Cut for Core Logic

The motivation for Core Logic is explained. Its system of proof is set out. It is then shown that, although the system has no Cut rule, its relation of deducibility obeys Cut with epistemic gain. §1. The debate over logical reform. There is much dispute over which logic is the right logic—indeed, over whether there could even be such a thing as the right logic, rather than a spectrum of logics variously suited for different applications in different areas. Absolutists about logic regard the use of the definite article as justified; pluralists have their principled doubts. For those who engage in the absolutist debate, those whom we can call the quietists are willing to accept the full canon C of classical logic. Their opponents, whom we can call the reformists—intuitionists and relevantists prominent among them— argue that certain rules of classical logic lack validity, and have no right to be in the canon. Intuitionists, on the one hand, originally drew inspiration for their critique of classical logic from the requirements of constructivity in mathematical proof. According to the intuitionist’s construal of existence, a mathematical existence claim of the form ‘there is a natural number n such that F(n)’ requires its asserter to be able to provide a justifying instance—a constructively determinable number t for which one can prove (intuitionistically!) that F(t):