Vagueness and Indeterminacy, Misc

This paper argues that “free will” is vague. The argument has two steps. First, I argue that free will is a matter of degrees and, second, that there are no sharp boundaries separating free decisions and actions and non‐free ones. After presenting the argument, I focus on one significant consequence of the thesis, although others are mentioned along the way. In short, considerations of vagueness help understand the logic behind so‐called manipulation arguments, but also show why these arguments are ultimately (...) flawed. (shrink)

I defend the hypothesis that the semantic paradoxes, the paradoxes about collections, and the sorites paradoxes, are all paradoxes of reference fixing: they show that certain conventionally adopted and otherwise functional reference-fixing principles cannot provide consistent assignments of reference to certain relevant expressions in paradoxical cases. I note that the hypothesis has interesting implications concerning the idea of a unified account of the semantic, collection and sorites paradoxes, as well as about the explanation of their “recalcitrance”. I also note that (...) it does not necessarily imply that one should not expect the sometimes hoped for “unique” solution to a paradox of these kinds. (shrink)

The presentation of the “dual picture of vagueness” in my earlier work is supplemented here with a number of additional considerations. I emphasize how the picture lends itself naturally to treatments of the contribution of a typical degree adjective to propositional content and to truth conditions. A number of reasonable refinements of the picture are presented, especially concerning occasions of use of a degree adjective in which a class containing a sorites series is somehow involved in content fixing, but in (...) such a way that an extension ultimately gets fixed for the predicate. I also develop the observation that standard theories of the logical and semantic paradoxes rely on the idea that the content-fixing mechanisms for the paradoxical expressions work well in general but fail in some localized cases, and that while standard theories of the sorites paradox are clearly not based on this idea at all, the dual theory is fully based on the idea. Next, I offer a few remarks on some recent theories of vagueness and the sorites that are broadly in the spirit of the dual picture, emphasizing the aspects where the latter compares favorably against those recent theories. Finally, the question whether the dual theory is statable with truth according to the theory itself is considered in more detail than in my earlier presentation. (shrink)

This dissertation explores the prospects of postdeflationary substantive theorizing about truth. Postdeflationary theories define the concept of truth or the property of being a true truthbearer in a way that respects the deflationary desiderata of clarity, purity, and permissiveness with truth-aptness, without a necessary commitment to the core negative thesis of the deflationary approach. Postdeflationary substantive theories further acknowledge the complexity and explanatory utility of truth in understanding and defining other concepts and phenomena. The motivation for pursuing this study arises (...) from the so-called contemporary crisis of truth, where a substantive understanding of truth is subjected to widespread skepticism, critique, and even cynicism both inside and outside of philosophy in formal and mundane discourse. To better understand this crisis, particular attention is directed towards the deflationary critique of substantive theories of truth, which is a prevalent point of discussion in contemporary literature on western analytic philosophy. By exploring the limits and philosophical sustainability of deflationary critique of substantive accounts of truth, valuable insight is gained about the contemporary crisis of truth and the potential for substantive theorizing about truth in general. This dissertation composes of an introduction and four original research publications that address two connected themes: exploration of the philosophical sustainability of deflationary critique of substantive theories of truth, and exploration of the prospects for development of the now popular substantive pluralist theories of truth. These themes constitute both negative and positive aspects in relation to analyzing the prospects of postdeflationary substantive theorizing about truth. The first part of this dissertation focuses on arguing against the widespread deflationary readings of W.V.O. Quine’s truth, who is widely interpreted as a prominent and influential deflationist in both the secondary literature on his philosophy and contemporary truth-theoretic debates more broadly conceived. The first essay demonstrates that Quine’s immanent conception of truth involves commitments that are incompatible with general and theory-specific framings of the deflationary thesis. The second essay demonstrates conflicts between Quine’s views and what has in recent literature been argued as strong and moderate variants of the deflationary thesis. In conclusion, these essays demonstrate that the widespread deflationary readings of Quine’s truth are mistaken, thus removing a prominent thinker from the deflationists ranks while simultaneously casting suspicion towards the philosophical sustainability of the deflationary approach in general. The second part of this dissertation explores the prospects of postdeflationary substantive theorizing about truth by analyzing the limits and prospects for development of the increasingly popular substantive pluralist theories. The third essay explores different ways in which semantic ambiguity poses trouble for current pluralist models. The fourth essay argues that to achieve the theoretical desiderata that pluralists ask from discourse domains, the latter ought to be individuated on ontological rather than topical grounds. In conclusion, these essays demonstrate that while current pluralist models involve shortcomings, they encompass potential for development and provide a viable prospect for sustainable postdeflationary substantive theorizing about truth. (shrink)

