# Justification Logic with Confidence

## Abstract

Justification logics are a family of modal logics whose non-normal modalities are parametrised by a type-theoretic calculus of terms. The first justification logic was developed by Sergei Artemov to provide an explicit modal logic for arithmetical provability in which these terms were taken to pick out proofs. But, justification logics have been given various other interpretations as well. In this paper, we will rely on an interpretation in which the modality $$\llbracket \tau \rrbracket \varphi$$ is read ‘S accepts $$\tau$$ as justification for $$\varphi$$’. Since it is often important to specify just how much confidence agents have in propositions on the basis of justifications, the logic will need to be extended if it is to provide a sufficiently general account. The primary purpose of this paper is to extend justification logic with the expressive resources to needed to do so. Thus, we will construct the justification logic with confidence ($${\mathsf {JC}}$$). While $${\mathsf {JC}}$$ will be extremely general, capable of accommodating a wide range of interpretations, we provide motivation in terms of the notion of confidence deriving recent work by Paul and Quiggin (Episteme 15(3):363–382, 2018). Under this understanding, confidence must only correspond to a partial ordering. We axiomatise $${\mathsf {JC}}$$ and provide a sound and complete semantics.

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## Acknowledgements

This work was generously supported by ARC Discovery Project Grant #DP160100903. We are grateful to audiences at the 2017 Reasoning Club Conference in Torino, the Melbourne Logic Seminar, the Economics and Philosophy Workshop at the Australian National University, and the 2017 New Zealand Association of Philosophers Conference in Dunedin for valuable discussions. We would also like to individually acknowledge Guillermo Badia, Rohan French, Lloyd Humberstone, Fabio Lampert, Toby Meadows, Laurie Paul, Shawn Standefer, Joe Ulkowski, Greg Restall, Zach Weber, and two anonymous referees for their particularly helpful feedback.

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Correspondence to Ted Shear.

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Shear, T., Quiggin, J. Justification Logic with Confidence. Stud Logica 108, 751–778 (2020). https://doi.org/10.1007/s11225-019-09874-1

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• DOI: https://doi.org/10.1007/s11225-019-09874-1

### Keywords

• Modal logic
• Justification logic
• Justification
• Confidence
• Bounded rationality