Rules Versus Theorems

Abstract

Contemporary critics of two-valued logic concentrate on the reasons for accepting the tertium non datur A V ¬A as a valid propositional schema. Brouwer explicitly states1 that only by unjustified extrapolation of logical principles from those which correctly describe the general relations among propositions on finite domains to those that allegedly regulate propositions on infinite domains, could it happen that A V ¬ A is accepted as valid. He was the first to observe that value-definite (decidably true or false) propositions do not generally transfer value-definiteness to their logical compounds. No better support could be found for the claim that the classical characterization of propositions as entities that are either true or false is inadequate. The union of the class of all true propositions and the class of all false propositions does not contain all logical compounds out of either true or false propositions; it does not contain, for example, certain as yet neither proven nor disproven universal propositions of elementary arithmetic. But nobody has seriously advanced the thesis that such propositions should not count as propositions at all.2 In fact, it is generally conceded that the usual way to form finite and infinite logical compounds makes sense even if nothing can be said about their truth-value.