A Model of Tolerance

I want to address a foundational issue in the contemporary vagueness debate. This is very much dominated by three main approaches—semantic, epistemic and psychological— which, however distinct they may be in other respects, share a crucial common feature, namely that of characterizing vagueness in terms of the notion of borderlineness. These approaches “only” differ in their explication of borderlineness, which is given in their respective favourite terms, but agree in characterizing vagueness by means of substantial non-logical notions (be them related either to semantic indeterminacy, or to unknowability, or to partial belief). In so doing, they obliterate one of the main phenomena in which vagueness is manifest in natural language, namely the intuitive truth of the major premise of sorites paradoxes. Such a premise can be expressed in purely logical terms (save of course for the relevant vague predicate and for some functor or relational predicate expressing a suitable ordering), without need of non-logical expressions like ‘borderline’, ‘definitely’, ‘roughly’ etc. I want to provide some philosophical and technical support for a theory (the naive theory of vagueness) which takes very seriously this phenomenon (and in so doing is close to contextualist positions, especially to Hans Kamp’s theory). According to the naive theory of vagueness, the vagueness of an expression consists (a) in the existence of both positive and negative cases of application of the expression and (b) in the non-existence of a sharp cut-off point between them. The sorites paradox shows the naive theory to be inconsistent in most logics proposed for a vague language. I will explore the prospects of saving the naive theory by revising the logic in a novel way, placing principled restrictions on the structural property of transitivity of the consequence relation. To this end, I will propose and to some extent develop a lattice-theoretical semantics for a whole family of (0th-order) “tolerant logics”. I will start by delineating the semantics of a very weak tolerant logic (the system T), which can be seen as providing a common basis for a very wide class of tolerant logics. Every element (“value”) of a lattice V is associated with a tolerance function tol, which, for every v ∈ V , picks out an upper subset of V containing v (intuitively, tol maps each