Sorites Paradox

This paper responds to Theron Pummer’s distinction between Sorites arguments and repugnant conclusion arguments by presenting a Sorites overpopulation argument. Also I present a Sorites argument in favour of myths of genius.

Neutralism is the broad view that philosophical progress can take place when (and sometimes only when) a thoroughly neutral, non-specific theory, treatment, or methodology is adopted. The broad goal here is to articulate a distinct, specific kind of sorites paradox (The Observational Sorites Paradox) and show that it can be effectively treated via Neutralism.

Contemporary discussions do not always clearly distinguish two different forms of vagueness. Sometimes focus is on the imprecision of predicates, and sometimes the indefiniteness of statements. The two are intimately related, of course. A predicate is imprecise if there are instances to which it neither definitely applies nor definitely does not apply, instances of which it is neither definitely true nor definitely false. However, indefinite statements will occur in everyday discourse only if speakers in fact apply imprecise predicates to such (...) indefinite instances. (What makes an instance indefinite is, it should be clear, predicate-relative.) The basic issue in the present inquiry is whether this indefiniteness ever really occurs; the basic question is, Why should it ever occur? (shrink)

Fine (2017) proposes a new logic of vagueness, CL, that promises to provide both a solution to the sorites paradox and a way to avoid the impossibility result from Fine (2008). The present paper presents a challenge to his new theory of vagueness. I argue that the possibility theorem stated in Fine (2017), as well as his solution to the sorites paradox, fail in certain reasonable extensions of the language of CL. More specifically, I show that if we extend the (...) language with any negation operator that obeys reductio ad absurdum, we can prove a new impossibility result that makes the kind of indeterminacy that Fine takes to be a hallmark of vagueness impossible. I show that such negation operators can be conservatively added to CL and examine some of the philosophical consequences of this result. Moreover, I demonstrate that we can define a particular negation operator that behaves exactly like intuitionistic negation in a natural and unobjectionable propositionally quantified extension of CL. Since intuitionistic negation obeys reductio, the new impossibility result holds in this propositionally quantified extension of CL. In addition, the sorites paradox resurfaces for the new negation. (shrink)

When you and I seriously argue over whether a man of seventy is old enough to count as an "old man", it seems that we are appealing neither to our own separate standards of oldness nor to a common standard that is already fixed in the language. Instead, it seems that both of us implicitly invoke an ideal, shared standard that has yet to be agreed upon: the place where we ought to draw the line. As with other normative standards, (...) it is hard to know whether such borderlines exist prior to our coming to agree on where they are. But epistemicists plausibly argue that they must exist whether we ever agree on them or not, as this provides the only logically acceptable response to the sorites paradox. This paper argues that such boundaries do typically exist as hypothetical ideals, but not as determinate features of the present actual world. There is in fact no general solution to the paradox, but attention to practice in resolving vague disagreements shows that its instances can be dealt with separately, as they arise, in many reasonable ways. (shrink)

What the Sorites has to tell us is a simple truth regarding our categories. It appears to saddle us with something other than a simple truth—something worse, a contradiction or a problem or a paradox—only when we insist on viewing it through a discrete logic of categories. Discrete categories and discrete logic are for robots. We aren’t robots, and the simple truth is that we don’t handle categories in the way any discrete logic would demand. For us non-robots, what the (...) Sorites has to offer is a straightforward truth regarding how incapable robots and their logic are of handling categories like ours. (shrink)

According to a view called 'nihilism', sentences containing vague expressions cannot strictly speaking be true or false, because they lack definite truth conditions. While most theorists of vagueness tend to regard nihilism as a hopeless view, a few isolated attempts have been made to defend it. This paper aims to develop such attempts in a new direction by showing how nihilism, once properly spelled out, can meet three crucial explanatory challenges that respectively concern truth, assertibility, and communication.

