Theories of Vagueness

This book concerns the foundations of epistemic modality and hyperintensionality and their applications to the philosophy of mathematics. I examine the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality and hyperintensionality relate to the computational theory of mind; metaphysical modality and hyperintensionality; the types of mathematical modality and hyperintensionality; to the epistemic status of large cardinal axioms, undecidable propositions, and abstraction principles in the philosophy of mathematics; to the modal and hyperintensional profiles of the logic of rational intuition; and to the types of intention, when the latter is interpreted as a hyperintensional mental state. Chapter 2 argues for a novel type of expressivism based on the duality between the categories of coalgebras and algebras, and argues that the duality permits of the reconciliation between modal and hyperintensional cognitivism and modal and hyperintensional expressivism. I also develop a novel, topic-sensitive truthmaker semantics for dynamic epistemic logic, and develop a novel, dynamic two-dimensional semantics grounded in two-dimensional hyperintensional Turing machines. Chapter 3 provides an abstraction principle for epistemic (hyper-)intensions. Chapter 4 advances a topic-sensitive two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter 5 applies the fixed points of the modal μ-calculus in order to account for the iteration of epistemic states in a single agent, by contrast to availing of modal axiom 4 (i.e. the KK principle). The fixed point operators in the modal μ-calculus are rendered hyperintensional, which yields the first hyperintensional construal of the modal μ-calculus in the literature and the first application of the calculus to the iteration of epistemic states in a single agent instead of the common knowledge of a group of agents. Chapter 6 advances a solution to the Julius Caesar problem based on Fine's `criterial' identity conditions which incorporate conditions on essentiality and grounding. Chapter 7 provides a ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapters 2 and 4 is availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting.

Chapters 8-12 provide cases demonstrating how the two-dimensional hyperintensions of hyperintensional, i.e. topic-sensitive epistemic two-dimensional truthmaker, semantics, solve the access problem in the epistemology of mathematics. Chapter 8 examines the interaction between my hyperintensional semantics and the axioms of epistemic set theory, large cardinal axioms, the Epistemic Church-Turing Thesis, the modal axioms governing the modal profile of Ω-logic, Orey sentences such as the Generalized Continuum Hypothesis, and absolute decidability. These results yield inter alia the first hyperintensional Epistemic Church-Turing Thesis and hyperintensional epistemic set theories in the literature. Chapter 9 examines the modal and hyperintensional commitments of abstractionism, in particular necessitism, and epistemic hyperintensionality, epistemic utility theory, and the epistemology of abstraction. I countenance a hyperintensional semantics for novel epistemic abstractionist modalities. I suggest, too, that higher observational type theory can be applied to first-order abstraction principles in order to make first-order abstraction principles recursively enumerable, i.e. Turing machine computable, and that the truth of the first-order abstraction principle for two-dimensional  hyperintensions is grounded in its being possibly recursively enumerable and the machine being physically implementable. Chapter 10 examines the philosophical significance of hyperintensional Ω-logic in set theory and discusses the hyperintensionality of metamathematics. Chapter 11 provides a modal logic for rational intuition and provides a hyperintensional semantics. Chapter 12 avails of modal coalgebras to interpret the defining properties of indefinite extensibility, and avails of hyperintensional epistemic two-dimensional semantics in order to account for the interaction between interpretational and objective modalities and the truthmakers thereof. This yields the first hyperintensional category theory in the literature. I invent a new mathematical trick in which first-order structures are treated as categories, and Vopenka's principle can be satisfied because of the elementary embeddings between the categories and generate Vopenka cardinals in the category of Set in category theory. Chapter 13 examines modal responses to the alethic paradoxes. Chapter 14 examines, finally, the modal and hyperintensional semantics for the different types of intention and the relation of the latter to evidential decision theory.

Susan Haack, “The Whole Truth and Nothing but the Truth.”

