# Disappearing Diamonds: Fitch-Like Results in Bimodal Logic

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## Abstract

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 *φ* → *ψ*, 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 *possibly**ψ*. 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.

## Keywords

Fitch’s Paradox Paradox of Knowability Modal logic Bimodal logic## Preview

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## Notes

### Acknowledgments

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

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