Modal Ω-Logic

In Matteo Vincenzo D'Alfonso & Don Berkich (eds.), On the Cognitive, Ethical, and Scientific Dimensions of Artificial Intelligence. Springer. pp. 65-82 (2019)
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Abstract

This essay examines the philosophical significance of $\Omega$-logic in Zermelo-Fraenkel set theory with choice (ZFC). The categorical duality between coalgebra and algebra permits Boolean-valued algebraic models of ZFC to be interpreted as coalgebras. The modal profile of $\Omega$-logical validity can then be countenanced within a coalgebraic logic, and $\Omega$-logical validity can be defined via deterministic automata. I argue that the philosophical significance of the foregoing is two-fold. First, because the epistemic and modal profiles of $\Omega$-logical validity correspond to those of second-order logical consequence, $\Omega$-logical validity is genuinely logical. Second, the foregoing provides a modal account of the interpretation of mathematical vocabulary.

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Attitudes de Dicto and de Se.David Lewis - 1979 - Philosophical Review 88 (4):513-543.
Warrant for Nothing (and Foundations for Free)?Crispin Wright - 2004 - Aristotelian Society Supplementary Volume 78 (1):167–212.

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