A consequence relation is strongly classical if it has all the theorems and entailments of classical logic as well as the usual meta-rules (such as Conditional Proof). A consequence relation is weakly classical if it has all the theorems and entailments of classical logic but lacks the usual meta-rules. The most familiar example of a weakly classical consequence relation comes from a simple supervaluational approach to modelling vague language. This approach is formally equivalent to an account of logical consequence according (...) to which α1, ..., αn entails β just in case □α1, ..., □αn entails □β in the modal logic S5. This raises a natural question: If we start with a different underlying modal logic, can we generate a strongly classical logic? This paper explores this question. In particular, it discusses four related technical issues: (1) Which base modal logics generate strongly classical logics and which generate weakly classical logics? (2) Which base logics generate themselves? (3) How can we directly characterize the logic generated from a given base logic? (4) Given a logic that can be generated, which base logics generate it? The answers to these questions have philosophical interest. They can help us to determine whether there is a plausible supervaluational approach to modelling vague language that yields the usual meta-rules. They can also help us to determine the feasibility of other philosophical projects that rely on an analogous formalism, such as the project of defining logical consequence in terms of the preservation of an epistemic status. (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)

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)

Most words in natural language are vague, that is to say, they lack sharp boundaries and, hence, they have (actual or potential) borderline cases, where the word in question neither definitely applies nor definitely fails to apply. Vagueness gives rise to paradoxes, the best known of which is the sorites (concerned with how many grains of sand are needed to make a heap). Besides offering a solution to such paradoxes, a theory of vagueness should systematically describe how the truth conditions (...) of sentences with vague terms are determined; and it should also define the right logical principles for reasoning with such sentences. This article offers an introduction to the main theories of vagueness and to the problems they have to face. (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.

In this paper I develop a framework for understanding ontic vagueness. The project of the paper is two-fold. I first outline a definitional account of ontic vagueness – one that I think is an improvement on previous attempts because it remains neutral on other, independent metaphysical issues. I then develop one potential manifestation of that basic definitional structure. This is a more robust (and much less neutral) account which gives a fully classical explication of ontic vagueness via modal concepts. The (...) overarching aim is to systematically investigate the puzzling question of what exactly it could be for the world itself to be vague. (shrink)

Paraconsistent approaches have received little attention in the literature on vagueness (at least compared to other proposals). The reason seems to be that many philosophers have found the idea that a contradiction might be true (or that a sentence and its negation might both be true) hard to swallow. Even advocates of paraconsistency on vagueness do not look very convinced when they consider this fact; since they seem to have spent more time arguing that paraconsistent theories are at least as (...) good as their paracomplete counterparts, than giving positive reasons to believe on a particular paraconsistent proposal. But it sometimes happens that the weakness of a theory turns out to be its mayor ally, and this is what (I claim) happens in a particular paraconsistent proposal known as subvaluationism. In order to make room for truth-value gluts subvaluationism needs to endorse a notion of logical consequence that is, in some sense, weaker than standard notions of consequence. But this weakness allows the subvaluationist theory to accommodate higher-order vagueness in a way that it is not available to other theories of vagueness (such as, for example, its paracomplete counterpart, supervaluationism). (shrink)

The use of vague language in law has important implications for legal theory. Legal philosophers have occasionally grappled with those implications, but they have not come to grips with the characteristic phenomenon of vagueness: the sorites paradox. I discuss the paradox, and claim that it poses problems for some legal theorists (David Lyons, Hans Kelsen, and, especially, Ronald Dworkin). I propose that a good account of vagueness will have three consequences for legal theory: (i) Theories that deny that vagueness in (...) formulations of the law leads to discretion in adjudication (including Dworkin's) cannot accommodate “higher-order” vagueness, (ii) A legal theory should accept that the law is partly indeterminate when it can be stated in vague language, (iii) However, the traditional formulation of the indeterminacy claim, that a vague statement is “neither true nor false” in a borderline case, is misconceived and should be abandoned. (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)

