If it’s clear, then it’s clear that it’s clear, or is it? Higher-order vagueness and the S4 axiom⋆
This paper challenges some widespread assumptions about the role of the modal axiom S4 in theories of vagueness. In the context of vagueness, S4 usually appears as the principle ‘If it’s clear (determinate, definite) that A, then it’s clear that it’s clear that A’, or, formally, CA→CCA. We argue first that contrary to common opinion, higher-order vagueness and S4 are perfectly compatible. This is in response to claims such as Williamson’s that if vagueness is defined using a clarity operator that obeys S4, higher-order vagueness disappears. Second, we argue that contrary to common opinion, bivalence-preservers (such as epistemicists) can without contradiction condone S4, and bivalence-discarders (such as open-texture theorists and supervaluationists) can without contradiction reject S4. To this end, we show how in the debate over S4 two different notions of clarity are in play, and elucidate their respective functions in accounts of higher-order vagueness. Third, we rebut several arguments produced by opponents of S4, including those by Williamson.
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