Carter (Noûs 55(1):171–198, 2021) argued that while most simple positive numerical sentences are literally false, they can communicate true contents because relevance has a weakening effect on their literal contents. This paper presents a challenge for his account by considering entailments between the imprecise contents of numerical sentences and the imprecise contents of comparatives. I argue that while Carter's weakening mechanism can generate the imprecise contents of plain comparatives such as `A is taller than B', it cannot generate the imprecise (...) contents of comparatives that quantify the difference between their arguments, such as `A is exactly n times as tall as B'. I then propose an alternative theory on which intervals serve as both thick degrees on a scale and denotations of numerical expressions. I argue that the alternative theory can account for the imprecise contents of both forms of comparatives as well as the data that motivate Carter's theory. (shrink)

This short article pairs the realms of “Mathematics”, “Philosophy”, and “Poetry”, presenting some corners of intersection of this type of scientocreativity. Poetry have long been following mathematical patterns expressed by stern formal restrictions, as the strong metrical structure of ancient Greek heroic epic, or the consistent meter with standardized rhyme scheme and a “volta” of Italian sonnets. Poetry was always connected to Philosophy, and further on, notable mathematicians, like the inventor of quaternions, William Rowan Hamilton, or Ion Barbu, the creator (...) of the Barbilian spaces, have written appreciated poems. We will focus here on an avant-garde movement in literature, art, philosophy, and science, called Paradoxism, founded in Romania in 1980 by a mathematician, philosopher and poet, and on the laboured writing exercises of the Oulipo group, founded in Paris in 1960 by mathematicians and poets, both of them still in act. (shrink)

Methodological deflationism is a policy about how we should conduct ourselves when it comes to theories of truth: in particular, a deflationary theory of truth should be taken as one’s starting point, and the notion of truth should be inflated only as necessary. This policy is motivated, in part, by the need to balance the theoretical virtue of parsimony with that of explanatory sufficiency. In this article, the case is made that the methodological deflationist is in no position to properly (...) balance those virtues—a point made evident by tracing the relationship between semantic theories and the explanatory needs of theories of truth. Furthermore, methodological deflationism threatens to unduly influence semantic theorizing and, in doing so, displays an inappropriate bias towards deflationary theories of truth. (shrink)

Dorothy Edgington’s work has been at the centre of a range of ongoing debates in philosophical logic, philosophy of mind and language, metaphysics, and epistemology. This work has focused, although by no means exclusively, on the overlapping areas of conditionals, probability, and paradox. In what follows, I briefly sketch some themes from these three areas relevant to Dorothy’s work, highlighting how some of Dorothy’s work and some of the contributions of this volume fit in to these debates.

This chapter discusses the defence of metaphysical indeterminacy by Elizabeth Barnes and Robert Williams and discusses a classical and bivalent theory of such indeterminacy. Even if metaphysical indeterminacy arguably is intelligible, Barnes and Williams argue in favour of it being so and this faces important problems. As for classical logic and bivalence, the chapter problematizes what exactly is at issue in this debate. Can reality not be adequately described using different languages, some classical and some not? Moreover, it is argued (...) that the classical and bivalent theory of Barnes and Williams does not avoid the problems that arise for rival theories. (shrink)

Miriam Schoenfield argues that moral realism and moral vagueness imply ontic vagueness. In particular, she argues that neither shifty nor rigid semantic accounts of vagueness can provide a satisfactory explanation of moral vagueness for moral realists. This paper constitutes a response. I argue that Schoenfield's argument against the shifty semantic account presupposes that moral indeterminacies can, in fact, be resolved determinately by crunching through linguistic data. I provide different reasons for rejecting this assumption. Furthermore, I argue that Schoenfield's rejection of (...) the rigid semantic account is based on a presupposition that ultimately implies the very same claim that is under dispute: the vagueness of moral predicates in imperfect languages persists in the perfect language, as well. (shrink)

