Structural Underdetermination and Rule Underdetermination in the Sorites Paradox

This paper argues that the Sorites paradox is not primarily a problem of vagueness understood as epistemic limitation, but rather a problem arising from structural and rule underdetermination. Standard responses, including Williamson’s epistemicism and Horgan’s account of robust vagueness, presuppose the stability of the tolerance principle and of the notion of “small change” across sorites sequences. Firstly, I develop a structural occupancy model in which classificatory judgments depend not only on quantity but also on spatial distribution. By constructing cases with identical hair quantities but different spatial configurations, I argue that “small change” is not structurally invariant. This undermines the traditional formulation of the tolerance principle. Secondly, I argue that even if spatial configurations are artificially constrained in order to recover tolerance-like stability, classificatory judgments remain underdetermined due to competing admissible rules. Using the “Mediterranean Baldness 2.0” case, I show that identical spatial structures can yield incompatible classifications under different rule systems. Finally, I develop the “ Mediterranean Baldness Model 3.0 ” in order to demonstrate that structural underdetermination within spatial occupancy configurations disrupts the transitivity and local continuity presupposed by the tolerance principle itself. Taken together, these arguments suggest that the Sorites paradox concerns not merely the location of sharp boundaries, but also the instability of the spatial-structural and rule-governed assumptions underlying the tolerance principle.