Substructural logics are logics obtained from a sequent formulation of intuitionistic or classical logic by rejecting some structural rules. The substructural logics considered here are linear logic, relevant logic and BCK logic. It is proved that first-order variants of these logics with an intuitionistic negation can be embedded by modal translations into S4-type extensions of these logics with a classical, involutive, negation. Related embeddings via translations like the double-negation translation are also considered. Embeddings into analogues of S4 are obtained with (...) the help of cut elimination for sequent formulations of our substructural logics and their modal extensions. These results are proved for systems with equality too. (shrink)

The ontological assumptions of the modal logic of Saul Kripke are analyzed. Answers to the questions are given: What are the modalities? What is thesemantic problem that Kripke faced? How was it resolved? How a substitution oflogic reforms the ontological analyzes? This paper is also intended to be anintroduction to the semantics of Kripke's propositional modal logic.

This paper contributes to the theory of hybrid substructural logics, i.e. weak logics given by a Gentzen-style proof theory in which there is only alimited possibility to use structural rules. Following the literture, we use an operator to mark formulas to which the extra structural rules may be applied. New in our approach is that we do not see this as a modality, but rather as themeet of the marked formula with a special typeQ. In this way we can make (...) the specific structural behaviour of marked formulas more explicit.The main motivation for our approach is that we can provide a nice, intuitive semantics for hybrid substructural logics. Soundness and completeness for this semantics are proved; besides this we consider some proof-theoretical aspects like cut-elimination and embeddings of the strong system in the hybrid one. (shrink)

This book sets out the foundations, methodology, and practice of a formal framework for the description of language. The approach embraces the trends of lexicalism and compositional semantics in computational linguistics, and theoretical linguistics more broadly, by developing categorial grammar into a powerful and extendable logic of signs. Taking Montague Grammar as its point of departure, the book explains how integration of methods from philosophy (logical semantics), computer science (type theory), linguistics (categorial grammar) and meta-mathematics (mathematical logic ) provides a (...) categorial foundation with coverage including intensionality, quantification, featural polymorphism, domains and constraints. For the first time, the book systematises categorial thinking into a unified program which is at once both logically secured, and a practical tool for pure lexical grammar development with type-theoretic semantics. It should be of interest to all those active in computational linguistics and formal grammar and is suitable for use at advanced undergraduate, postgraduate, and research levels. (shrink)