Recently, an improvement in respect of simplicity was found by Rohan French over extant translations faithfully embedding the smallest congruential modal logic (E) in the smallest normal modal logic (K). After some preliminaries, we explore the possibility of further simplifying the translation, with various negative findings (but no positive solution). This line of inquiry leads, via a consideration of one candidate simpler translation whose status was left open earlier, to isolating the concept of a minimally congruential context. This amounts, roughly (...) speaking, to a context exhibiting no logical properties beyond those following from its being congruential (i.e., from its yielding provably equivalent results when provably equivalent formulas are inserted into the context). On investigation, it turns out that a context inducing a translation embedding E faithfully in K need not be minimally congruential in K. Several related minimality conditions are noted in passing, some of them of considerable interest in their own right (in particular, minimal normality). The paper is exploratory, raising more questions than it settles; it ends with a list of open problems. (shrink)

This paper offers a semantic study in multi-relational semantics of quantified N-Monotonic modal logics with varying domains with and without the identity symbol. We identify conditions on frames to characterise Barcan and Ghilardi schemata and present some related completeness results. The characterisation of Barcan schemata in multi-relational frames with varying domains shows the independence of BF and CBF from well-known propositional modal schemata, an independence that does not hold with constant domains. This fact was firstly suggested for classical modal systems (...) by Stolpe, 557–575, 2003), but unfortunately that work used only models and not frames. (shrink)

This paper extends David Lewis' result that all first degree modal logics are complete to weakly aggregative modal logic by providing a filtration-theoretic version of the canonical model construction of Apostoli and Brown. The completeness and decidability of all first-degree weakly aggregative modal logics is obtained, with Lewis's result for Kripkean logics recovered in the case k = 1.