Abstract: This is a history of relevant and substructural logics, written for the Handbook of the History and Philosophy of Logic, edited by Dov Gabbay and John Woods. 1 1

this paper, however, that the techniques of nonmonotonic logic may provide a better theoretical framework---at least for the formalization of commonsense normative reasoning---than the usual modal treatment. After reviewing some standard approaches to deontic logic, I focus on two areas in which nonmonotonic techniques promise improved understanding: reasoning in the presence of conflicting obligations, and reasoning with conditional obligations. 2 Modal techniques in deontic logic

Negation raises three thorny problems for anyone seeking to interpret relevant logics. The frame semantics for negation in relevant logics involves a `point shift' operator . Problem number one is the interpretation of this operator. Relevant logics commonly interpreted take the inference from A and ¸A B to B to be invalid, because the corresponding relevant conditional A (¸ A B) ! B is not a theorem. Yet we often make the inference from A and ¸A B to B, and we seem to be reasoning validly when we do so. Problem number two is explaining what is really going on here. Finally, we can add an operation which Meyer has called Boolean negation to our logic, which is evaluated in the traditional way: x j= \GammaA if and only if x 6j= A. Problem number three involves deciding which is the `real' negation. How can we decide between orthodox negation and the new, `Boolean' negation. In this paper, I present a new interpretation of the frame semantics for relevant logics which will allow u...

Australian Realist analytic philosophy is full of claims about truthmakers and truthmaking. In this paper, I seek to show that a number of intuitions about truthmaking are jointly inconsistent, and that some common attempts at resolving the inconsistency are unsatisfying. Finally, I propose an account of truthmaking which resolves the tensions as best as possible. This account has great affinities with both relevant entailment and situation semantics. This note can be seen as an apologetic for relevant entailment for those who are familiar with truthmaking, or as an introduction to truthmaking for those familiar with logic. Either way, it is an attempt to apply modern logical methods and insights to a philosophical problem.

We give a categorial characterization of how labelled deduction systems for logics with a propositional basis behave under unconstrained fibring and under fibring that is constrained by symbol sharing. At the semantic level, we introduce a general semantics for our systems and then give a categorial characterization of fibring of models. Based on this, we establish the conditions under which our systems are sound and complete with respect to the general semantics for the corresponding logics, and establish requirements on logics and systems so that completeness is preserved by both forms of fibring.

The purpose of this paper is to question some commonly accepted patterns of reasoning involving nonmonotonic logics that generate multiple extensions. In particular, I argue that the phenomenon of floating conclusions indicates a problem with the view that the skeptical consequences of such theories should be identified with the statements that are supported by each of their various extensions.

We present a framework for machine implementation of families of non-classical logics with Kripke-style semantics. We decompose a logic into two interacting parts, each a natural deduction system: a base logic of labelled formulae, and a theory of labels characterizing the properties of the Kripke models. By appropriate combinations we capture both partial and complete fragments of large families of non-classical logics such as modal, relevance, and intuitionistic logics. Our approach is modular and supports uniform proofs of correctness and proof normalization. We have implemented our work in the Isabelle Logical Framework.

This paper is an exercise in formal and philosophical logic. I will show how intuitionistic propositional logic can be extended with a new two-place connective, not expressible in the traditional language of intuitionistic logic (the language of conjunction, disjunction, negation and implication). The new system will be shown to be a conservative extension of intuitionistic logic. After examining the formal properties of this extension, the task will be to consider whether this is an `acceptable' extension of intuitionistic logic. It will turn out that on some

iii This thesis describes a theory for representing, manipulating, and reasoning about structured pieces of knowledge in open collaborative systems. The theory’s design is motivated by both its general model as well as its target user commu-nity. Its model is structured information, with emphasis on classification, relative structure, equivalence, and interpretation. Its user community is meant to be non-mathematicians and non-computer scientists that might use the theory via computational tool support once inte-grated with modern design and development tools. This thesis discusses a new logic called kind theory that meets these challenges. The core of the work is based in logic, type theory, and universal algebras. The theory is shown to be efficiently implementable, and several parts of a full realization have already been constructed and are reviewed. Additionally, several software engineering concepts, tools, and technologies have been con-structed that take advantage of this theoretical framework. These constructs are discussed as well, from the perspectives of general software engineering and applied formal methods. Acknowledgements iv I am grateful to my initial primary adviser, Prof. K. Mani Chandy, for bringing me to Caltech and his willingness to let me explore many unfamiliar research fields of my own choosing. I am also appreciative of my second adviser, Prof. Jason Hickey, for his support, encouragement, feedback, and patience through the later years of my work. If Jason had not appeared at Caltech in Autumn of 1999, I may well have not finished my Ph.D. I am very much in debt to Joseph Goguen whose inspiring work started me on the path of using algebras and categories. José Meseguer and Francisco (Paco) Duran have been of tremendous help and inspiration in my use of Maude and rewriting logic.

There is widespread acknowledgement that the law of non-contradiction is an important logical principle. However, there is less-than-universal agreement on exactly what the law amounts to. This unclarity is brought to light by the emergence of paraconsistent logics in which contradictions are tolerated: From the point of view of proofs, not everything need follow from a contradiction --- from the point of view of models, there are "worlds" in which contradictions are true. In this sense, the law of non-contradiction is violated in these logics. However, in many paraconsistent logics, statement (A ^ A) (it is not the case that A and not-A) is still provable. In this sense, the law of non-contradiction is upheld. This paper attempts to clarify the different readings of the law of non-contradiction, in particular taking cues from the tradition of relevant logics. A further guiding principle will be the natural duality between the law of non-contradiction and rejection on the one hand ...