Logical Revision by Counterexamples: A Case Study of the Paraconsistent Counterexample to Ex Contradictione Quodlibet

摘要

It is often said that a correct logical system should have no counterexample to its logical rules and the system must be revised if its rules have a counterexample. If a logical system (or theory) has a counterexample to its logical rules, do we have to revise the system? In this paper, focussing on the role of counterexamples to logical rules, we deal with the question. We investigate two mutually exclusive theories of arithmetic - intuitionistic and paraconsistent theories. The paraconsistent theory provides a (strong) counterexample to Ex Contradiction Quodlibet (ECQ). On the other hand, the intuitionistic theory gives a (weak) counterexample to the Double Negation Elimination (DNE) of the paraconsistent theory. If any counterexample undermines the legitimate use of logical rules, both theories must be revised. After we investigate a paraconsistent counterexample to ECQ and the intuitionist's answer against it, we arrive at the unwelcome conclusion that ECQ has both a justification and a counterexample. Moreover, we argue that if a logical rule were abolished whenever it has a counterexample, a promising conclusion would be logical nihilism which is the view that there is no valid logical inference, and so a correct logical system does not exist. Provided that the logical revisionist is not a logical nihilist, we claim that not every counterexample is the ground for logical revision. While logical rules of a given system have a justification, the existence of a counterexample loses its role for logical revision unless the rules and the counterexample share the same structure.