‘Puzzles’, ‘word games’, ‘logical anomalies’, whatever we call them, they perplex us and challenge our familiar patterns of reasoning. One of these puzzles, among many others, originated from the mind of an ancient Megarian logician, Eubulides of Miletus, and endures to the modern day.1 Its name, ‘sorites’, can be traced to the Greek word soros, meaning ‘heap.’ The answer to whether one grain of sand ‘is a heap’ or ‘is not a heap’ seems quite simple: it is not a heap. (...) However, as we add grains to the one, at what future point does the non-heap become a heap? Our decision is fraught with uncertainty. Are the objects or the language we are using to describe them vague? In academic philosophy, the ancient Greek puzzle has gained the status of a paradox, as philosophers apply stoic and modern logic to these propositions considered to have vague predicates. The current debate has developed quite serious and wide-ranging implications, such as whether sorites issues provide adequate grounds for abandoning our standard ontology (or our understanding of what really exists), 2 and (germinating into another discipline) whether vagueness in the language of legal rules can generate disagreement as to whether there are right answers to questions of law. 3 Several unique solutions to the paradox have been proposed, yet all suffer from specific inadequacies that might, upon further reflection, disappear in the event that we endeavour to produce a synthesis. When we say that we are attempting to derive a synthesis, we seek to combine elements of two solutions for the sake of creating a new (and hopefully, though not necessarily, a better) one. The paradox of the sorites, or ‘the heap’, appeals to us because it challenges our assumption that we may categorically describe clear cases and their negations, yet fail to distinguish the borderline ones. I will demonstrate that some combinations of solutions are easy fits while others are extremely weak and awkward as attempts at resolving the paradox. Nonetheless, the process of synthesis produces three noteworthy combinations. (shrink)

This paper deals with the question of what it is for a quantifier expression to be vague. First it draws a distinction between two senses in which quantifier expressions may be said to be vague, and provides an account of the distinction which rests on independently grounded assumptions. Then it suggests that, if some further assumptions are granted, the difference between the two senses considered can be represented at the formal level. Finally, it outlines some implications of the account provided (...) which bear on three debated issues concerning quantification. (shrink)

The topic of this paper is whether there is metaphysical vagueness. It is shown that it is important to distinguish between the general phenomenon of indeterminacy and the more narrow phenomenon of vagueness (the phenomenon that paradigmatically rears its head in sorites reasoning). Relatedly, it is important to distinguish between metaphysical indeterminacy and metaphysical vagueness. One can wish to allow metaphysical indeterminacy but rule out metaphysical vagueness. As is discussed in the paper, central argument against metaphysical vagueness, like those of (...) Gareth Evans and Mark Sainsbury, would if successful rule out metaphysical indeterminacy. One way to argue specifically against the possibility of metaphysical vagueness might be thought to be to argue for a specific theory of the nature of vagueness according to which vagueness is a semantic phenomenon. But it is shown that there are complications also pertaining to arguments with that structure. Toward the end of the paper, I discuss Trenton Merricks’ well-known argument against a semantic view on vagueness and for a metaphysical view. (shrink)

Suppose that we repair a wooden ship by replacing its planks one by one with new ones while at the same time reconstructing it using the discarded planks. Some defenders of vague or indeterminate identity claim that: (1) although the reconstructed ship is distinct from the repaired ship, it is indeterminate whether the original ship is the reconstructed ship and indeterminate whether it is the repaired ship, and (2) the indeterminacy is due to the world and not just an imprecision (...) in the language used to describe the situation. I argue that such a description is incoherent. The argument has two features. First, it differs in spirit from Gareth Evans's more general famous proof against the possibility of indeterminate identity. This is because I rely on facts regarding counting and sets. Second, I focus on Terence Parsons's recent defence of indeterminate identity. I argue that his attempts at making sense of counting objects involving indeterminate identities fail on technical and philosophical grounds. (shrink)

The semantics of our event talk is a complex affair. What is it that we are talking about when we speak of Brutus’s stabbing of Caesar? Exactly where and when did it take place? Was it the same event as the killing of Caesar? Some take questions such as these to be metaphysical questions. I think they are questions of semantics—questions about the way we talk and about what we mean. And I think that this conflict between metaphysic and semantic (...) concerns is indicative of a “deep indeterminacy” (Bennett’s phrase) in our event concept. We do talk about events; but what events a statement is about is not something that can be inferred from the event names we use; it depends heavily (more heavily than with ordinary material objects) on local context and unprincipled intuitions. This paper illustrates this view in connection with two examples: the phenomenon of vagueness and the dispute over identity statements. (shrink)

