Degree-theoretical approaches to vagueness attempt to flesh out the idea that properties referred to by vague predicates come in degrees, and that sentences containing such predicates can be true to a degree in between absolute truth and absolute falsity. This many-valued semantics is wedded either to some fuzzy logic, or to a non-truth-functional logic, or even to classical logic. The first part of the chapter is devoted to surveying these different alternatives. Subsequently, we discuss the standard fuzzy approach (SFA) to the Sorites, based on infinite-valued Łukasiewicz logic. The mainstream objections to the SFA are then dispelled from a perspective that views classical logic as an ambiguous logic. Next, we address the status of the Tolerance principle in the SFA. We provide a semantics for vague predicates within Rational Pavelka Logic (RPL), contending that the conditional premisses in a Sorites are ambiguous between a reading as Łukasiewicz conditionals and a reading as 'tolerance conditionals'. In conclusion, we formalise in RPL a purely structural version of the paradox, where no logical constant is involved. We ascribe this paradox to an equivocation over consequence.