Paraconsistent assertions

Abstract. Classical logic predicts that everything (thus nothing useful at all) follows from inconsistency. A paraconsistent logic is a logic where inconsistency does not lead to such an explosion. We argue that paraconsistent logics are especially advantageous in order to deal with assertions made by intelligent agents. Other propositional attitudes like knowledge and beliefs can in principle be treated along the same lines. We propose a many-valued paraconsistent logic based on a simple notion of indeterminacy. The proposed paraconsistent logic has a semantics that extends the one of classical logic and it is described using key equalities for the logical operators. A case study is included. We briefly compare with logics based on bilattices. We finally investigate how to translate the paraconsistent logic into classical predicate logic thereby allowing us to make use of automated deduction of classical logic in the future. We base our initial translation on recent work by Muskens. Our final translation is polynomial in the size of the translated formula and follows the semantics for the paraconsistent logic directly. The major motivation behind paraconsistent logic has always been the thought that in certain circumstances we may be in a situation where our information or theory is inconsistent, and yet where we are required to draw inferences in a sensible fashion... Numerous examples of inconsistent information/theories from which one might want to draw inferences in a controlled way have been offered by paraconsistent logicians. For example: 1. information in a computer data base; 2. various scientific theories; 3. constitutions and other legal documents; 4. descriptions of fictional (and other non-existent) objects; 5. descriptions of counterfactual situations. The first of these is fairly obvious..