This paper proposes a new framework to solve the surprise examination paradox. I survey preliminary the main contributions to the literature related to the paradox. I introduce then a distinction between a monist and a dichotomic analysis of the paradox. With the help of a matrix notation, I also present a dichotomy that leads to distinguish two basically and structurally different notions of surprise, which are respectively based on a conjoint and a disjoint structure. I describe then how Quine's solution (...) and Hall's reduction apply to the version of the paradox corresponding to the conjoint structure. Lastly, I expose a solution to the version of the paradox based on the disjoint structure. (shrink)

A plausible thought about vagueness is that it involves semantic incompleteness. To say that a predicate is vague is to say (at the very least) that its extension is incompletely specified. Where there is incomplete specification of extension there is indeterminacy, an indeterminacy between various ways in which the specification of the predicate might be completed or sharpened. In this paper we show that this idea is bound to founder by presenting an argument to the effect that there are vague (...) predicates which cannot be sharpened in such a way as to meet certain basic constraints (of penumbral connection and public accessibility) that must be imposed on the very notion of a sharpening. (shrink)

In ‘Something to Do with Vagueness ...’, Linda Burns defends an analogy between the informational and the borderline-case variety of vagueness. She argues that the latter is in fact less extraordinary and less disastrous than people in the tradition of Michael Dummett and Crispin Wright have told us. However, her account involves presuppositions that cannot be taken for granted. Here is to take a closer look at some of these presuppositions and argue hat they may--when left unguarded--undermine much of Burns’ (...) general account. (shrink)

Book Information Vagueness and Contradiction. Vagueness and Contradiction Roy Sorensen Oxford Clarendon Press 2001 208 £25 By Roy Sorensen. Clarendon Press. Oxford. Pp. 208. £25.

[Author's note: I am posting this dissertation since it may be of interest to some people working on vagueness and related topics. It does not represent my current views on the topic. I have never attempted to publish any of this work, though I hope some day to return to it.] -/- Philosophers have devoted a lot of attention to vagueness in recent years, but there is still no general consensus about how to resolve the Sorites paradox. Timothy Williamson‘s epistemic (...) view, which claims that our vague terms have unknown sharp boundaries, is the most popular and most controversial current account. No one has shown exactly what is wrong the epistemic view and no one has provided a satisfying alternative to it. These two projects – articulating what is wrong with the epistemic view, and providing a plausible alternative – are the primary goals of this dissertation. -/- Additionally, I survey ordinary intuitions that underlie Sorites paradoxes, and I note how these intuitions inform, and are informed by, a number of deeper philosophical debates in metaphysics, epistemology, philosophy of language, and ethics. In part, this serves as an explanation of why the Sorites paradox has remained so difficult to resolve. -/- The most common objection to the epistemic view – that it provides an unsatisfactory account of the connection between meaning and use – has not been successful in undermining the view. My own objection is a metaphysical, and not a semantic, objection: the epistemic view fails to provide the best explanation of what objects and properties exist. Instead, an eliminativist account of macro-level objects and properties, according to which there are no mountains and there is no property of being lavender-colored, is a better metaphysical account than one that claims that there are mountains and color properties that have sharp boundaries. -/- Of course, this eliminativist view is intuitively unappealing, and to show how statements in ordinary language can in some way be taken to be true, I introduce the normative choice account. According to this view, although non-normative facts about linguistic behavior and about the external world do not determine a precise reference for our terms, our choices may do so. I claim that this provides all that is needed for there to be semantic normativity. First, we are still guided in our choices to some extent by psychological tendencies, and second, there are resources in semantic deliberation to respond to aberrant uses of language. (shrink)

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 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. (shrink)

One of the hardest problems in philosophy, one that has been around for over two thousand years without generating any significant consensus on its solution, involves the concept of vagueness: a word or concept that doesn't have a perfectly precise meaning. There is an argument that seems to show that the word or concept simply must have a perfectly precise meaning, as violently counterintuitive as that is. Unfortunately, the argument is usually so compressed that it is difficult to see why (...) exactly the problem is so hard to solve. In this article I attempt to explain just why it is that the problem – the sorites paradox – is so intractable.Export citation. (shrink)

