On a conservative extension argument of Dana Scott

Exegesis, analysis and discussion of an argument deployed by Dana Scott in his 1973 paper ‘Background to Formalization’, rovide an ideal setting for getting clear about some subtleties in the apparently simple idea of conservative extension. There, Scott claimed in respect of two fundamental principles concerning implication (principles sufficing for the pure implicational fragment of intuitionistic logic, in fact) that any generalized (or ‘multiple-conclusion’) consequence relation respecting these principles is always extended conservatively by some similarly fundamental principles concerning conjunction and disjunction. This claim appears on the face of it to conflict with cases in the literature in which adding principles governing conjunction or disjunction or both provides a non-conservative extension of the logic to which they are added, even if that logic does satisfy the intuitionistic conditions on implication. We explore the extent to which such cases can be transformed into counterexamples to Scott’s claim. Once one part of this claim is suitably disambiguated, we find no conflict after all, though we also find that Scott occasionally understates what the argument he provides in support of this claim actually establishes. The main goal, apart from getting straight about Scott’s argument, is to give an airing to various issues and distinctions in the general area of conservativity of extensions; as a side benefit, some semantic light will be thrown (in the final section) on a fragmentary intermediate logic of R. A. Bull, which A. N. Prior showed to be extended non-conservatively by the addition of conjunction, governed by the usual axioms. We will see exactly why, despite appearances, this is not a counterexample to Scott’s claim.