Remarks on the Scott-Lindenbaum Theorem

In the late 1960s and early 1970s, Dana Scott introduced a kind of generalization (or perhaps simplification would be a better description) of the notion of inference, familiar from Gentzen, in which one may consider multiple conclusions rather than single formulas. Scott used this idea to good effect in a number of projects including the axiomatization of many-valued logics (of various kinds) and a reconsideration of the motivation of C.I. Lewis. Since he left the subject it has been vigorously prosecuted by a number of authors under the heading of abstract entailment relations where it has found an important role in both algebra and theoretical computer science. In this essay we go back to the beginnings, as presented by Scott, in order to make some comments about Scott's cut rule, and show how much of Scott's main result may be applied to the case of single-conclusion logic.