In his essay ‘“Wang’s Paradox”’, Crispin Wright proposes a solution to the Sorites Paradox (in particular, the form of it he calls the ‘Paradox of Sharp Boundaries’) that involves adopting intuitionistic logic when reasoning with vague predicates. He does not give a semantic theory which accounts for the validity of intuitionistic logic (and the invalidity of stronger logics) in that area. The present essay tentatively makes good the deficiency. By applying a theorem of Tarski, it shows that intuitionistic logic is (...) the strongest logic that may be applied, given certain semantic assumptions about vague predicates. The essay ends with an inconclusive discussion of whether those semantic assumptions should be accepted. (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)

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)

Noniterative approaches to the sorites paradox accept single steps of soritical reasoning, but deny that these can be combined into valid chains of soritical reasoning. The distributed sorites is a puzzle designed to undermine noniterative approaches to the sorites paradox, by deriving an inconsistent conclusion using only single steps, but not chains, of soritical reasoning. This paper shows how a dialetheist version of the noniterative approach, the strict-tolerant approach, also solves the distributed sorites paradox, at no further cost, by accepting (...) the inconsistent conclusion. (shrink)

In his paper ‘The possibility of vagueness’ (Fine in Synthese 194(10):3699–3725, 2017), Kit Fine proposes a new logic of vagueness, CL, that promises to provide both a solution to the sorites paradox and a way to avoid the impossibility result from Fine (Philos Perspect 22(1):111–136, 2008). The present paper presents a challenge to his new theory of vagueness. I argue that the possibility theorem stated in Fine (Synthese 194(10):3699–3725, 2017), as well as his solution to the sorites paradox, fail in (...) certain reasonable extensions of the language of CL. More specifically, I show that if we extend the language with any negation operator that obeys reductio ad absurdum, we can prove a new impossibility result that makes the kind of indeterminacy that Fine takes to be a hallmark of vagueness impossible. I show that such negation operators can be conservatively added to CL and examine some of the philosophical consequences of this result. Moreover, I demonstrate that we can define a particular negation operator that behaves exactly like intuitionistic negation in a natural and unobjectionable propositionally quantified extension of CL. Since intuitionistic negation obeys reductio, the new impossibility result holds in this propositionally quantified extension of CL. In addition, the sorites paradox resurfaces for the new negation. (shrink)

”It is often assumed that knowledge claims must be justified. But what kind of justification is required for knowledge ? . . . ” (*) -/- presupposition: the kind of epistemic justification depends on the type of the knowledge claim and its respective knowledge claim tradeoff ’vague vs. precise’. -/- procedere: in two - almost purely logical - case studies I account for this tradeoff and question in each case what (if any) were its general outcome wrt justification -/- first (...) for basic measurement statements of the form ′′ϕ(x) = r ∈ IR ± δ′′ (wrt ”measurement accuracy realism” debate) -/- and secondly predication statements of the form ′′x is ( a case of )P′′ (wrt ”epistemic vagueness” debate). (shrink)

The Consequence Argument is the classic argument for the incompatibility of determinism and our ability to do otherwise. Daniel C. Dennett objects that the Consequence Argument suffers from the same error as a clearly unconvincing argument that there are no mammals. In this paper, I argue that these arguments do not suffer from the same error. The argument that there are no mammals is unconvincing as it takes the form of a sorites, whereas the Consequence Argument does not. Accordingly, Dennett's (...) objection misses its mark. (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)

A review of Kit Fine's book, Vagueness: A Global Approach.

By pooling together exhaustive analyses of certain philosophical paradoxes, we can prove a series of fascinating results regarding philosophical progress, agreement on substantive philosophical claims, knockdown arguments in philosophy, the wisdom of philosophical belief, the epistemic status of metaphysics, and the power of philosophy to refute common sense. As examples, this Element examines the Sorites Paradox, the Liar Paradox, and the Problem of the Many – although many other paradoxes can do the trick too.

