Abstract
Predicates are term-to-sentence devices, and operators are sentence-to-sentence devices. What Kaplan and Montague's Paradox of the Knower demonstrates is that necessity and other modalities cannot be treated as predicates, consistent with arithmetic; they must be treated as operators instead. Such is the current wisdom.
A number of previous pieces have challenged such a view by showing that a predicative treatment of modalities neednot raise the Paradox of the Knower. This paper attempts to challenge the current wisdom in another way as well: to show that mere appeal to modal operators in the sense of sentence-to-sentence devices is insufficient toescape the Paradox of the Knower. A family of systems is outlined in which closed formulae can encode other formulae and in which the diagonal lemma and Paradox of the Knower are thereby demonstrable for operators in this sense.
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I am deeply indebted to Robert F. Barnes and Evan W. Conyers, without whom these ideas might not have germinated and certainly would not have grown. Many of the results offered here evolved in the course of mutual discussion and correspondence. I am also grateful to an anonymous reviewer forSynthese for many very helpful suggestions.
The current paper contains the technical results promised in Footnote 25 of Grim (1988) and Footnote 26, Chapter 3, of Grim (1991).
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Grim, P. Operators in the paradox of the knower. Synthese 94, 409–428 (1993). https://doi.org/10.1007/BF01064487
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DOI: https://doi.org/10.1007/BF01064487