Abstract: “Much truth is spoken, that more may be concealed,” wrote Mr. Justice Darling in 1879. Opening with an articulation of the distinction between truth (the concept) and truths (particular true propositions), this paper is largely devoted to: developing an account of the dual meaning of “partially true” – “true-in-part” vs. “part of the truth”; and (ii) understanding the reasons for, and the dangers of, the very common tendency to tell only part of the relevant truth. 


Keywords: truth, truths, partial truth, Ramsey, laconicism, vagueness, deception

(Proceedings of the 17th Amsterdam colloquium conference on Logic, language and meaning)
This paper provides an analysis of statements with predicates of personal taste (tasty, fun, etc.) Rather than directly relativizing semantic interpretation to a judge (cf., Lasersohn, 2005), this paper aims to capture the phenomenon called ‘faultless disagreement’ (the fact that one can deny a speaker’s subjective utterance without challenging the speaker‘s opinion) by means of pragmatic restrictions on quantification domains. Using vagueness models, a statement like the cake is tasty is analyzed as true in a partial context c iff it is true in the set of completions t consistent with c (Kamp, 1975), wherein tasty denotes different, contextually possible, taste measures (Kennedy, 1999). Phrases like for me restrict the set of completions to those with taste measures consistent with the speaker’s taste. Faultless disagreement naturally follows assuming speakers accommodate or reject implicit restrictions of this sort (Lewis, 1979).

This chapter suggests that the musical concept of timbre can function as an ideational “trading zone,” or Unscharfer Begriff. It explores some diverse treatments of timbre in Western and in Chinese music performance practice, in composition, and in musicological scholarship. The chapter discusses a parallel between the respective epistemic function of “atmosphere” and of “timbre,” as vague concepts in music scholarship. A combination of phenomenal dimensions together, atmospherically, form a gateway to such experiential possibilities. That they can be loosely conceived of as a single conceptual entity, timbre, Klangfarbe or yinse, speaks to the epistemological productiveness and affective power of vagueness. The chapter argues that in the West the notion of timbre has served as a mid-level conceptual “placeholder,” in the absence of more concrete analyses of acoustics and perception. As such it has travelled between music, aesthetics, psychology, physiology, acoustics and studies of culture more generally.

Within the sociology of science there is a constant debate of how we acquire knowledge of reality and whether this knowledge can be considered " true " and if, how reliable this knowledge might be. This is often referred to as the epistemological problem. In the process of aiming at true knowledge, if true knowledge exists, I will in this paper take a closer look at concepts in social science, characteristics of vagueness in concepts, how we define concepts in social science and what consequences choices regarding definitions might have for the outcome of research in education. When discussing vagueness and the definition of concepts I will use the concept national identity as an example. My interest in the concept national identity is grounded in my own research project looking at how secondary school teachers in South Sudan develop the sense of nationalism, patriotism and national unity, as stated in the Syllabus for Southern Sudan Certificate of Secondary Education (GOSS, 2007), among their students when teaching the recent history.

12 Ovviamente dovrei parlare del significato ordinario di "mountain" dal momento che la lingua dei cartografi era l'inglese. Prescindo dai dettagli dell'esempio per una questione di comodità (cfr. supra n. 4). 13 Vedi infra §3.2.1 e §4.1. 14 Sto qui usando intercambiabilmente "concetto vago" con "espressione [per per quel concetto] vaga" perché Waissman parla di concetti. Si potrebbe però sostenere che le espressioni di una lingua siano vaghe senza che i concetti corrispondenti, intesi come entità mentali, lo siano (o viceversa). Per semplificare tralascerò queste possibilità.

There are passages in Wittgenstein where he compares his method to psychotherapy and one or two where he seems to suggest that the ‘patient’ has the last word on his ‘illness’ and ‘cure’.  This paper tries to take these seriously, especially in connexion with the idea that victims of philosophical confusion talk nonsense.

Philosophical accusations of talking nonsense made in a polemical spirit are bound to fail, since the only way one can effectively criticise another’s utterance is by attaching some meaning to it.  But if the other acknowledges the possibility that his puzzlement arises from his having been taken in by nonsense, then perhaps a co-operative investigation of this suggestion might be fruitful.  Indeed self-diagnosis ought in principle to be possible.  Ought it not to have happened at least once in the history of philosophy if philosophical nonsense has ever been exposed? 