I show that the Lottery Paradox is just a version of the Sorites, and argue that this should modify our way of looking at the Paradox itself. In particular, I focus on what I call “the Cut-off Point Problem” and contend that this problem, well known by Sorites scholars, ought to play a key role in the debate on Kyburg’s puzzle. Very briefly, I show that, in the Lottery Paradox, the premises “ticket n°1 will lose”, “ticket n°2 will lose”… “ticket (...) n°1000 will lose” are equivalent to soritical premises of the form “~(the winning ticket is in {…, (tn)}) ⊃ ~(the winning ticket is in {…, tn, (tn + 1)})” (where “⊃” is the material conditional, “~” is the negation symbol, “tn” and “tn + 1” are “ticket n°n” and “ticket n°n + 1” respectively, and “{}” identify the elements of the lottery tickets’ set. The brackets in “(tn)” and “(tn + 1)” are meant to point out that in the antecedent of the conditional we do not always have a “tn” (and, as a result, a “tn + 1” in the consequent): consider the conditional “~(the winning ticket is in {}) ⊃ ~(the winning ticket is in {t1})”). As a result, failing to believe, for some ticket, that it will lose comes down to introducing a cut-off point in a chain of soritical premises. In this paper I explore the consequences of the different ways of blocking the Lottery Paradox with respect to the Cut-off Point Problem. A heap variant of the Lottery Paradox is especially relevant for evaluating the different solutions. One important result is that the most popular way out of the puzzle, i.e., denying the Lockean Thesis, becomes less attractive. Moreover, I show that, along with the debate on whether rational belief is closed under classical logic, the debate on the validity of modus ponens should play an important role in discussions on the Lottery Paradox. -/- . (shrink)

Ethics in the tradition of Derek Parfit’s Reasons and Persons is riddled with sorites-like arguments, which lead us by what seem innocent steps to seemingly false conclusions. Take, for example, spectrum arguments for the Repugnant Conclusion that appeal to slight differences in quality of life. Several authors have taken the view that, since spectrum arguments are structurally analogous to sorites arguments, the correct response to spectrum arguments is structurally analogous to the correct response to sorites arguments. This sorites analogy is (...) here argued against. There are potential structural disanalogies between spectrum arguments and sorites arguments. But even if these arguments are relevantly structurally analogous, they differ in their content in ways that show the sorites analogy to be implausible. Two content-based disanalogies are here explored—one is inspired by Parfit’s work on reductionism, and the other involves hypersensitivity. The chapter concludes with a methodological lesson. (shrink)

A constant of Frege’s writing is his rejection of indeterminate predicates as found in natural language. This paper follows Frege’s remarks on vagueness from the early "Begriffsschrift” to his mature works, drawing brief parallels with the main contemporary theories of vagueness. I critically examine Frege’s arguments for the inconsistency of natural language and argue that the inability to accommodate vagueness in his mature ontology is mainly due to heuristic rules of thumb which Frege took as essential, not to a deep (...) problem in his fundamental apparatus. (shrink)

Is there an interesting relation between the Preface paradox and the Sorites paradox that might be used to illuminate either or both of those paradoxes and the phenomena of rationality and vagueness with which they, respectively, are bound up? In particular, if we consider the analogy alongside a familiar response to the Preface Paradox that employs degrees of belief, does this give any support to the thought that we should adopt some kind of degree-theoretic treatment of vagueness and the sorites? (...) This chapter argues that it does not; indeed exploring the disanalogies contributes to a case against such a treatment of vagueness more generally. Among other views, it considers Edgington’s account of vagueness that employs a probabilistic structure of “verities”. It then contends that appeal to the framework of supervaluationism yields a better guide to reasoning in vague language than the degree-theoretic treatment can sustain. (shrink)

Graham Priest has influentially claimed that the Sorites paradox is an Inclosure paradox, concluding that his favored dialetheic solution to the Inclosure paradoxes should be extended to the Sorites paradox. We argue that, given Priest’s dialetheic solution to the Sorites paradox, the argument purporting to show that that paradox is an Inclosure is unsound, and discuss some issues surrounding this fact.

Essentialists suppose that for every individual, if that individual exists at any possible world, then necessarily that individual exemplifies some non-trivial qualitative property essential to it, as such. Anti-essentialists deny this. One important argument leveled by some anti-essentialists against essentialism takes the form of a thought experiment, one originally introduced by Roderick Chisholm, sometimes referred to as Chisholm's Paradox (CP). In this essay, I defend essentialism against CP. I begin by presenting the argument and showing how it leads to a (...) contradiction of the essentialist thesis. I then consider one of the most popular solutions to CP to date, that given by Nathan Salmon. Next, I critique Salmon's proposal and show that it is an insufficient response on behalf of the essentialist. And finally, I propose a novel solution to the paradox and discuss why it is that many metaphysician in the past have found CP plausible, despite being fallacious. (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)

The PhD thesis advances a new approach to vagueness as dispersion, comparing it with the main philosophical theories of vagueness in the analytic tradition.

