Harnessing Vagueness with Productivity
Harnessing Vagueness with Productivity

Summary/Abstract: The sorites paradox, exemplified by the gradual loss of hair in the "bald man" scenario, illustrates the fundamental problem of vagueness in language and logic. In this work, I analyze this phenomenon through the lens of productive sets - sets for which a productive function f systematically generates elements outside any recursively enumerable subset. I model the concept of "bald" as a productive set, where f produces borderline cases (e.g., an individual with n + 1 hairs), thereby undermining the possibility of precise, algorithmic determination of semantic thresholds. Classical forms of induction fail to capture the open-ended nature of such concepts, revealing structural limitations of formal systems in handling vagueness. I argue that vagueness is ontological rather than merely epistemic, arising from the inherent non-algorithmic character of certain predicates. This perspective calls for a reconsideration of classical approaches in logic and the philosophy of language and encourages further exploration of non-recursive models for understanding vague concepts.

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