Classificatory Sorites, Probabilistic Supervenience, and Rule-Making
This work addresses foundational issues in logic and philosophy for researchers in formal systems, but it is incremental as it builds on existing theories of supervenience and sorites.
The paper tackles the sorites paradox by analyzing it through the lens of stimuli and responses, hypothesizing that supervenience in empirical systems is probabilistic, which stabilizes response probabilities and reduces higher-order distributions to ordinary ones. It concludes that arbitrariness in rule-making for soritical situations does not challenge classical logic.
We view sorites in terms of stimuli acting upon a system and evoking this system's responses. Supervenience of responses on stimuli implies that they either lack tolerance (i.e., they change in every vicinity of some of the stimuli), or stimuli are not always connectable by finite chains of stimuli in which successive members are `very similar'. If supervenience does not hold, the properties of tolerance and connectedness cannot be formulated and therefore soritical sequences cannot be constructed. We hypothesize that supervenience in empirical systems (such as people answering questions) is fundamentally probabilistic. The supervenience of probabilities of responses on stimuli is stable, in the sense that `higher-order' probability distributions can always be reduced to `ordinary' ones. In making rules about which stimuli ought to correspond to which responses, the main characterization of choices in soritical situations is their arbitrariness. We argue that arbitrariness poses no problems for classical logic.