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This book in categorial proof theory formulates in terms of
category theory a generalization close to linear algebra of the
notions of distributive lattice and Boolean algebra. These notions
of distributive lattice category and Boolean category codify a
plausible nontrivial notion of identity of proofs in classical
propositional logic, which is in accordance with Gentzen's
cut-elimination procedure for multiple-conclusion sequents modified
by admitting new principles called union of proofs and zero proofs.
It is proved that these notions of category are coherent in the
sense that there is a faithful structure-preserving functor from
freely generated distributive lattice categories and Boolean
categories into the category whose arrows are relations between
finite ordinals-a category related to generality of proofs and to
the notion of natural transformation. These coherence results yield
a simple decision procedure for equality of proofs. Coherence in
the same sense is also proved for various more general notions of
category that enter into the notions of distributive lattice
category and Boolean category. Some of these coherence results,
like those for monoidal and symmetric monoidal categories are well
known, but are here presented in a new light. The key to this
categorification of the proof theory of classical propositional
logic is distribution of conjunction over disjunction that is not
an isomorphism as in cartesian closed categories.
The new area of logic and computation is now undergoing rapid
development. This has affected the social pattern of research in
the area. A new topic may rise very quickly with a significant body
of research around it. The community, however, cannot wait the
traditional two years for a book to appear. This has given greater
importance to thematic collections of papers, centred around a
topic and addressing it from several points of view, usually as a
result of a workshop, summer school, or just a scientific
initiative. Such a collection may not be as coherent as a book by
one or two authors yet it is more focused than a collection of key
papers on a certain topic. It is best thought of as a thematic
collection, a study in the area of logic and computation. The new
series Studies in Logic and Computation is intended to provide a
home for such thematic collections. Substructural logics are
nonclassical logics, which arose in response to problems in
foundations of mathematics and logic, theoretical computer science,
mathematical linguistics, and category theory. They include
intuitionistic logic, relevant logic, BCK logic, linear logic, and
Lambek's calculus of syntactic categories. Substructural logics
differ from classical logics, and from each other, in their
presuppositions about Gentzen's structural rules, although their
presuppositions about the deductive role of logical constants are
invariant. Substructural logics have been a subject of study for
logicians during the last sixty years. Specialists have often
worked in isolation, however, largely unaware of the contributions
of others. This book brings together new papers by some of the most
eminent authorities in these varioustraditions to produce a unified
view of substructural logics.
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