Vagueness, tolerance and contextual logic

Synthese, 2011

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: semantic indeterminacy-as exemplified by the existence of borderline cases-and tolerance: the insensitivity of a predicate to sufficiently small changes in the objects of predication (a one-foot increment of a walking distance is a walking distance); tolerance is the source of the Sorites paradox. In each of these subjects, the philosophical analysis is followed by formal implementation. I have tried to organize and present the material, so as to make the main philosophical points accessible, without requiring technical sophistication. The considerable literature on vagueness has bundled together the existence of borderlines and the Sorites paradox. This is understandable, given that most standard examples display both indeterminateness and tolerance, and there is, indeed, a connection. Yet, I will argue, these are two separate aspects. It is not difficult to establish by direct analysis, and by using examples from natural language, that vagueness per se (that is, indeterminacy that is manifested in borderline cases) does not imply tolerance. The non-implication from tolerance to vagueness is more difficult, since, in natural language, tolerance is manifested in predicates that are vague. Nonetheless, an analysis, aided by a specially constructed example, will show that, in principle, tolerance need not imply vagueness. Tolerance turns out to be a contextual phenomenon, whose rigorous reconstruction requires a framework of contextual logic, one that provides explicit representation of certain contexts attaching to declarative sentences. Within this logic we can fully maintain tolerance without being led to the Sorites contradiction. I will propose such a formal system: a semantics and a sound and complete deductive system, which extends first-order logic and fully preserves the classical rules, and I will show how to apply it to vocabularies that include tolerant predicates, such as 'poor', 'walking distance', 'noonish', etc. 1 2 The full system incorporates two machineries: one, which takes care of borderline phenomena, including higher order vagueness, and another, which handles tolerance by * Most of this work represents research that has been carried on since 1996. The first versions of contextual logic were presented in two meetings of a joint workshop on vagueness, held in 1996 at NYU and Columbia University. It was also presented in the 1997 fall meeting of the New York Conference on Science and Methods. The system, in its present revised and simplified form that fully preserves classical logic, was presented in an invited talk at the 2001 meeting of the Association of Symbolic Logic, held in Philadelphia. The results on borderlines, higher order vagueness, and KTB represent more recent research done in the summer of 2001 and discussed in my seminar on vagueness, in the fall of 2001 at Columbia University. I wish to thank my colleague Achille Varzi and the participants of this seminar for illuminating discussions.-1 D. Raffman [1994] has proposed a contextual approach to the Sorites. My paper is independent of her work, which came to my attention after the presentations of my preliminary results in 1996. Raffman's approach is psychologistic, rather than semantic, and does not involve a logical system. See section 5 for further comments on the difference. 2 I should also mention here Kamp's work [1981], of which I became aware in 1997 and which is an unfinished attempt at a logical system for handling the Sorites. His approach would not have lead to the present system. I think it may succeed on the sentential level but would fail when it comes to quantifiers.

Approaching Vagueness, 1983

It is argued in this paper that the vagueness of natural language predicates arises from the fact that they are learned and used always in limited contexts and hence are incompletely defined. A semantics for natural language must take this into account by making the interpretation of predicates context-dependent. It is shown that a context dependent semantics also provides the means for an account of vagueness. These notions are first developed and argued for in abstract terms and are then applied to a solution of the prototype of vagueness puzzles: the paradox of the heap.

The Australasian Journal of Logic, 2010

This paper presents F, substructural logic designed to treat vagueness. Weaker than Lukasiewicz’s infinitely valued logic, it is presented first in a natural deduction system, then given a Kripke semantics in the manner of Routley and Meyer's ternary relational semantics for R and related systems, but in this case, the points are motivated as degrees to which the truth could be stretched. Soundness and completeness are proved, not only for the propositional system, but also for its extension with first-order quantifiers. The first-order models allow not only objects with vague properties, but also objects whose very existence is a matter of degree.

philosophy.stanford.edu

2011

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 the predicates 'being a stick of 1.45 meters' and 'being a stick of 1.50 meters' are inconsistent under a precise interpretation, the predicates might well be applicable to the same object when a margin of error is taken into account, i.e., when the predicates are interpreted tolerantly. 2 Thus, although the standard, i.e. precise, semantic meanings of two predicates might be incompatible, when one or both of these observational predicates are more realistically interpreted in a tolerant way, they might well be compatible.

