Disappearing Diamonds: Fitch-Like Results in Bimodal Logic

Journal of Philosophical Logic:1-14 (forthcoming)
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Augment the propositional language with two modal operators: □ and ■. Define \ to be the dual of ■, i.e. \. Whenever is of the form φ → ψ, let ) be \. ) can be thought of as the modally qualified counterpart of —for instance, under the metaphysical interpretation of \, where says φ implies ψ, ) says φ implies possiblyψ. This paper shows that for various interesting instances of, fairly weak assumptions suffice for ) to imply —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.
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