But there are difficulties.  For example:
a) We must ask whether or not philosophical nonsense, when exposed as such, is supposed to lose all appearance of meaningfulness.  Either answer is problematic.
b) The acknowledged similarity of the methodology under investigation to that of psychoanalysis exposes it to many of the familiar objections to the methodology of psychoanalysis, especially those that draw attention to the misleading influence of suggestion by an authority-figure.
c) The proposal that relentless cross-questioning of a suspected nonsense-talker will eventually lead her to ‘dry up’ (if she really is talking nonsense) seems to involve a version of the so-called Socratic Fallacy.

.Taking as my starting-point Cora Diamond’s paper ‘What nonsense might be’, I extend her ‘austere’ conception of nonsense to encompass the talking of nonsense.  We need to focus on the utterer as well as the utterance.  This brings out how difficult it is going to be to DIAGNOSE the talking of (philosophical) nonsense, without falling into the ‘sense that is senseless’ trap (PI, I, 500).  It is not clear to what extent Diamond sees the problem here.  I suggest that if there is a way round the difficulty, it will involve adopting a therapeutic approach.

What sort of logic do we get if we adopt a supervaluational semantics for vagueness? As it turns out, the answer depends crucially on how the standard notion of validity as truth preservation is recasted. There are several ways of doing that within a supervaluational framework, the main alternative being between “global” construals (e.g., an argument is valid iff it preserves truth-under-all-precisifications) and “local” construals (an argument is valid iff, under all precisifications, it preserves truth). The former alternative is by far more popular, but I argue in favor of the latter, for (i) it does not suffer from a number of serious objections, and (ii) it makes it possible to restore global validity as a defined notion.

The thesis defended is that at a certain arbitrary level of granularity, mountains have sharp, bona fide boundaries. In reply to arguments advanced by Varzi (2001), Smith & Mark (2001, 2003) I argue  that the lower limit of a mountain is neither vague nor fiat. Relying on early works by Cayley (1859), Maxwell (1870) and Jordan (1872), this lower limit consists in the  lines of watercourse which are defined as the lines of slope starting at passes. Such lines are metaphysically sharply delineated although they are not always easy to get at when facing a mountain. Hence, the indetermination is only epistemic.
In the second part of the paper, I try to combine this claim about the lower limit of a mountain with more recent claims advanced by alpinists on the right way to measure the height of a mountain, so as to capture its topographic prominence. I argue following them that the proper height of a mountain is the difference of altitude between its summit and its key-saddle, defined as the highest saddle one needs to cross in order to reach the closest higher summit.
Combining this two plausible views about the lower limit and height of a mountain leads to the surprising result that the key-saddle needed to measure the height of a mountain is not necessarily located on the lower-boundary of that mountain.

In this essay I will discuss the issue of vagueness when defining the concept of open work within the philosophy of Umberto Eco (1932-2016), particularly considering its relevance for the development of his original semiotic view. The analysis of vagueness allows us to stress the importance of Eco’s concept of open work not only in Opera Aperta (1962) but in different phases of his thought. This paper is divided into five sections. Section 1 briefly outlines the Peircean notion of vagueness, trying to understand it as a pivotal concept to define the structure and dynamical form of the open work. Section 2 dwells on the concept of quality in both John Dewey and Charles Sanders Peirce: here the possibility of articulating an interpretation is considered embodied in the vague immediacy of the qualitative experience. Section 3 analyses to what extent an artwork can be considered open and in motion, according to the Pareysonian concepts of form and interpretation. Section 4 stresses the hypothesis that vagueness produces not only the condition of possibility of an open work and of its multiple interpretations, but also an increase of information due to its aesthetic quality. Section 5 calls into question the distinction between use and interpretation and specifies that interpreting an open work does not mean using it for your own purposes but interpreting a developmental movement towards its own fulfilment; a text is nothing other than the rule that constitutes the universe of its interpretations.

