A standard approach to indeterminacy treats ‘determinately’ and ‘indeterminately’ as modal operators. Determinacy behaves like necessity; indeterminacy like contingency. This raises two questions. What is the appropriate modal system for these operators? And how should we interpret that system? Ken Akiba has developed an account of ontic indeterminacy that interprets possible worlds as worldly precisifications. He argues that this account vindicates S4 as the logic of indeterminacy. In this paper I explore one significant and surprising consequence of this view. I do this in two stages. First, I prove a technical result, which I call the Infinite Indeterminacy Theorem. Roughly put, this theorem states that, at any given precisification w, either every instance of indeterminacy admits of never-ending higher-order indeterminacy, or else there's indeterminacy in an infinite amount of distinct atomic formulas. In slogan form: either all indeterminacy is infinitely ascending or else there's infinitely wide indeterminacy. Second, I unpack the metaphysical consequences of this technical result (under its intended interpretation) and assess their plausibility. I conclude that these consequences commit Akiba's theory to highly contentious–and arguably untenable–views about the metaphysics of indeterminacy and related matters.
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Acknowledgements This study is funded by SFB-TR36, Berlin Institute of Health and the German Cancer Consortium (DKTK). The authors thank Margarete Gries, Nicole Hellmig, Kiara Freitag, Petra Matylewski, Randi Koll, Fabienne Pritsch, and Francisca Egelhofer for excellent technical assistance. NCG is supported by a grant of the Berlin Institute of Health (BIH) CRG-TP7. JR is a participant of the BIH-Charité Clinical Scientist Program funded by the Charité-Universitätsmedizin Berlin and the Berlin Institute of Health (BIH).
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- higher-order indeterminacy
- modal logic
- ontic indeterminacy