A Paraconsistent Higher Order Logic

Classical logic predicts that everything (thus nothing useful at all) follows from inconsistency. A paraconsistent logic is a logic where an inconsistency does not lead to such an explosion, and since in practice consistency is difficult to achieve there are many potential applications of paraconsistent logics in knowledge-based systems, logical semantics of natural language, etc. Higher order logics have the advantages of being expressive and with several automated theorem provers available. Also the type system can be helpful. We present a concise description of a paraconsistent higher order logic with countable infinite indeterminacy, where each basic formula can get its own indeterminate truth value (or as we prefer: truth code). The meaning of the logical operators is new and rather different from traditional many-valued logics as well as from logics based on bilattices. The adequacy of the logic is examined by a case study in the domain of medicine. Thus we try to build a bridge between the HOL and MVL communities. A sequent calculus is proposed based on recent work by Muskens.Comment: Originally in the proceedings of PCL 2002, editors Hendrik Decker, Joergen Villadsen, Toshiharu Waragai (http://floc02.diku.dk/PCL/). Correcte

The Strict/Tolerant Family Continued: Quantifiers and Modalities

This paper continues my earlier work, which showed there is a broad family of propositional many valued logics that have a strict/tolerant counterpart.  Here we generalize those results from propositional to a range of both modal and quantified many valued logics, providing strict/tolerant counterparts for all.  This paper is not self-contained; some results from earlier papers are called on, and are not reproved here.  The key new machinery added to earlier work, allowing modalities and quantifiers to be handled in similar ways, is the central use of bilattices that are function spaces, and more generally lattices that are function spaces.  Two versions of the central proofs are considered, one at length and the other in outline

The Strict/Tolerant Family Continued: Quantifiers and Modalities

This paper continues my earlier work, which showed there is a broad family of propositional many valued logics that have a strict/tolerant counterpart.  Here we generalize those results from propositional to a range of both modal and quantified many valued logics, providing strict/tolerant counterparts for all.  This paper is not self-contained; some results from earlier papers are called on, and are not reproved here.  The key new machinery added to earlier work, allowing modalities and quantifiers to be handled in similar ways, is the central use of bilattices that are function spaces, and more generally lattices that are function spaces.  Two versions of the central proofs are considered, one at length and the other in outline