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 countably infinite indeterminacy, where each basic formula can get its own indeterminate truth value. The meaning of the logical operators is new and rather different from traditional many-valued logics as well as from logics based on bilattices. Thus we try to build a bridge between the communities of higher order logic and many-valued logic. A case study is studied and a sequent calculus is proposed based on recent work by Muskens.