Semantics without Toil? Brady and Rush meet Halldén

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 auto-matic availability – of soundness and completeness results for alogic in a metalanguage employing at least as much logical vo-cabulary 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 dif-ficulties in the face of Halldén incompleteness, for essentially the same reasons that Halldén thought this phenomenon raised seman-tic 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.