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.

Abstract: "The constructive negation rule has been introduced by Chan [5,6] to overcome the main drawbacks of the negation-as-failure rule: the unsoundness of floundering programs and, consequently, the inability of providing answers for non-ground negative queries. In this paper we define a bottom-up semantics for constructive negation which we prove sound and complete with respect to the three-valued completion of the program. The semantics describes answers as well as undefined computations for both positive and negative queries. Its construction closely follows the basic idea of constructive negation whereby answers to a negative query are obtained by negating a frontier of the computation tree for the corresponding positive query. Therefore, the proposed semantics can be considered as a natural base for reasoning on the operational semantics for constructive negation defined in the literature. Moreover, we show how the semantics can be effectively used to perform a bottom-up computation of the answers of a normal query."

Negation is one of the most discussed phenomena within linguistics, on all language levels though it never seems to be exhausted. This operator establishes complex sentence structures and constantly challenges – from a cognitive, syntactical, semantic and morphologic viewpoint – presuppositions on language internal relations as rational and logic. It therefore arouses interest through all fields within language sciences. From a pragmatic perspective, where negation is conceived a marked structure, using negation often produces meanings beyond the one of a reversed affirmation "it is not the case that X”. This book explores the various uses and pragmatic meanings of negation in authentic communication, in different text types and in different languages, predominately romance languages. The multilingual composition marries a macro-micro perspective where aspects of genre, sociocultural context, memory, rhetoric and argumentation interplay with the negative morpheme’s nature and embedded instructions. This broad approach makes this book a unique contribution to negation studies and to pragmatics in general. The book is important and enriching reading for scholars in all linguistic domains, but particularly for researchers in semantics, pragmatics, argumentation and, discourse analysis.

Routley-Meyer Ternary Relational Semantics for Intuitionistic-type Negations examines how to introduce intuitionistic-type negations into RM-semantics. RM-semantics is highly malleable and capable of modeling families of logics which are very different from each other. This semantics was introduced in the early 1970s, and was devised for interpreting relevance logics. In RM-semantics, negation is interpreted by means of the Routley operator, which has been almost exclusively used for modeling De Morgan negations. This book provides research on particular features of intuitionistic-type of negations in RM-semantics, while also defining the basic systems and many of their extensions by using models with or without a set of designated points. - Provides a clear development of the fundamentals of RM-semantics in a new application - Covers the most general research on ternary relational semantics - Includes scrutiny of constructive negation from the ternary relational perspective

Perspectives on Negation and Polarity Items contains a selection of papers on the semantics, acquisition and licensing behavior of negation. Negation, being one of the prevalent features of any human language, has many facets of interest to linguists, psychologists and philosophers alike. In recent years, much attention has been paid to the complicated distributional patterns of polarity items. Many of the contributions in this volume are devoted to the study of one or more of these items in langages such as English (Laurence Horn, Anita Mittwoch, Chris Kennedy), Dutch (Jack Hoeksema and Hotze Rullmann, Henny Klein, Gertjan Postma), German (Gabriel Falkenberg), Hindi (Utpal Lahiri) and Greek (Anastasia Giannakidou). In addition, some general issues surrounding negation are addressed, such as the characterization of the notion “strength of negation” (Jay Atlas), the problem of NEG-raising (Lucia Tovena), the interaction of negation and modality (Johan van der Auwera) and the acquisition of negation (Kenneth Drozd).

Abstract: "A standard approach to negation in logic programming is negation as failure. Its major drawback is that it cannot produce answer substitutions to negated queries. This paper presents an approach to overcoming this limitation. The approach is based on construction of finitely failed trees for some instances of a negated query. The method is applicable to normal programs, it is a proper extension of SLDNF- resolution. Its semantics is given by Clark's completion. A version for the well-founded semantics is also presented."