Every reduct of two-element boolean algebra can be finitely axiomatized by modus ponens or substitution rule

摘要

A general method yielding a Hilbert-type finite axiomatization of any fragment of the two-valued propositional logic is described in this paper. The method extends and simplifies a L. Henkin's method of axiomatization of only those propositional connectives which define implication. It also simplifies an earlier author's result. We start with conjunctive normal forms of formulas rather than with their zero-one definitions as L. Henkin does. This gives us the possibility of reducing the number of axioms characterizing an arbitrary propositional connective.