Books in the Studies in Logic (logic & Cognitive Systems) series

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.

This volume is both a tribute to Ulrich Felgner's research inalgebra, logic, and set theory and a strong researchcontribution to these areas. Felgner's former students, friendsand collaborators have contributed sixteen papers to thisvolume that highlight the unity of these three fields in the spiritof Ulrich Felgner's own research. The interested reader will find excellent original research surveys and papers that span the field from set theory without the axiom of choice via model-theoretic algebra to the mathematics of intonation.

This volume takes a reflective position with respect to the conferenceseries "Foundations of the Formal Sciences" (FotFS) and asks:* What are the Formal Sciences?* Can we develop a theoretical classification of the sciences thatjuxtaposes the formal sciences to the natural sciences, social sciences,and humanities? Can we do this solely by identifying common methodologicalfeatures?* Can we identify changes of the notion of formal sciences over time?How were the areas that we now conceived as the "Foundations of the FormalSciences" classified throughout history?Investigating the "History of the Concept of the Formal Sciences" to findanswers to an array of questions with this wide scope, you need anenthusiastic group of researchers interested in going beyond thetraditional boundaries of their subjects covering at once thephilosophical, historical and logical issues at hand, like the authors ofthis volume.The papers in this volume stand witness to our success in touching thementioned questions. It will be of interest to philosophers,sociologists, historians, and logicians, and covers many aspects of thehistory of the formal sciences from the Bronze Age to the early XXIstcentury.

Paraconsistent logics are logics which allow solid deductivereasoning under contradictions by offering a mathematical andphilosophical support to contradictory yet non-trivial theories.Due to its role in models of scientific reasoning and to itsphilosophical implications, as well as to its connections totopics such as abduction, automated reasoning, logic programming,and belief revision, paraconsistency has becoming a fast growingarea.During the III World Congress onParaconsistency (WCP3) held in Toulouse, France, in July, 2003, it became apparent that there is a need for a Handbook covering the most recent results on several aspects of paraconsistent logic, including philosophical debates onparaconsistency and its connections to philosophy of language,argumentation theory, computer science, information theory, andartificial intelligence.This book is a basic tool for those who want to know more about paraconsistent logic, its history and philosophy, the various systems of paraconsistent logic and their applications.The present volume is edited by Jean-Yves Beziau, Walter Carnielli and Dov Gabbay, expert logicians versed in a variety of logics.