If there are fundamental laws of nature, can they fail to be exact? In this paper, I consider the possibility that some fundamental laws are vague. I call this phenomenon 'fundamental nomic vagueness.' I characterize fundamental nomic vagueness as the existence of borderline lawful worlds and the presence of several other accompanying features. Under certain assumptions, such vagueness prevents the fundamental physical theory from being completely expressible in the mathematical language. Moreover, I suggest that such vagueness can be regarded as 'vagueness in the world.' For a case study, we turn to the Past Hypothesis, a postulate that (partially) explains the direction of time in our world. We have reasons to take it seriously as a candidate fundamental law of nature. Yet it is vague: it admits borderline (nomologically) possible worlds. An exact version would lead to an untraceable arbitrariness absent in any other fundamental laws. However, the dilemma between fundamental nomic vagueness and untraceable arbitrariness is dissolved in a new quantum theory of time's arrow.

Published in 1908, C.S. Peirce's 'A Neglected Argument for the Reality of God' is one of his most difficult articles. Presenting a peculiar entanglement of scientific method and theology, it sketches a 'humble' argument for the reality—and not the existence—of God for Musers, that is, those who pursue the activity he calls 'Musement'. In Musement, Peirce claims, we can achieve a kind of perception of the intertwinement of the three universes of experience: of feeling, of brute fact, and of reason. He also somehow relates each universe to a distinct phase of inquiry, which is described by the use of induction, deduction, and abduction or retroduction. The way that he develops his claims allows him to outline God as an abductive vague hypothesis to explain how those three universes make up a single whole. The hypothesis being vague means that the principle of contradiction does not hold for it. In this presentation, we aim at throwing some light at these points, focusing on the concept of Musement and what is understood as the vagueness of the hypothesis.

Scott Soames has recently argued that the fact that lawmakers and other legal practitioners regard vagueness as having a valuable power-delegating function in the law gives us good reason to favor one theory of vagueness over another. If Soames is right, then facts about legal practice can in an important sense adjudicate between rival theories of vagueness, which is an exciting conclusion, both from the point of view of philosophy of law and philosophy of language. In this paper, I argue that in the light of recent arguments made by Mark Greenberg, Lawrence Solum, and Dale Smith we have to give up the one premise of Soames’s argument that he seems to take to be uncontroversial: that the legal content of a statute or constitutional clause is identical with, or constituted by, its communicative content. I go on to provide a preliminary sketch of my own account of legal content and show how it provides a principled and unified response to the problems raised by Greenberg, Solum, and Smith. This account, however, does not suffice to vindicate Soames’s argument: since legal content is neither identical with nor constituted by communicative content, facts about legal practice are not able to adjudicate between rival theories of vagueness – at least not in the way envisioned by Soames. I conclude the paper by arguing that my point about Soames’s argument is generalizable.

In this paper we study the vague cosets and their properties.These concepts are used in the development of some important results and theorems about vague groups and vague normalgroups.Also some of their important properties have been investigated. Index Terms— vague set,vague group, vague normal group, vague coset.

ABSTRACT: Intuitionistic logic provides an elegant solution to the Sorites Paradox. Its acceptance has been hampered by two factors. First, the lack of an accepted semantics for languages containing vague terms has led even philosophers sympathetic to intuitionism to complain that no explanation has been given of why intuitionistic logic is the correct logic for such languages. Second, switching from classical to intuitionistic logic, while it may help with the Sorites, does not appear to offer any advantages when dealing with the so-called paradoxes of higher-order vagueness. We offer a proposal that makes strides on both issues. We argue that the intuitionist’s characteristic rejection of any third alethic value alongside true and false is best elaborated by taking the normal modal system S4M to be the sentential logic of the operator ‘it is clearly the case that’. S4M opens the way to an account of higher-order vagueness which avoids the paradoxes that have been thought to infect the notion. S4M is one of the modal counterparts of the intuitionistic sentential calculus (IPC) and we use this fact to explain why IPC is the correct sentential logic to use when reasoning with vague statements. We also show that our key results go through in an intuitionistic version of S4M. Finally, we deploy our analysis to reply to Timothy Williamson’s objections to intuitionistic treatments of vagueness.

