Vagueness, tolerance and contextual logic

Abstract

The goal of this paper is a comprehensive analysis of basic reasoning patterns that are characteristic of vague predicates. The analysis leads to rigorous reconstructions of the phenomena within formal systems. Two basic features are dealt with. One is tolerance: the insensitivity of predicates to small changes in the objects of predication (a one-increment of a walking distance is a walking distance). The other is the existence of borderline cases. The paper shows why these should be treated as different, though related phenomena. Tolerance is formally reconstructed within a proposed framework of contextual logic, leading to a solution of the Sorites paradox. Borderline-vagueness is reconstructed using certain modality operators; the set-up provides an analysis of higher order vagueness and a derivation of scales of degrees for the property in question.

Vagueness: A Conceptual Spaces Approach

“…Generator points that share a Voronoi edge/face are said to be adjacent. 5 An example of a Voronoi diagram of a bounded twodimensional space is given in Fig. 1.…”

“…Whether or not the resulting account (or rather accounts) would be empirically adequate is not something we will try to determine here. 5 See Okabe et al [11,Ch. 2] for these definitions.…”

Temporal Propositions as Vague Predicates

“…-feasible contexts in §2 for Sorites (Gaifman 2002) amounting to comparison classes -selected events in §3 that induce temporal moments (applying the RussellWiener-Kamp construction on event structures with pre-and post-events) -strings in §4 that generalize event occurrences to event types, and are interpretable as incomplete samples from open intervals in the real line R of bounded granularity > 0.…”

“…(Landman 1991, page 138) Different notions of context are explored below. We start in section 2 with the use of context in Gaifman 2002 to sidestep the Sorites paradox before returning in the succeeding sections to the special case of time. A basic aim is to critically examine the intuition that the temporal extent of an event is an intervalan intuition developed in Kamp 1979, Allen 1983and Thomason 1989 A concrete linguistic question concerning intervals is brought out by (2).…”

Strict Finitism, Feasibility, and the Sorites

“…(Parikh, 1983, p. 259) Theories of vagueness based on similar considerations have been proposed more recently by Sazonov (1995), Gaifman (2010), andMagidor (2011). But Parikh's original formulation is of particular interest because the prior passage was informed by an earlier paper (Parikh, 1971) in which he not only anticipated Dummett's critique of strict finitism, but also defines the notion of an almost consistent theory which can be understood to contain the germ of a reply.…”