Comparing Some Substructural Strategies Dealing with Vagueness

It is well-known that in combination with further premises that look less controversial, the tolerance principle – the constraint that if Pa holds, and a and b are similar in P -relevant respects, Pb holds as well – leads to contradiction, namely to the sorites paradox. According to many influential views of the sorites paradox (e.g. Williamson 1994), we therefore ought to reject the principle of tolerance as unsound. There are reasons to think of such a view as too drastic and as missing out on the role that such a principle plays in categorization and in ordinary judgmental and inferential practice. Taking a different perspective, the tolerance principle ought not to be discarded that fast, even when viewed normatively. Instead, it corresponds to what might be called a soft constraint, or a default, namely a rule that we can use legitimately in reasoning, but that must be used with care. One family of approaches represents the tolerance principle by a certain conditional sentence, of the form: Pa ∧ a ∼P b → Pb, and bestows special properties to the conditional to turn it into a soft constraint. One natural option is to use fuzzy logic, where vM(A) can be anywhere in [0, 1] and vM(A → B) = Min{1, 1 − vM(A) + vM(B)}. One can demand that the tolerance conditional may never have a value below 1− ε for some small ε. Given an appropriate sorites sequence, it will be possible to have: vM(Pa1 → Pa2) = 1−ε, vM(Pa2 → Pa3) = 1− ε, without having vM(Pa1 → Pa3) = 1− ε. A different option is to treat the tolerance conditional as expressing a defeasible rule (like when ‘→’ expresses a counterfactual conditional). Say that Pa ∧ a ∼P b → Pb is true provided Pb is true in all ‘optimal’ (Pa ∧ a ∼P b)worlds. Call a world (Pa ∧ a ∼P b)-optimal if a is P -similar to b but is not close to a borderline case of P . From Pa ∧ a ∼P b, Pa ∧ a ∼P b → Pb, it need not follow that Pb, since a world may satisfy Pa ∧ a ∼P b without being