Deriving Natural Deduction Rules from Truth Tables

Abstract

We develop a method for deriving natural deduction from the truth table for a connective. The method applies to both constructive and classical logic. This implies we can derive constructively valid for any classical connective. We show this constructive validity by giving a Kripke semantics, that is shown to be sound and complete for the constructive rules. For the well-known connectives $$\vee $$, $$\wedge $$, $$\rightarrow $$, $$\lnot $$ the constructive we derive are equivalent to the natural deduction we know from Gentzen and Prawitz. However, they have a different shape, because we want all our to have a standard format, to make it easier to define the notions of cut and to study proof reductions. In style they are close to the general elimination rules studied by Von Plato [13] and others. The also shed some new light on the classical connectives: e.g. the classical we derive for $$\rightarrow $$ allow to prove Peirce's law. Our method also allows to derive for connectives that are usually not treated in natural deduction textbooks, like the if-then-else, whose truth table is clear but whose constructive deduction are not. We prove that if-then-else, in combination with $$\bot $$ and $$\top $$, is functionally complete all other constructive connectives can be defined from it. We define the notion of cut, generally for any constructive connective and we describe the process of cut-elimination.