According to contextualism about vagueness, the content of a vague predicate is context sensitive. On this view, when item a is in the penumbra of the vague predicate ‘F’, speakers may utter ‘Fa’, or they may utter ‘not-Fa’, without contravening the literal meaning of ‘F’. Unlike its more popular variants, the version of contextualism I defend rejects the principle of tolerance, a principle according to which small differences should not affect the applicability of a vague predicate. My goal is to (...) show how such a rejection allows for a plausible treatment of higher-order vagueness, and for a dissolution of paradoxes of higher-order vagueness. (shrink)

. According to Horgans transvaluationist approach, the robustness that characterizes vague terms is inherently incoherent. He analyzes that robustness into two conceptual poles, individualistic and collectivistic, and ascribes the incoherence to the former. However, he claims vague terms remain useful nonetheless, because the collectivistic pole can be realized with a suitable non-classical logic and can quarantine the incoherence arising out of the individualistic pole. I argue, on the contrary, that the nonclassical logic fails to resolve the difficulty and that the (...) incoherence afflicts Horgans collectivistic pole as well, consequently invalidating the entire transvaluationist approach. An alternative, coherent conception of robustness is suggested. (shrink)

In this paper we explain our pretense account of truth-talk and apply it in a diagnosis and treatment of the Liar Paradox. We begin by assuming that some form of deflationism is the correct approach to the topic of truth. We then briefly motivate the idea that all T-deflationists should endorse a fictionalist view of truth-talk, and, after distinguishing pretense-involving fictionalism (PIF) from error- theoretic fictionalism (ETF), explain the merits of the former over the latter. After presenting the basic framework (...) of our PIF account of truth-talk, we demonstrate a few advantages it offers over T-deflationist accounts that do not explicitly acknowledge pretense at work in the discourse. In turning to the Liar Paradox, we explain how the quasi-anaphoric functioning that our account attributes to truth-talk provides a diagnosis of the Liar Paradox (and other instances of semantic pathology) as having no content—in the sense of not specifying any of what we call M-conditions. At the same time, however, we vindicate the intuition that we can understand liar sentences, thereby avoiding one standard objection to “meaningless strategy” responses to the Liar Paradox. With this diagnosis in place, we then, by way of treatment, introduce a new predicate, ‘semantically defective’, and show how the explanation we give for its application allows for a consistent, yet revenge-immune, (dis)solution of the Liar Paradox, and semantic pathology generally. (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)

The Pyrrhonian sceptic Favorinus of Arelata personified indeterminacy, cultivating his (or her) borderline status to undermine dogmatism. Inspired by the techniques of Favorinus, I show, by example, that ‘vague’ has borderline cases. These concrete steps lead to a more abstract argument that ‘vague’ has borderline borderline cases and borderline borderline borderline cases. My specimens are intended supplement earlier non-constructive proofs of the vagueness of ‘vague’.

In natural language we rarely use relation-words with more than three argument places. This paper studies one systematic device, rooted in natural language, by which relations of greater adicity can be expressed. It is based on a higher-order relation between 1-place, 2-place, and 4-place relations (and so on) of which the relation between the positive and comparative degrees of a predicate is a special case. Two formal languages are presented in this connection, one of which represents the language of communication (...) and the other the contextual information against which the first language is interpreted. A semantical theory is described, which treats the two languages in an interdependent way. Logical consequence is non-compact. Connections with issues about vagueness are made. (shrink)

In this paper I present two arguments against the thesis that we experience qualia. I argue that if we experienced qualia then these qualia would have to be essentially vague entities. And I then offer two arguments, one a reworking of Gareth Evans' argument against the possibility of vague objects, the other a reworking of the Sorites argument, to show that no such essentially vague entities can exist. I consider various objections but argue that ultimately they all fail. In particular (...) I claim that the stock responses to the Sorites argument fail against my reworking of the argument because they require us to make a distinction between a determinate reality and how that reality appears to us, whereas in the case of qualia we can make no such distinction. I conclude that there can be no such things as qualia. (shrink)