This paper intends to further the understanding of the formal properties of (higher-order) vagueness by connecting theories of (higher-order) vagueness with more recent work in topology. First, we provide a “translation” of Bobzien's account of columnar higher-order vagueness into the logic of topological spaces. Since columnar vagueness is an essential ingredient of her solution to the Sorites paradox, a central problem of any theory of vagueness comes into contact with the modern mathematical theory of topology. Second, Rumfitt’s recent topological reconstruction (...) of Sainsbury’s theory of prototypically defined concepts is shown to lead to the same class of spaces that characterize Bobzien’s account of columnar vagueness, namely, weakly scattered spaces. Rumfitt calls these spaces polar spaces. They turn out to be closely related to Gärdenfors’ conceptual spaces, which have come to play an ever more important role in cognitive science and related disciplines. Finally, Williamson’s “logic of clarity” is explicated in terms of a generalized topology (“locology”) that can be considered an alternative to standard topology. Arguably, locology has some conceptual advantages over topology with respect to the conceptualization of a boundary and a borderline. Moreover, in Williamson’s logic of clarity, vague concepts with respect to a notion of a locologically inspired notion of a “slim boundary” are (stably) columnar. Thus, Williamson’s logic of clarity also exhibits a certain affinity for columnar vagueness. In sum, a topological perspective is useful for a conceptual elucidation and unification of central aspects of a variety of contemporary accounts of vagueness. (shrink)

Alessandro Torza argues that Ted Sider’s Lewisian argument against vague existence is insufficient to rule out the possibility of what he calls ‘super-vague existence’, that is the idea that existence is higher-order vague, for all orders. In this chapter it is argued that the possibility of super-vague existence is ineffective against the conclusion of Sider’s argument since super-vague existence cannot be consistently claimed to be a kind of linguistic vagueness. Torza’s idea of super-vague existence seems to be better suited to (...) model vague existence under the assumption that vague existence is instead a form of ontic indeterminacy, contra what Ted Sider and David Lewis assume. (shrink)

This paper is an expanded written version of my reply to Rosanna Keefe’s paper ‘Modelling higher-order vagueness: columns, borderlines and boundaries’ (Keefe 2015), which in turn is a reply to my paper ‘Columnar higher-order vagueness, or Vagueness is higher-order vagueness’ (Bobzien 2015). Both papers were presented at the Joint Session of the the Aristotelian Society and the Mind Association in July, 2015. At the Joint Session meeting, there was insufficient time to present all of my points in response to Keefe’s (...) paper. In addition, the audio of the session, which is available online, becomes inaudible at the beginning of my reply to Keefe’s comments due to a technical defect. The following is a full version of my remarks. (shrink)

Most work on the semantic paradoxes within classical logic has centered around what this essay calls “linguistic” accounts of the paradoxes: they attribute to sentences or utterances of sentences some property that is supposed to explain their paradoxical or nonparadoxical status. “No proposition” views are paradigm examples of linguistic theories, although practically all accounts of the paradoxes subscribe to some kind of linguistic theory. This essay shows that linguistic accounts of the paradoxes endorsing classical logic are subject to a particularly (...) acute form of the revenge paradox: that there is no exhaustive classification of sentences into “good” and “bad” such that the T-schema holds when restricted to the “good” sentences unless it is also possible to prove some “bad” sentences. The foundations for an alternative classical nonlinguistic approach is outlined that is not subject to the same kinds of problems. Although revenge paradoxes of different strengths can be formulated, they are found to be indeterminate at higher orders and not inconsistent. (shrink)

There is a trade-off between specificity and accuracy in existing models of belief. Descriptions of agents in the tripartite model, which recognizes only three doxastic attitudes—belief, disbelief, and suspension of judgment—are typically accurate, but not sufficiently specific. The orthodox Bayesian model, 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 model by using sets of functions is also inaccurate, since it requires us to (...) have interval-valued credences with perfectly precise endpoints. We can see this problem as 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 models. I construct a new model of belief—the minimal model—that allows us to characterize agents with much greater specificity than the tripartite model, and yet which remains, unlike existing Bayesian models, perfectly accurate. (shrink)

Most descriptions of higher-order vagueness in terms of traditional modal logic generate so-called higher-order vagueness paradoxes. The one that doesn't is problematic otherwise. Consequently, the present trend is toward more complex, non-standard theories. However, there is no need for this.In this paper I introduce a theory of higher-order vagueness that is paradox-free and can be expressed in the first-order extension of a normal modal system that is complete with respect to single-domain Kripke-frame semantics. This is the system QS4M+BF+FIN. It corresponds (...) to the class of transitive, reflexive and final frames. With borderlineness defined logically as usual, it then follows that something is borderline precisely when it is higher-order borderline, and that a predicate is vague precisely when it is higher-order vague.Like Williamson's, the theory proposed here has no clear borderline cases in Sorites sequences. I argue that objections that there must be clear borderline cases ensue from the confusion of two notions of borderlineness—one associated with genuine higher-order vagueness, the other employed to sort objects into categories—and that the higher-order vagueness paradoxes result from superimposing the second notion onto the first. Lastly, I address some further potential objections. (shrink)