What is the best way to address the problem of combinatory vagueness in legal adjudication? According to a view which is often held by philosophers of language, courts must make reference to what the legislature intended to say. Intentional content makes up for the lack of determinacy of meaning and contextual content, and can be used to determine whether a combinatory vague norm applies to borderline cases. In this paper I argue that this view does not provide a convincing account of how legal language works since it does not take into consideration the specific characteristics of legislative intention and the institutional dimension of linguistic communication in law. I then put forward a different explanation of combinatory vagueness based on an inferentialist approach to semantics and pragmatics. In this account, combinatory vagueness is a feature of language which depends on a specific form of disagreement between the participants in an exchange of reasons. By looking at the linguistic interplay between parties and judge in a legal dispute, the paper shows how combinatory vagueness arises and how it is reduced by courts in order to settle borderline cases.

I consider the dispute, not with a view to reaching a final verdict, but for the light it sheds on the notion of philosophical nonsense.  How does it look to a sceptic about philosophical nonsense?  It is generally agreed that the Tractatus is an unsatisfactory work, if the traditional reading of, for example, G. E. M. Anscombe or Peter Hacker is correct.  But is it any better on the ‘resolute’ reading of Cora Diamond or James Conant?  I maintain that it is more difficulty to handle the concept of nonsense than nonsensicalists of any stripe realise.  If the ‘ineffable truths’ and ‘important nonsense’ of traditional readings raise problems, so too do the contrast between meaningful frame and nonsensical core, Diamond’s idea that one can imaginatively identify with the nonsense-talker and the mysterious new entity, ‘transitional’ nonsense.  Meaning and communication, I suggest, keep reappearing precisely where they should not.

PREFACE


The Second Bologna Workshop on Vugueness took piace on the
9th and 10th of January 2004 at the Dipartimento di Discipline della Comunicazione of the Università degli Studi di Bologna.
The event was jointly organized by the Dipartimento di Discipline
della Comunicazione and the Dipartimento di Filosofia, and it had been preceded by a pilot workshop on vagueness which took piace on the 22nd and 23rd November 2001, field at the Dipartimento di Filosofia and organized by Annalisa coliva and Eva Picardi. We want to express our thanks to the speakers, Roy Cook, Patrick Greenough, Agustin Rayo, Sven Rosenkranz and Crispin Wright and to the discussants who delivered their comments on that occasion, Antonio Capuano, Manuel Gatto, Vittorio Murato and Luca Morena. We also would like to thank the Department of Phi losophy of the University of St. Andrews and the AHRC Arché Centre for the Philosophy of Logic. Language, Mathematics and Mind for their contribution in funding the participation of most of the invited speakers. Finally, our gratitude goes to Paolo Leonardi and Eva Picardi for their support in organizing this event.
This book has been published within the research program jointly
funded by the Ministero dell'Università e della Ricerca Scientifica
Tecnologica (MURST) and by the Università degli Studi di Bologna. The national project,  "Mente e linguaggio", as well as the Local project, on "La naturalizzazione della mente", were coordinated by Paolo Leonardi

SEBASTIANO MORIJZZI & ANDREA SERENI

TABLE OF CONTENTS

Pretace 7

Introduction 9 (Andrea Sereni; Sebastiano Moruzzi)

51 Vagueness, Response Dependence and Rule-Following
Some Reflections (Crispin Wright)

75 Clarifying the Logic of Clear Cases (Roy T Cook)

105 The Unexplained Supervenience Objection (Agustin Rayo)

117 Anti-Realism, Borderline Cases and Knowledge by Observation (Sven Rosenkranz)

131 Vagueness and Agnosticism (Sebastiano Moruzzi)

155 List of Contributors

157 Index of Names

ABSTRACT: This paper argues that the so-called paradoxes of higher-order vagueness are the result of a confusion between higher-order vagueness and the distribution of the objects of a Sorites series into extensionally non-overlapping non-empty classes.

