Semantics without Toil? Brady and Rush Meet Hallden

Abstract

The present discussion takes up an issue raised in Section 5 of Ross Brady and Penelope Rush's paper 'Four Basic Logical Issues' concerning the (claimed) triviality-in the sense of automatic availability-of soundness and completeness results for a logic in a metalanguage employing at least as much logical vocabulary as the object logic, where the metalogical behaviour of the common logical vocabulary is as in the object logic. We shall see-in Propositions 4.5-4.7-that this triviality claim faces difficulties in the face of Halldén incompleteness, for essentially the same reasons that Halldén thought this phenomenon raised semantic difficulties for the modal logics of C. I. Lewis exhibiting it. To counter any inclination to dismiss the phenomenon as providing at best a marginal range of counterexamples to the triviality claim, a Postscript assembles some reminders of the extent of-and the varied considerations favouring-Halldén incompleteness.