Here is an account of recent investigations into the two main concepts of negation developed in the constructive logic: the negation as reduction to absurdity, and the strong negation. These concepts are studied in the setting of paraconsistent logic.
This book presents the author's recent investigations of the two main concepts of negation developed in the constructive logic: the negation as reduction to absurdity (L.E.J. Brouwer) and the strong negation (D. Nelson) are studied in the setting of paraconsistent logic.
The paraconsistent logics are those, which admit inconsistent but non-trivial theories, i.e., the logics which allow making inferences in non-trivial fashion from an inconsistent set of hypotheses. Logics in which all inconsistent theories are trivial are called explosive . In the intuitionistic logic Li , the negation is defined as reduction to absurdity. The concept of strong negation is realized in the Nelson logic N3 . Both logics are explosive and have paraconsistent analogs: Johansson's logic Lj and paraconsistent Nelson's logic N4 . It will be shown that refusing the explosion axiom "contradiction implies everything" does not lead to decrease of the expressive power of a logic. To understand, which new expressive possibilities have the logics Lj and N4 as compared to the explosive logics Li and N3 , we study the lattices of extensions of the logics Lj and N4 . This is the first case when lattices of paraconsistent logics are systematically investigated. The study is based on algebraic methods, demonstrates the remarkable regularity and the similarity of structures of both lattices of logics, and gives essential information on the paraconsistent nature of logics Lj and N4 .
The methods developed in this book can be applied for investigation of other classes of paraconsistent logics.
Reductio ad Absurdum.- Minimal Logic. Preliminary Remarks.- Logic of Classical Refutability.- The Class of Extensions of Minimal Logic.- Adequate Algebraic Semantics for Extensions of Minimal Logic.- Negatively Equivalent Logics.- Absurdity as Unary Operator.- Strong Negation.- Semantical Study of Paraconsistent Nelson's Logic.- N4?-Lattices.- The Class of N4?-Extensions.- Conclusion.
From the reviews:
"This book is much more than a collection of papers, everything has been organized very smoothly and it forms a coherent whole about the study of a big class of logics. The presentation of this systematic work in a book is a very good point. It will turn research on paraconsistent logic and negation more accessible. This kind of book can easily be used as a textbook for advanced courses. It is clearly written, gathering a variety of scattered concepts and results." (Jean-Yves Beziau, Studia Logica, Vol. 100, 2012)
"This is the first book-length algebraic study of constructive paraconsistent logics. ... The monograph under review has a very clear structure. ... This monograph is indispensable for anybody interested in the algebraic study of constructive paraconsistent logics in particular, but it is also most rewarding for anyone interested in non-classical logics in general." (Heinrich Wansing, Zentralblatt MATH, Vol. 1161, 2009)
Thetitleofthisbookmentionstheconceptsofparaconsistencyandconstr- tive logic. However, the presented material belongs to the ?eld of parac- sistency, not to constructive logic. At the level of metatheory, the classical methods are used. We will consider two concepts of negation: the ne- tion as reduction to absurdity and the strong negation. Both concepts were developed in the setting of constrictive logic, which explains our choice of the title of the book. The paraconsistent logics are those, which admit - consistent but non-trivial theories, i. e. , the logics which allow one to make inferences in a non-trivial fashion from an inconsistent set of hypotheses. Logics in which all inconsistent theories are trivial are called explosive. The indicated property of paraconsistent logics yields the possibility to apply them in di?erent situations, where we encounter phenomena relevant (to some extent) to the logical notion of inconsistency. Examples of these si- ations are (see ): information in a computer data base; various scienti?c theories; constitutions and other legal documents; descriptions of ?ctional (and other non-existent) objects; descriptions of counterfactual situations; etc. The mentioned survey by G. Priest  may also be recommended for a ?rst acquaintance with paraconsistent logic. The study of the paracons- tency phenomenon may be based on di?erent philosophical presuppositions (see, e. g. , ). At this point, we emphasize only one fundamental aspect of investigations in the ?eld of paraconsistency. It was noted by D. Nelson in [65, p.
Is the first book in the field of paraconsistent logic devoted to the study of lattices of logics. Studying not particlar systems but classes of logics is one of the most important features of modern non-classical logic
Studies two main concepts of constructive negation in the setting of paraconsistent logic
Is the first book containing systematic and transparent presentation of algebraic semantics for logics with strong negation
Develops methods which could be applied to other classes of paraconsistent logic