Timothy Williamson has shown that the B axiom for 'definitely' (α → Δ¬Δ¬α) guarantees that if a sentence is second-order vague in a Kripke model, it is nth order vague for every n. More recently, Anna Mahtani has argued that Williamson's epistemicist theory of vagueness does not support the B axiom, and conjectured that if we consider models in which the “radius of accessibility” varies between different points, we will be able to find sentences that are nth-order vague but (n+1)th-order (...) precise, for any n. This paper bolsters Mahtani's argument, shows her conjecture to be true, and shows that imposing certain further natural constraints on "variable radius" models does not change the situation. (shrink)

The conclusion of this chapter is that higher-order vagueness is universal: no sentence whatsoever is definitely true, definitely definitely true, definitely definitely definitely true, and so on ad infinitum. The argument, of which there are several versions, turns on the existence of Sorites sequences of possible worlds connecting the actual world to possible worlds where a given sentence is used in such a way that its meaning is very different. The chapter attempts to be neutral between competing accounts of the (...) nature of vagueness and definiteness. (shrink)

Epistemicism about vagueness is the view that vagueness, or indeterminacy, is an epistemic matter. Truthmaker-gap epistemicism is the view that indeterminate truths are indeterminate because their truth is not grounded by any worldly fact. Both epistemicism in general and truthmaker-gap epistemicism originated in Roy Sorensen's work on vagueness. My aim in this paper is to give a characterization of truthmaker-gap epistemicism and argue that the view is incompatible with higher-order vagueness: vagueness in whether some case of the form ‘it is (...) determinate that A’ or ‘it is indeterminate whether A’ is true. Since it is highly likely that there is higher-order vagueness, truthmaker-gap epistemicism is in an uncomfortable position. (shrink)

A number of authors have objected to the application of non-classical logic to problems in philosophy on the basis that these non-classical logics are usually characterised by a classical metatheory. In many cases the problem amounts to more than just a discrepancy; the very phenomena responsible for non-classicality occur in the field of semantics as much as they do elsewhere. The phenomena of higher order vagueness and the revenge liar are just two such examples. The aim of this paper is (...) to show that a large class of non-classical logics are strong enough to formulate their own model theory in a corresponding non-classical set theory. Specifically I show that adequate definitions of validity can be given for the propositional calculus in such a way that the metatheory proves, in the specified logic, that every theorem of the propositional fragment of that logic is validated. It is shown that in some cases it may fail to be a classical matter whether a given sentence is valid or not. One surprising conclusion for non-classical accounts of vagueness is drawn: there can be no axiomatic, and therefore precise, system which is determinately sound and complete. (shrink)

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.

The purpose of this paper is to challenge some widespread assumptions about the role of the modal axiom 4 in a theory of vagueness. In the context of vagueness, axiom 4 usually appears as the principle ‘If it is clear (determinate, definite) that A, then it is clear (determinate, definite) that it is clear (determinate, definite) that A’, or, more formally, CA → CCA. We show how in the debate over axiom 4 two different notions of clarity are in play (...) (Williamson-style "luminosity" or self-revealing clarity and concealeable clarity) and what their respective functions are in accounts of higher-order vagueness. On this basis, we argue first that, contrary to common opinion, higher-order vagueness and S4 are perfectly compatible. This is in response to claims like that by Williamson that, if vagueness is defined with the help of a clarity operator that obeys axiom 4, higher-order vagueness disappears. Second, we argue that, contrary to common opinion, (i) bivalence-preservers (e.g. epistemicists) can without contradiction condone axiom 4 (by adopting what elsewhere we call columnar higher-order vagueness), and (ii) bivalence-discarders (e.g. open-texture theorists, supervaluationists) can without contradiction reject axiom 4. Third, we rebut a number of arguments that have been produced by opponents of axiom 4, in particular those by Williamson. (The paper is pitched towards graduate students with basic knowledge of modal logic.). (shrink)

This paper discusses Fara's so-called 'Paradox of Higher-Order Vagueness' concerning supervaluationism. In the paper I argue that supervaluationism is not committed to global validity, as it is largely assumed in the literature, but to a weaker notion of logical consequence I call 'regional validity'. Then I show that the supervaluationist might solve Fara's paradox making use of this weaker notion of logical consequence. The paper is discussed by Delia Fara in the same volume.