Studia Logica, 2008

It is a commonplace that the extensions of most, perhaps all, vague predicates vary with such features as comparison class and paradigm and contrasting cases. My view proposes another, more pervasive contextual parameter. Vague predicates exhibit what I call open texture: in some circumstances, competent speakers can go either way in the borderline region. The shifting extension and anti-extensions of vague predicates are tracked by what David Lewis calls the "conversational score", and are regulated by what Kit Fine calls penumbral connections, including a principle of tolerance. As I see it, vague predicates are response-dependent, or, better, judgement-dependent, at least in their borderline regions. This raises questions concerning how one reasons with such predicates. In this paper, I present a model theory for vague predicates, so construed. It is based on an overall supervaluationist-style framework, and it invokes analogues of Kripke structures for intuitionistic logic. I argue that the system captures, or at least nicely models, how one ought to reason with the shifting extensions (and anti-extensions) of vague predicates, as borderline cases are called and retracted in the course of a conversation. The model theory is illustrated with a forced march sorites series, and also with a thought experiment in which vague predicates interact with so-called future contingents. I show how to define various connectives and quantifiers in the language of the system, and how to express various penumbral connections and the principle of tolerance. The project fits into one of the topics of this special issue. In the course of reasoning, even with the external context held fixed, it is uncertain what the future extension of the vague predicates will be. Yet we still manage to reason with them. The system is based on that developed, more fully, in my Vagueness in Context, Oxford, Oxford University Press, 2006, but some criticisms and replies to critics are incorporated.

Philosophical Perspectives, 2008

Communications in computer and information science, 2016

It is well-known that in combination with further premises that look less controversial, the tolerance principle-the constraint that if P a holds, and a and b are similar in P-relevant respects, P b holds as well-leads to contradiction, namely to the sorites paradox. According to many influential views of the sorites paradox (e.g. Williamson 1994), we therefore ought to reject the principle of tolerance as unsound. There are reasons to think of such a view as too drastic and as missing out on the role that such a principle plays in categorization and in ordinary judgmental and inferential practice. Taking a different perspective, the tolerance principle ought not to be discarded that fast, even when viewed normatively. Instead, it corresponds to what might be called a soft constraint, or a default, namely a rule that we can use legitimately in reasoning, but that must be used with care. One family of approaches represents the tolerance principle by a certain conditional sentence, of the form: P a ∧ a ∼ P b → P b, and bestows special properties to the conditional to turn it into a soft constraint. One natural option is to use fuzzy logic, where v M (A) can be anywhere in [0, 1] and v M (A → B) = Min{1, 1 − v M (A) + v M (B)}. One can demand that the tolerance conditional may never have a value below 1 − ε for some small ε. Given an appropriate sorites sequence, it will be possible to have: v M (P a 1 → P a 2) = 1−ε, v M (P a 2 → P a 3) = 1 − ε, without having v M (P a 1 → P a 3) = 1 − ε. A different option is to treat the tolerance conditional as expressing a defeasible rule (like when '→' expresses a counterfactual conditional). Say that P a ∧ a ∼ P b → P b is true provided P b is true in all 'optimal' (P a ∧ a ∼ P b)worlds. Call a world (P a ∧ a ∼ P b)-optimal if a is P-similar to b but is not close to a borderline case of P. From P a ∧ a ∼ P b, P a ∧ a ∼ P b → P b, it need not follow that P b, since a world may satisfy P a ∧ a ∼ P b without being Except for the last section, this paper is an abridged version of a longer paper entitled "The Tolerance Principle: Nontransitive Reasoning or Nonmonotonic Reasoning?". We are indebted to two anonymous reviewers for helpful comments.

2012

Robert van Rooij presents an original framework dealing with the formalization of vague predicates. His solution is based on a non-classical entailment, which he calls tolerant entailment. The author thoroughly discusses the relation of his solution to the major approaches to vagueness (supervaluationism, contextualism, epistemic approaches, . . . ) and gives an analysis of higher order vagueness. The paper discusses several topics related to the analysis of vagueness. We shall not comment on all of them but instead concentrate on the notion of tolerant truth and its connections to modal logic.

Ordinary intuitions that vague predicates are tolerant, or cannot have sharp boundaries, can be formalized in first-order logic in at least two non-equivalent ways, a stronger and a weaker. The stronger turns out to be false in domains that have a significant central gap for the predicate in question, i.e. where a sufficiently large middle segment of the ordering relation (such as taller for 'tall') is uninstantiated. The weaker principle is true in such domains, but does not in those domains induce the sorites conclusion.