Organon F

Volume 26, August 2019, Issue 3, Pages 340–404

ISSN 2585-7150 (online) ISSN 1335-0668 (print)

Research Article | Special issue on Reflecting on the Legacy of C.I. Lewis: Contemporary and Historical Perspectives on Modal Logic

Semantics without Toil? Brady and Rush Meet Hallden

Lloyd Humberstone


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 Hallden incompleteness, for essentially the same reasons that Hallden 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 – Hallden incompleteness.


Nonclassical logic, semantic completeness, homophonic model theory, Hallden completeness, choice of meta-logic.

Author(s) and affiliation(s)


Lloyd Humberstone


Monash University


Department of Philosophy and Bioethics, Monash University, Victoria 3800, Australia


About this article


23 August 2018


3 November 2018

Published online

24 August 2019


Institute of Philosophy of the Slovak Academy of Sciences

Institute of Philosophy of the Czech Academy of Sciences

Cite as


Humberstone, L. (2019). Semantics without Toil? Brady and Rush Meet Hallden. Organon F, 26(3), 340–404.


Humberstone, Lloyd. 2019. "Semantics without Toil? Brady and Rush Meet Hallden." Organon F 26 (3): 340–404.


Humberstone, L. (2019). Semantics without Toil? Brady and Rush Meet Hallden. Organon F, 26(3), pp. 340–404.

Copyright information

© Lloyd Humberstone

Response page

The above URL is linked with the article's response page. The response page is a permanent location that is associated with the article's DOI number.

This article is distributed under the Creative Commons Attribution-NonCommercial 4.0 International Public License (CC BY-NC 4.0).


Institute of Philosophy
Slovak Academy of Sciences
Klemensova 19
813 64 Bratislava
Slovak Republic
(+421 2) 5292 1215
FAX (+421 2) 5292 1215

Organon F takes part on the long-term preservation of the digital cultural heritage carried out by the University Library in Bratislava.