Disappearing Diamonds: Fitch-Like Results in Bimodal Logic

  • Weng Kin SanEmail author


Augment the propositional language with two modal operators: □ and ■. Define \(\blacklozenge \) to be the dual of ■, i.e. \(\blacklozenge \equiv \neg \blacksquare \neg \). Whenever (X) is of the form φψ, let (X\(^{\blacklozenge } \) ) be \(\varphi \rightarrow \blacklozenge \psi \). (X\(^{\blacklozenge } \) ) can be thought of as the modally qualified counterpart of (X)—for instance, under the metaphysical interpretation of \(\blacklozenge \), where (X) says φ implies ψ, (X\(^{\blacklozenge } \) ) says φ implies possiblyψ. This paper shows that for various interesting instances of (X), fairly weak assumptions suffice for (X\(^{\blacklozenge } \)) to imply (X)—so, the modally qualified principle is as strong as its unqualified counterpart. These results have surprising and interesting implications for issues spanning many areas of philosophy.


Fitch’s Paradox Paradox of Knowability Modal logic Bimodal logic 


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I owe thanks to Jeff Russell, Gabriel Uzquiano, Tim Williamson, and an anonymous reviewer for their helpful comments.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.School of PhilosophyUniversity of Southern CaliforniaLos AngelesUSA

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