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

Meaning and partiality revised

Muskens presents in Meaning and Partiality a semantics of possibly contradictory beliefs and other propositional attitudes. We propose a different partial logic based on a few key equalities for the connectives and with four values (truth, falsehood, and undefinedness with negative and positive polarity; only the first truth value is designated). Aim and Scope In logical semantics the grammar and meaning of natural language sentences are defined and the logical consequences of the sentences are tested against our intuition [5,8]. A set of sentences provides a model of the world as observed by a person or, more generally, an agent, and the agent is part of the world as are other agents. In such cases it is important to be able to reason about the knowledge, beliefs, assertions and other propositional attitudes of agents. For instance, (1) is a consequence of (2), but (3) is not a consequence of (1) or (2): Mary believes that John cheats. (1) Mary believes that John cheats and smiles. (2) John cheats. (3