Philosophers disagree about whether vagueness requires us to admit truth-value gaps, about whether there is a gap between the objects of which a given vague predicate is true and those of which it is false on an appropriately constructed sorites series for the predicate—a series involving small increments of change in a relevant respect between adjacent elements, but a large increment of change in that respect between the endpoints. There appears, however, to be widespread agreement that there is some sense (...) in which vague predicates are gappy which may be expressed neutrally by saying that on any appropriately constructed sorites series for a given vague predicate there will be a gap between the objects of which the predicate is definitely true and those of which it is definitely false. Taking as primitive the operator ‘it is definitely the case that’, abbreviated as ‘D’, we may stipulate that a predicate F is definitely true (or definitely false) of an object just in case ‘DF (a)’, where a is a name for the object, is true (or false) simpliciter.1 This yields the following conditional formulation of a ‘gap principle’: (DΦ(x) ∧ D¬Φ(y)) → ¬R(x, y). Here ‘Φ’ is to be replaced with a vague predicate, while ‘R’ is to stand for a sorites relation for that predicate: a relation that can be used to construct a sorites series for the predicate—such as the relation of being just one millimetre shorter than for the predicate ‘is tall’. Disagreements about the sense in which it is correct to say that vague predicates are gappy can then be recast as disagreements about how to understand the definitely operator. One might give it, for example, a pragmatic construal such as ‘it would not be misleading to assert that’; or an epistemic construal such as ‘it is known that’ or ‘it is knowable that’; or a semantic construal such as ‘it is true that’. (shrink)

ABSTRACT: Stewart Shapiro recently argued that there is no higher-order vagueness. More specifically, his thesis is: (ST) ‘So-called second-order vagueness in ‘F’ is nothing but first-order vagueness in the phrase ‘competent speaker of English’ or ‘competent user of “F”’. Shapiro bases (ST) on a description of the phenomenon of higher-order vagueness and two accounts of ‘borderline case’ and provides several arguments in its support. We present the phenomenon (as Shapiro describes it) and the accounts; then discuss Shapiro’s arguments, arguing that (...) none is compelling. Lastly, we introduce the account of vagueness Shapiro would have obtained had he retained compositionality and show that it entails true higher-order vagueness. (shrink)

ABSTRACT: Recently a bold and admirable interpretation of Chrysippus’ position on the Sorites has been presented, suggesting that Chrysippus offered a solution to the Sorites by (i) taking an epistemicist position1 which (ii) made allowances for higher-order vagueness. In this paper I argue (i) that Chrysippus did not take an epistemicist position, but − if any − a non-epistemic one which denies truth-values to some cases in a Sorites-series, and (ii) that it is uncertain whether and how he made allowances (...) for higher-order vagueness, but if he did, this was not grounded on an epistemicist position. (shrink)

The conclusion of this chapter is that higher-order vagueness is universal: no sentence whatsoever is definitely true, definitely definitely true, definitely definitely definitely true, and so on ad infinitum. The argument, of which there are several versions, turns on the existence of Sorites sequences of possible worlds connecting the actual world to possible worlds where a given sentence is used in such a way that its meaning is very different. The chapter attempts to be neutral between competing accounts of the (...) nature of vagueness and definiteness. (shrink)

