Abstract
The knower paradox states that the statement ‘We know that this statement is false’ leads to inconsistency. This article presents a fresh look at this paradox and some well-known solutions from the literature. Paul Égré discusses three possible solutions that modal provability logic provides for the paradox by surveying and comparing three different provability interpretations of modality, originally described by Skyrms, Anderson, and Solovay. In this article, some background is explained to clarify Égré’s solutions, all three of which hinge on intricacies of provability logic and its arithmetical interpretations. To check whether Égré’s solutions are satisfactory, we use the criteria for solutions to paradoxes defined by Susan Haack and we propose some refinements of them. This article aims to describe to what extent the knower paradox can be solved using provability logic and to what extent the solutions proposed in the literature satisfy Haack’s criteria. Finally, the article offers some reflections on the relation between knowledge, proof, and provability, as inspired by the knower paradox and its solutions.
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Acknowledgements
We would like to thank Paul Égré for illuminating e-mail discussions about his paper on the knower paradox. We are also grateful to Marc Pauly for proofreading an earlier version of this article. This work was partially supported by the Netherlands Organisation for Scientific Research (NWO) Vici grant NWO 277-80-001, awarded to Rineke Verbrugge for the project ‘Cognitive systems in interaction: Logical and computational models of higher-order social cognition’. Finally, we would like to thank the anonymous reviewers and the editor Frank Veltman for their helpful suggestions which led to many important improvements, as well as for their patience.
Funding
This work was partially supported by the Netherlands Organisation for Scientific Research (NWO) Vici grant NWO 277-80-001, awarded to Rineke Verbrugge for the project ‘Cognitive systems in interaction: Logical and computational models of higher-order social cognition’.
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BK, RV and MdV devised the research question and the research strategy. MdV did most of the initial research and made all pictures and tables. BK, RV and MdV all wrote parts of the original submitted manuscript, its two revisions, and the replies to reviewers.
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de Vos, M., Verbrugge, R. & Kooi, B. Solutions to the Knower Paradox in the Light of Haack’s Criteria. J Philos Logic 52, 1101–1132 (2023). https://doi.org/10.1007/s10992-023-09699-3
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DOI: https://doi.org/10.1007/s10992-023-09699-3