Vagueness, tolerance and non-transitive entailment

Abstract

1 Tolerance and vagueness Vagueness is standardly opposed to precision. Just as gradable adjectives like ‘tall ’ and a quantity modifier like ‘a lot ’ are prototypical vague expressions, mathematical adjectives like ‘rectangular’, and measure phrases like ‘1.80 meter ’ are prototypically precise. But what does it mean for these latter expressions to be precise? On first thought it just means that they are precise, because they have an exact mathematical definition. However, if we want to use these terms to talk about observable objects, it is clear that these mathematical definitions would be useless: if they exist at all, we cannot possibly determine what are the rectangular objects in the precise geometrical sense, or objects that are exactly 1.80 meters long. For this reason, one allows for a margin of measurement error, or a threshold, in physics, psychophysics and other sciences. Assuming that the predicates we use are observational predicates gives rise to another consequence as well. If statements like ‘the length of stick S is 1.45 meters ’ come with a large enough margin of error, the circumstances in which this statement can be made appropriately (or truly, if you don’t want the notion of truth to be empty) might overlap with the circumstances in which the statement ‘the The main ideas of this paper were first presented in a workshop on vagueness at

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