The Logic for Metaphysical Conceptions of Vagueness

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Philosophical Perspectives, 2016

How does vagueness interact with metaphysical modality and with restrictions of it, such as nomological modality? In particular, how do definiteness, necessity (understood as restricted in some way or not), and actuality interact? This paper proposes a model-theoretic framework for investigating the logic and semantics of that interaction. The framework is put forward in an ecumenical spirit: it is intended to be applicable to all theories of vagueness that express vagueness using a definiteness (or: determinacy) operator. We will show how epistemicists, supervaluationists, and theorists of metaphysical vagueness like Barnes and Williams (2010) can interpret the framework.  We will also present a complete axiomatization of the logic we recommend to both epistemicists and local supervaluationists. 

Forthcoming in Canadian Journal of Philosophy I will here present a number of problems concerning the idea that there is ontological vagueness, and the related claim that appeal to this idea can help solve some vagueness-related problems. A theme underlying the discussion will be the distinction between vagueness specifically and indeterminacy more generally (and, relatedly, the distinction between ontological vagueness and ontological indeterminacy). Even if the world is somehow ontologically indeterminate it by no means follows that it is, properly speaking, ontologically vague. 1

In Williamson's 'Vagueness', he presents an epistemic view of vagueness. However, this view causes an anti-intuitive consequence. Neither epistemicists nor semanticists have given a satisfactory solution. Higher-order vagueness is one of the core issue in the research area of vagueness, on characterizing which Williamson rejects semantic views of vagueness mainly by their failure. However he himself does not characterize higher-order vagueness good enough. In this paper I find the essential reason of that anti-intuitive consequence, and offer a margin model of knowledge. This model is not only suitable for describing higher-order vagueness, but also can avoid the anti-intuitive consequence. 1 Epistemicism Epistemicism is a view about vagueness. Vagueness is phenomenon causing the sorites paradox. A typical example of the sorites paradox is: a grain of wheat does not make a heap, and if so then two grains of wheat do not make a heap, and if so then three grains of wheat do not make a heap, and so on ad infinitum. Thus we can say that ten thousand grains of wheat do not make a heap, which is clearly false. The fact is that we know one grain of wheat does not make a heap and we also know that ten thousand grains of wheat do make a heap, but we do not know whether a hundred grains of wheat make a heap. Therefore we call a hundred grains of wheat a borderline case of the predict 'heap'. The semantic view used to be the study's mainstream in the area of vagueness, which treats vagueness as a semantic phenomenon. Generally, people who hold the view think that the vague predicates such as 'tall', 'red', 'heap' and 'bold' are partially or unusually defined, which means the truth of propositions like 'a hundred grains of wheat make a heap' are void or something except Truth and Falsity. However, a different view on vagueness has become very active due to Williamson's monograph Vagueness published in 1994. In this book, he defends a view named epistemicism, and in this view, the semantics of vagueness predicts are precise but unknowable. There are two main standpoints of epistemicism: one is that the bivalence principle remains valid for vague propositions, and the other is that KK principle fails when knowledge is inexact for inexact knowledge is governed by the margin for error principles. In this paper I will not talk about the defense of the bivalence principle. We can just presuppose it. The margin for error principles and the failure of KK principle, on the contrary, will be explicated introduced for our purpose. The margin for error principles are actually reliability conditions for inexact knowledge. Williamson holds that when our knowledge is inexact, only if we leave a margin for error, the belief saying something is the case could be reliable enough to be knowledge. A belief has a margin for error means that in all situations similar enough to the actual situation, the thing is still the case. Let us look at an example, and we will see why inexact knowledge should be governed by the margin for error principles.

The paper develops a formal model for interpreting vague languages based on a variant of supervaluation semantics. Two modes of semantic variability are modelled, corresponding to different aspects of vagueness: one mode arises where there can be multiple definitions of a term; and the other relates to the threshold of applicability of a vague term with respect to the magnitude of relevant observable values. The truth of a proposition depends on both the possible world and the precisification with respect to which it is evaluated. Structures representing possible worlds and precisifications are both specified in terms of primitive functions representing observable measurements, so that the semantics is grounded upon an underlying theory of physical reality. On the basis of this semantics, the acceptability of a proposition to an agent is characterised in terms of a combination of agent's beliefs about the world and their attitude to admissible interpretations of vague predicates.

Proceedings of the 7th conference of the European Society for Fuzzy Logic and Technology (EUSFLAT-2011), 2011

In this paper we discuss the multifaceted nature of vagueness, the limits of (standard) set theory in dealing with the foundational aspects that a really innovating theory of vagueness should manifest, and the difficulties in outlining the possible features that such a type of new formalism should exhibit in order to be able to deal with such innovative aspects. We shall highlight some aspects of the role that Fuzzy Set Theory (FST) can play in this process.

I develop a new theory of vagueness, which repudiates the notion of local indeterminacy (or borderline case) and replaces it with the notion of global indeterminacy.

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.

Philosophical Studies, 2007

This paper presents an new epistemicist account of vagueness, one that avoids standard arbitrariness worries by exploiting a plenitudinous metaphysic. There are two natural objections to epistemicist accounts of vagueness that one frequently encounters in conversation (objections that are frequently run together). 1 One objection is that it is hard to live without an informative answer to the question as to how the non-semantic facts-non-relational and relational-about a given individual determine the semantic profile of that individual. Let us call this the Bruteness worry. A second objection is that it seems metaphysically arbitrary that just one of the many candidate cut-offs for, say, baldness, should be associated with some ordinary pattern of use of the term 'bald'. Let us call this the Arbitrariness worry. In this paper we sketch an epistemicist approach to vagueness that provides a distinctive perspective on these foundational issues. On the one hand, it affords epistemicism protection against the Arbitrariness worry, by combining the evenhandedness typically associated with supervaluationism with the logic and semantics distinctive of epistemicism. 2 And on the other, it renders the Bruteness objection less pressing by assimilating it to a kind of bruteness that many of us have

2012

During the last couple of decades, several attempts have been made to come up with a theory that can handle the various semantic, logical and philosophical problems raised by the vagueness of natural languages. One of the most influential ideas that have come into fashion in recent years is the idea that vagueness should be analysed as a form of context sensitivity. Such contextualist theories of vagueness have gained some popularity, but many philosophers have remained sceptical of the prospects of finding a tenable contextualist solution to the problems of vagueness. This paper provides an introduction to the most popular contextualist accounts, and a discussion of some of the most important arguments for and against them.