It is intuitive to suppose that the question of whether I persist through a given period will always have a metaphysically substantive ‘yes’ or ‘no’ answer. Derek Parfit challenges this intuition. Given the truth of Reductionism, he argues, identity can be indeterminate. The main argument Parfit marshals in support of this claim employs his Sorites-style Combined Spectrum thought experiment. Despite its influence, there are conspicuous gaps in his argument. Notably, he claims that identity is indeterminate when questions about persistence are (...) ‘empty’. But indeterminacy in Sorites puzzles is traditionally held to result from vagueness – a topic Parfit avoids. Without an account of the relationships between vagueness, indeterminacy, and question-emptiness, his argument remains incomplete. I begin by outlining Parfit’s argument as it stands. I then propose that we can provide the missing details by supplementing Parfit’s treatment of the Combined Spectrum with Sider’s notion of nonsubstantivity. This gives us a clearer understanding of Parfit’s indeterminacy claim, and affords the room to accommodate vagueness in his argument. Once we spell out what this means, however, it becomes clear that a more radical conclusion is justified by Parfit’s reasoning: it is always an empty question whether a person persists. (shrink)

There are some properties, like being bald, for which it is vague where the boundary between the things that have it and the things that do not lies. A number of arguments threaten to show that such properties can still be associated with determinate and knowable boundaries: not between the things that have it and those that don’t, but between the things such that it is borderline at some order whether they have it and the things for which it is (...) not. I argue that these arguments, if successful, turn on a contentious principle in the logic of determinacy: Brouwer’s Principle, that every truth is determinately not determinately false. Other paradoxes which do not appear to turn on this principle often tacitly make assumptions about assertion, knowledge and higher-order vagueness. In this paper I’ll show how one can avoid sharp higher-order boundaries by rejecting these assumptions. (shrink)

Graham Solomon, to whom this collection is dedicated, went into hospital for antibiotic treatment of pneumonia in Oc- ber, 2001. Three days later, on Nov. 1, he died of a massive stroke, at the age of 44. Solomon was well liked by those who got the chance to know him—it was a revelation to?nd out, when helping to sort out his a?airs after his death, how many “friends” he had whom he had actually never met, as his email included correspondence (...) with philosophers around the world running sometimes to hundreds of messages. He was well respected in the philosophical community more broadly. He was for several years a member of the editorial board for the Western Ontario Series in Philosophy of Science. While he was employed at Wilfrid Laurier University in Waterloo, Ontario, several of us at the University of Wat- loo always regarded our own department as a sort of second academic home for him. We therefore decided that it would be appropriate to hold a memorial conference in his honour. Thanks to the generous?nancial support of the Humphrey Conference Fund, we were able to do so in May 2003. Many of the papers in this volume were presented at that conf- ence. (shrink)

In this book, I present the results of an investigation which began with an extended stay at Oxford's Balliol College during the first half of 1995. My visit to Oxford was made possible by a grant from the Spanish Ministerio de Educaci6n y Ciencia. My sincere thanks go to Joseph Raz who served as my supervisor in Oxford. For several points of the present study, conversations with Timothy Endicott in Oxford were also of great help. The book is part of (...) a larger project of investigation, directed by Albert Calsami glia, which is a joint effort of a group of legal philosophers from the Universitat Pom peu Fabra and the Universitat de Girona, and which also receives financial support from the Spanish Ministerio de Educaci6n y Ciencia. An earlier version of the manuscript was presented in June 1996 to the selection committee for a tenured professorship in Legal Philosophy at the Universitat de Girona. The members of the committee were Francisco Laporta, Albert Calsamiglia, Gregorio Peces-Barba, Camilo J. Cela Conde, and Francesca Puigpelat. I am grateful to all of them for their comments, which have been extremely useful in preparing the final ver sion of this book. (shrink)

Sorensen has argued that one can exploit the vagueness of an ordinary predicate like ‘small’ to induce a sort of vagueness in ‘vague’, by constructing a series of predicates of the form ‘n-small’, where x is n- small if and only if x is small or x n. The resulting ‘Sorensen’ed’ predicates present a Sorites case for ‘vague’ ; hence the vagueness of ‘vague’. Hyde argues that this demonstrates that all vague predicates are higher-order vague. Others doubt whether Sorensen’s series (...) really delivers such a result, claiming Hyde’s argument to be either: unsound, because it misidentifies the true source of vagueness in Sorensen’s Sorites ; invalid, because it fails to generalize to all vague predicates ; or circular, because it presupposes the very thing it tries to prove, namely, higher-order vagueness. This paper contributes to the Sorensen-Hyde vs. Tye-Deas-Hull-Varzi debate by clarifying the relations between vague vagueness and higher-order vagueness. I show how claims of higher-order vagueness are derivable from claims of vague vagueness. This is the missing piece needed to complete Hyde’s argument and overcome the objections presented by Deas, Hull, Tye, and Varzi. I then show how Sorensen’s considerations, once properly generalized, pose more far-reaching consequences about the vagueness of ‘vague’ than either defenders or critics of Hyde’s argument have appreciated. (shrink)