Is There Such A Thing As Philosophical Nonsense?

For the best part of a century now philosophers have been accusing each other of talking nonsense.  This practice presupposes that people can be wrong in thinking they mean anything by what they say, that there can be an illusion of meaning.  But the assumption that illusions of meaning are possible has not, I believe, been seriously examined; nor has the problem of how such illusions could be diagnosed been seriously addressed.

This book extends the work of a number of recent writers on Wittgenstein who have advocated an'austere' view of nonsense, but in a more sceptical direction than they have taken.  It argues that nonsensicalism (the theory and practice of accusing your opponents of talking nonsense) is radically misguided and should be abandoned.

The possibility that illusions of meaning occur outside philosophy - in dreams, in certain hoaxes, in madness or under the influence of drugs - is also examined.

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.

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. 

What kind of semantics should someone who accepts the epistemicist theory of vagueness defended in Timothy Williamson's Vagueness (1994) give a defi niteness operator? To impose some interesting constraints on acceptable answers to this question, I will assume that the object language also contains a metaphysical necessity operator and a metaphysical actuality operator. I will suggest that the answer is to be found by working within a three-dimensional model theory. I will provide sketches of two ways of extracting an epistemicist semantics from that model theory, one of which I will fi nd to be more plausible than the other.

ABSTRACT: This short paper shows that the following common assumption is false: that in modal-logical representations of higher-order vagueness, for there to be borderline cases to borderline cases ad infinitum, the number of possible distinct modalities in a modal system must be infinite.

http://www.disputatio.com/wp-content/uploads/2014/11/Bobzien_Higher-Order-Vagueness-and-Numbers-of-Distinct-Modalities.pdf

Instead than denouncing the misunderstandings of legal formalism, mostly an invented formalism, we have to deepen our work on the forms of the law within which, paradoxically, one must count for the very forms of vagueness in the law, their varieties, their fidelity to the law and reasoning with vague concept in law. The law treats vagueness in an original way which constitutes a challenge to its analysis and technical approaches in the philosophy of language

Wittgenstein has shown that that life, in the sense that applies in the first place to human beings, is inherently linguistic. In this paper, I ask what is involved in language, given that it is thus essential to life, answering that language – or concepts – must be both alive and the ground for life. This is explicated by a Wittgensteinian series of entailments of features. According to the first feature, concepts are not intentional engagements. The second feature brings life back to concepts by describing them as inflectible: Attitudes, actions, conversations and other engagements inflect concepts, i.e. concepts take their particular characters in our actual engagements. However, inflections themselves would be reified together with the life they ground unless they could preserve the openness of concepts: hence the third feature of re-inflectibility. Finally, the openness of language must be revealed in actual life. This entails the possibility of conceptual ambivalence.

There is a trade-off between specificity and accuracy in existing accounts of belief. Descriptions of agents in the tripartite account, which recognizes only three doxastic attitudes—belief, disbelief, and suspension of judgment—are typically accurate, but not sufficiently specific. The orthodox Bayesian account, which requires real-valued credences, is perfectly specific, but often inaccurate: we often lack precise credences. I argue, first, that a popular attempt to fix the Bayesian account by using sets of functions is also inaccurate, since it postulates interval-valued credences with perfectly precise endpoints. This is analogous to the problem of higher order vagueness. Ultimately, I argue, the only way to avoid these problems is to endorse Insurmountable Unclassifiability. This principle has some surprising and radical consequences. For example, it entails that the trade-off between accuracy and specificity is in-principle unavoidable: sometimes it is simply impossible to characterize an agent's doxastic state in a way that is both fully accurate and maximally specific. What we can do, however, is improve on both the tripartite and existing Bayesian accounts. I construct a new framework which allows descriptions that are much more specific than those of the tripartite account, and yet which remain, unlike existing Bayesian accounts, perfectly accurate.