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2012 Minimally Congruential Contexts: Observations and Questions on Embedding E in K
Lloyd Humberstone
Notre Dame J. Formal Logic 53(4): 581-598 (2012). DOI: 10.1215/00294527-1722773

Abstract

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.

Citation

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Lloyd Humberstone. "Minimally Congruential Contexts: Observations and Questions on Embedding E in K." Notre Dame J. Formal Logic 53 (4) 581 - 598, 2012. https://doi.org/10.1215/00294527-1722773

Information

Published: 2012
First available in Project Euclid: 8 November 2012

zbMATH: 1271.03030
MathSciNet: MR2995422
Digital Object Identifier: 10.1215/00294527-1722773

Subjects:
Primary: 03B45

Rights: Copyright © 2012 University of Notre Dame

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