Philosophers disagree about whether vagueness requires us to admit truth-value gaps, about whether there is a gap between the objects of which a given vague predicate is true and those of which it is false on an appropriately constructed sorites series for the predicate—a series involving small increments of change in a relevant respect between adjacent elements, but a large increment of change in that respect between the endpoints. There appears, however, to be widespread agreement that there is some sense (...) in which vague predicates are gappy which may be expressed neutrally by saying that on any appropriately constructed sorites series for a given vague predicate there will be a gap between the objects of which the predicate is definitely true and those of which it is definitely false. Taking as primitive the operator ‘it is definitely the case that’, abbreviated as ‘D’, we may stipulate that a predicate F is definitely true (or definitely false) of an object just in case ‘DF (a)’, where a is a name for the object, is true (or false) simpliciter.1 This yields the following conditional formulation of a ‘gap principle’: (DΦ(x) ∧ D¬Φ(y)) → ¬R(x, y). Here ‘Φ’ is to be replaced with a vague predicate, while ‘R’ is to stand for a sorites relation for that predicate: a relation that can be used to construct a sorites series for the predicate—such as the relation of being just one millimetre shorter than for the predicate ‘is tall’. Disagreements about the sense in which it is correct to say that vague predicates are gappy can then be recast as disagreements about how to understand the definitely operator. One might give it, for example, a pragmatic construal such as ‘it would not be misleading to assert that’; or an epistemic construal such as ‘it is known that’ or ‘it is knowable that’; or a semantic construal such as ‘it is true that’. (shrink)

It is generally supposed that borderline cases account for the tolerance of vague terms, yet cannot themselves be sharply bounded, leading to infinite levels of higher order vagueness. This higher order vagueness subverts any formal effort to make language precise. However, it is possible to show that tolerance must diminish at higher orders. The attempt to derive it from indiscriminability founders on a simple empirical test, and we learn instead that there is no limit to how small higher order tolerance (...) may become. That means there is no limit to how precisely we may draw the boundaries of borderline cases, thus forestalling any requirement for higher order vagueness. (shrink)

Is 'vague' vague? Is the meaning of 'true' vague? Is higher-order vagueness unavoidable? Is it possible to say precisely what it is to say something precisely? These questions, deeply interrelated and of fundamental importance to logic and semantics, have been addressed recently by Achille Varzi in articles focused on an ingenius attempt by Roy Sorensen ("An Argument for the Vagueness of 'Vague'") to demonstrate that 'vague' is vague.

Defences of Williamson’s Anti-Luminosity Argument (ALA) that employ doxastic propagation principles—i.e., rules by which cases of beliefs and/or dispositions to believe are inferred from other such cases—risk running into sorites. Since these principles are explainable by an ineffective capacity to phenomenally discriminate between two adjacent cases, luminist rejections of the ALA can halt sorites by denying doxastic propagation, thereby reaffirming these discriminative capacities as appropriately effective. One potent method of resisting the luminist involves recharacterizing discriminative capacities in terms of a (...) distinction between beliefs and their underlying dispositions to reinstate the plausibility of doxastic propagation. To this effect, I propose a novel coarse-grained approach favouring the ALA that leverages a modal analysis of the belief/ disposition distinction. This motivates a sharp threshold between belief and absent belief that neither succumbs to sorites nor begs the question against the luminist. The upshot of this approach is substantial: by conferring a dialectical advantage to the anti-luminist, the luminist is held to highly problematic positions, regarding the trivialisation of safety and the relationship between beliefs and dispositions, if they deny the coarse-grained approach at many of its components. (shrink)

In his article, Vsevolod Ladov poses an important question – might paradoxes admit of a uniform solution? As an answer, I propose looking at an approach in which it is argued that we must recognize the truth predicate as merely analogous to a vague predicate. The proponents of this approach (V. McGee, J. Tappenden, H. Field, G. Priest and D. Hyde) insist that there is a structural relation between sorites paradoxes and self-reference paradoxes and that they should have a unified (...) solution. (shrink)

ABSTRACT: This paper offers a generic revenge-proof solution to the Sorites paradox that is compatible with several philosophical approaches to vagueness, including epistemicism, supervaluationism, psychological contextualism and intuitionism. The solution is traditional in that it rejects the Sorites conditional and proposes a modally expressed weakened conditional instead. The modalities are defined by the first-order logic QS4M+FIN. (This logic is a modal companion to the intermediate logic QH+KF, which places the solution between intuitionistic and classical logic.) Borderlineness is introduced modally as (...) usual. The solution is innovative in that its modal system brings out the semi-determinability of vagueness. Whether something is borderline and whether a predicate is vague or precise is only semi-determinable: higher-order vagueness is columnar. Finally, the solution is based entirely on two assumptions. (1) It rejects the Sorites conditional. (2) It maintains that if one specifies borderlineness in terms of the ‒suitably interpreted‒ modal logic QS4M+FIN, then one can explain why the Sorites appears paradoxical. From (1)+(2) it results that one can tell neither where exactly in a Sorites series the borderline zone starts and ends nor what its extension is. Accordingly, the solution is also called agnostic. (shrink)