is wrong. Yet they cannot seem to convince the other side that their view is even remotely plausible, thus leading some to an instrumentalist dismissal of the whole debate.I argue that the defenders of the two systems are right to suggest that something deeper is at issue between them than mere usefulness in particular applied contexts. Once we see what this is, however, we will see that they are, after all, talking past one another. When we succeed in clarifying the crucial underlying issue, we can then succeed in clarifying the crucial distinction between two dimensions of the underlying concept. Both the classicalist and the relevantist, then, will be seen to have accurately captured one dimension of the root concept. (shrink)

What is the logic of entailment? The latter half of the twentieth century has seen, for even the simplest languages, a proliferation of distinct formal entailment systems, each having those willing to defend its status as the answer. Among those defenders, and among the most adamant and mutually critical, are the champions of strict implication and relevance logic. To an outsider, this debate must seem singularly odd. Here we have a group of philosophers who cannot agree on the validity of (...) a simple, easily comprehensible inference scheme. They seem to think that they are not talking past one another---that there is some reality each seeks to describe---for they continue to write articles declaring that the opposing view is wrong. Yet they cannot seem to convince the other side that their view is even remotely plausible, thus leading some to an instrumentalist dismissal of the whole debate.I argue that the defenders of the two systems are right to suggest that something deeper is at issue between them than mere usefulness in particular applied contexts. Once we see what this is, however, we will see that they are, after all, talking past one another. When we succeed in clarifying the crucial underlying issue, we can then succeed in clarifying the crucial distinction between two dimensions of the underlying concept. Both the classicalist and the relevantist, then, will be seen to have accurately captured one dimension of the root concept. (shrink)