The paper discusses the transition from informal concepts to mathematically precise notions; examples are given, and in some detail the case of lawless sequences, a concept of intuitionistic mathematics, is discussed. A final section comments on philosophical discussions concerning intuitionistic logic in connection with a "theory of meaning". What I have to tell here is not a new story, and it does not contain any really new ideas. The main difference with my earlier discussions of the same topics ([TD88, chapter16],[Tro91]) is in the emphasis. This paper starts with some examples of the transition from informal concepts to mathematically precise notions, followed by a more detailed discussion of one of these examples, the intuitionistic notion of a choice sequence, arguing for the lasting interest of this notion for the philosophy of mathematics. In a final section, I describe my own position relative to some of the philosophical discussions concerning intuitionistic logic in the wr...

Many different kinds of items have been called vague, and so-called for a variety of different reasons. Traditional wisdom distinguishes three views of why one might apply the epitaph "vague" to an item; these views are distinguished by what they claim the vagueness is due to. One type of vagueness, The Good, locates vagueness in language, or in some representational system-- for example, it might say that certain predicates have a range of applicability. On one side of the range are those cases to which the predicate clearly applies and on the other side of the range are those cases where the negation of the predicate clearly applies. But there is no sharp cutoff place along the range where the one range turns into the other. Most examples of The Good are those terms which describe some continuum-- such as bald describes a continuum of the ratio of hairs per cm2 on the head. But not all work this way. Alston (1968) points to terms like religion invoking a number of criteria the joint applicability of which ensures that the activity in question is a religion and the failure of all to apply ensures that it is not a religion. But when only some middling number of the criteria are fulfilled, the term religion neither applies nor fails to apply. Some accounts of "family resemblance " and "open texture " might also fit this picture. Such a view is often called a "representational account of vagueness".

This paper brings together several approaches to vagueness, and ends by suggesting a new approach. The common thread in these approaches is the crucial role played by context. In Section 2, we treat game-theoretic rationales for vagueness, and for the related concepts of ambiguity and generality. Common

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

Tolerant, classical, strict In this paper we investigate a semantics for first-order logic originally proposed by R. van Rooij to account for the idea that vague predicates are tolerant, that is, for the principle that if x is P, then y should be P whenever y is similar enough to x. The semantics, which makes use of indifference relations to model similarity, rests on the interaction of three notions of truth: the classical notion, and two dual notions simultaneously defined in terms of it, which we call tolerant truth and strict truth. We characterize the space of consequence relations definable in terms of those and discuss the kind of solution this gives to the sorites paradox. We discuss some applications of the framework to the pragmatics and psycholinguistics of vague predicates, in particular regarding judgments about borderline cases. Our aim in this paper is to explore a semantic framework originally proposed by [van Rooij, 2010b] in order to deal with the sorites paradox, and intended to formalize the idea that vague predicates are tolerant. Standardly, the idea of tolerance is expressed

Vagueness is standardly opposed to precision. Just as gradable adjectives like ‘tall’ and a quantity modifier like ‘a lot ’ are prototypically vague expressions, mathematical adjectives like ‘rectangular’, and measure phrases like ‘1.80 meters ’ are prototypically precise. But what does it mean for these latter expressions to be precise? On first thought it just means that 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 existing (non-mathematical) 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. The assumption 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 is appropriate (or true, if you don’t want the notion of truth to be empty) might overlap with the appropriate circumstances for uttering statements like ‘the length of stick S is 1.50 meters’. Thus, although

Abstract In this paper we investigate a semantics for first-order logic originally proposed by R. van Rooij to account for the idea that vague predicates are tolerant, that is, for the principle that if x is P, theny should be P whenever y is similar enough to x. The semantics, which makes use of indifference relations to model similarity, rests on the interaction of three notions of truth: the classical notion, and two dual notions simultaneously defined in terms of it, which we call tolerant truth and strict truth. We characterize the space of consequence relations definable in terms of those and discuss the kind of solution this gives to the sorites paradox. We discuss some applications of the framework to the pragmatics and psycholinguistics of vague predicates, in particular regarding judgments about borderline cases.