The naive theory of vagueness holds that the vagueness of an expression consists in its failure to draw a sharp boundary between positive and negative cases. The naive theory is contrasted with the nowadays dominant approach to vagueness, holding that the vagueness of an expression consists in its presenting borderline cases of application. The two approaches are briefly compared in their respective explanations of a paramount phenomenon of vagueness: our ignorance of any sharp boundary between positive and negative cases. These (...) explanations clearly do not provide any ground for choosing the dominant approach against the naive theory. The decisive advantage of the former over the latter is rather supposed to consist in its immunity to any form of sorites paradox. But another paramount phenomenon of vagueness is higher-order vagueness: the expressions introduced in order to express in the object language the vagueness of the object language are themselves vague. Two highly plausible claims about higher-order vagueness are articulated and defended: the existence of “definitely ω ” positive and negative cases and the “radical” character of higher-order vagueness itself. Using very weak logical principles concerning vague expressions and the ‘definitely’-operator, it is then shown that, in the presence of higher-order vagueness as just described, the dominant approach is subject to higher-order sorites paradoxes analogous to the original ones besetting the naive theory, and therefore that, against the communis opinio, it does not fare substantially better with respect to immunity to any form of sorites paradox. (shrink)

The tolerance principle, the idea that vague predicates are insensitive to sufficiently small changes, remains the main bone of contention between theories of vagueness. In this paper I examine three sources behind our ordinary belief in the tolerance principle, to establish whether any of them might give us a good reason to revise classical logic. First, I compare our understanding of tolerance in the case of precise predicates and in the case of vague predicates. While tolerance in the case of (...) precise predicates results from approximation, tolerance in the case of vague predicates appears to originate from two more specific sources: semantic indeterminacy on the one hand, and epistemic indiscriminability on the other. Both give us good and coherent grounds to revise classical logic. Epistemic indiscriminability, it is argued, may be more fundamental than semantic indeterminacy to justify the intuition that vague predicates are tolerant. (shrink)

This 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. (Open access journal).

It is often assumed that the supervaluationist theory of vagueness is committed to a global notion of logical consequence, in contrast with the local notion characteristic of modal logics. There are, at least, two problems related to the global notion of consequence. First, it brings some counterexamples to classically valid patterns of inference. Second, it is subject to an objection related to higher-order vagueness . This paper explores a third notion of logical consequence, and discusses its adequacy for the supervaluationist (...) theory. The paper proceeds in two steps. In the first step, the paper provides a deductive notion of consequence for global validity using the tableaux method. In the second step, the paper provides a notion of logical consequence which is an alternative to global validity, and discusses i) whether it is acceptable to the supervaluationist and ii) whether it plays a better role in a theory of vagueness in the face of the problems related to the global notion. (shrink)

Does vagueness exclude knowledge? After arguing for an affirmative answer to this question, I consider a fascinating objection. Barnett offers purported counterexamples to the following: Vagueness as to whether p entails that nobody knows whether p. These putative counterexamples, were they successful, would establish that standard accounts of vagueness are mistaken. I defend three central theses: First, whenever it is vague whether p competent speakers would be ambivalent about whether p when considering whether p, and such ambivalence would exclude knowledge (...) of whether p. Second, it is impossible for there to be vagueness regarding which of two polar opposite mental states obtains when only one of such states obtains. Finally, this type of impossibility constitutes evidence for dualism; i.e., the thesis that mental states are neither identical to physical states nor obtain in virtue of the obtaining of physical states. (shrink)

According to columnar higher-order vagueness, all orders of vagueness coincide: any borderline case is a borderline borderline case, and a third-order borderline case, etc. Bobzien has worked out many details of such a theory and models it with a modal logic closely related to S4. I take up a range of questions about the framework and argue that it is not suitable for modelling the structure of vagueness and higher-order vagueness.