Notre Dame Journal of Formal Logic

Strong Noncontingency: On the Modal Logics of an Operator Expressively Weaker Than Necessity

Jie Fan

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Abstract

Operators can be compared in at least two respects: expressive strength and deductive strength. Inspired by Hintikka’s treatment of question embedding verbs, the variations of noncontingency operator, and also the various combinations of modal operators and Boolean connectives, we propose a logic with (deductively) strong noncontingency operator as the only primitive modality. The novel operator is deductively but not expressively stronger than both noncontingency operator and essence operator, and expressively but not deductively weaker than the necessity operator. The frame-definability power of this new logic is in between standard modal logic and noncontingency logic. A notion of bisimulation is proposed to characterize this logic within standard modal logic and first-order logic. Axiomatizations over various frame classes are presented, among which the minimal logic is related to the treatment of an alternative semantics of the agreement operator proposed by Lloyd Humberstone.

Article information

Source
Notre Dame J. Formal Logic, Volume 60, Number 3 (2019), 407-435.

Dates
Received: 3 September 2015
Accepted: 17 May 2017
First available in Project Euclid: 12 July 2019

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1562918419

Digital Object Identifier
doi:10.1215/00294527-2019-0010

Mathematical Reviews number (MathSciNet)
MR3985619

Subjects
Primary: 03B45: Modal logic (including the logic of norms) {For knowledge and belief, see 03B42; for temporal logic, see 03B44; for provability logic, see also 03F45}
Secondary: 03B42: Logics of knowledge and belief (including belief change)

Keywords
strong noncontingency completeness expressivity frame definability bisimulation

Citation

Fan, Jie. Strong Noncontingency: On the Modal Logics of an Operator Expressively Weaker Than Necessity. Notre Dame J. Formal Logic 60 (2019), no. 3, 407--435. doi:10.1215/00294527-2019-0010. https://projecteuclid.org/euclid.ndjfl/1562918419


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