The elimination theorem when modality is present

Abstract

The present paper contains an emendation to the theory of formal deducibility which was presented in lectures at the University of Notre Dame in the spring of 1948 [4]. The method of proof developed in these lectures for the elimination theorem, which corresponds to the “Hauptsatz” of Gentzen's thesis [6], failed when modal connectives were present. This deficiency was remedied soon thereafter; but it was not possible to include the new proof in those lectures. This new proof is presented below.Before proceeding to this proof, the elimination theorem will be formulated, and its proof outlined, in an abstract form. Certain general conditions on the rules are formulated in §1; then, in §2, it is shown that the first two stages of the elimination theorem go through for any rules satisfying the conditions. The modifications necessary to extend this proof to cases where modal rules are present will then be taken care of in the later sections. A section on the extension of the elimination theorem to the singular form of LC is added at the end.