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A UNIFORM APPROACH TO AXIOMATIZING BUNDLED OPERATORS

Part of: General logic

Published online by Cambridge University Press:  04 November 2025

JIE FAN Open the ORCID record for JIE FAN [Opens in a new window]
JIE FAN*
Affiliation:
INSTITUTE OF PHILOSOPHY CHINESE ACADEMY OF SCIENCES BEIJING CHINA SCHOOL OF HUMANITIES UNIVERSITY OF CHINESE ACADEMY OF SCIENCES BEIJING CHINA
*
E-mail: jiefan@ucas.ac.cn
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Abstract

Recent years have witnessed extensive logical studies of bundled operators. A known difficulty of such studies is how to axiomatize bundled operators. In this paper, we propose a uniform approach to axiomatizing such operators, which we call ‘the method based on almost-definability schemas’. The approach is useful in finding axioms and inference rules, and in defining a suitable canonical relation, thus in the completeness proof of logical systems. This is, hopefully, good news for modal logicians who are interested in axiomatizing bundled operators. To explicate this approach, we choose four bundled operators—the operator N of purely physical necessity, the operator $N'$ called ‘All$_1$ and Only$_2$’, the operator $N"$, and the operator $N"'$, in the literature, where $N\varphi := \Box _1\varphi \land \neg \Box _2\varphi $, $N'\varphi := \Box _1\varphi \land \Box _2\neg \varphi $, $N"\varphi := \Box _1\varphi \vee \neg \Box _2\varphi $, $N"'\varphi := \Box _1\varphi \vee \Box _2\neg \varphi $. This approach can uniformly deal with axiomatizing these bundled operators. Among other contributions, we also answer several open questions, and obtain alternative axiomatizations which are deductively equivalent to the existing ones in the literature.

Keywords

bundled operatorsaxiomatizationsalmost-definability schemas

MSC classification

Primary: 03B45: Modal logic (including the logic of norms)

Information

Type
Research Article
Information
The Review of Symbolic Logic , First View , pp. 1 - 47
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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