O fenômeno da vagueza é quase onipresente na linguagem natural. Por um lado, a vagueza abrange variadas categorias lógicas, incluindo predicados, termos singulares e quantificadores. Por outro, é argumentável que a maioria das expressões em cada uma destas categorias, principalmente termos singulares e predicados, é vaga. Isto não seria um problema, não tivesse o fenômeno da vagueza relacionado ao paradoxo sorites, que supostamente mostra a incoerência destas expressões. O Supervalorativismo fornece uma explicação do fenômeno, acompanhado de uma solução do paradoxo, (...) que é interessante por pelo menos duas razões. Primeiro, é argumentável que esta teoria é a que melhor acomoda certa imagem intuitiva da vagueza, conhecida como indecisão semântica. Segundo, ela é relativamente conservadora com respeito à lógica clássica. Ao contrário de algumas rivais, como a Teoria Trivalente e o Gradualismo, o Supervalorativismo nos permite manter tanto o terceiro excluído como a não contradição. A despeito de suas aparentes vantagens, no entanto, a teoria carrega alguns sérios problemas. Com foco na categoria dos predicados vagos, meu objetivo neste artigo é mostrar que o Supervalorativismo falha em satisfazer pelo menos um de três critérios de adequação para uma teoria ideal da vagueza: o critério da precisão. O artigo é dividido em quatro seções. Na primeira, apresento o paradoxo sorites e três critérios de adequação para uma teoria ideal da vagueza: o critério do sorites, o critério da coerência e o critério da precisão. Na segunda e terceira, apresento o Supervalorativismo e sua solução para o paradoxo sorites. Por fim, argumento que, mesmo que a teoria satisfaça os dois primeiros critérios de adequação, ela certamente não satisfaz o terceiro. (shrink)

The first part of the chapter surveys some of the main ways in which the Sorites Paradox has figured in arguments in practical philosophy in recent decades, with special attention to arguments where the paradox is used as a basis for criticism. Not coincidentally, the relevant arguments all involve the transitivity of value in some way. The second part of the chapter is more probative, focusing on two main themes. First, I further address the relationship between the Sorites Paradox and (...) the main arguments discussed in the first part, by elucidating in what sense they rely on (something like) tolerance principles. Second, I briefly discuss the prospect of rejecting the respective principles, aiming to show that we can do so for some of the arguments but not for others. The reason is that in the latter cases the principles do not function as independent premises in the reasoning but, rather, follow from certain fundamental features of the relevant scenarios. I also argue that not even adopting what is arguably the most radical way to block the Sorites Paradox – that of weakening the consequence relation – suffices to invalidate these arguments. (shrink)

On some accounts of vagueness, predicates like “is a heap” are tolerant. That is, their correct application tolerates sufficiently small changes in the objects to which they are applied. Of course, such views face the sorites paradox, and various solutions have been proposed. One proposed solution involves banning repeated appeals to tolerance, while affirming tolerance in any individual case. In effect, this solution rejects the reasoning of the sorites argument. This paper discusses a thorny problem afflicting this approach to vagueness. (...) In particular, it is shown that, on the foregoing view, whether an object is a heap will sometimes depend on factors extrinsic to that object, such as whether its components came from other heaps. More generally, the paper raises the issue of how to count heaps in a tolerance-friendly framework. (shrink)

This article argues that resolutions to the sorites paradox offered by epistemic and supervaluation theories fail to adequately account for vagueness. After explaining the paradox, I examine the epistemic theory defended by Timothy Williamson and discuss objections to his semantic argument for vague terms having precise boundaries. I then consider Rosanna Keefe's supervaluationist approach and explain why it fails to accommodate the problem of higher-order vagueness. I conclude by discussing how fuzzy logic may hold the key to resolving the sorites (...) paradox without positing indefensible borders to the correct application of vague terms. (shrink)

Larry Temkin famously argues that what he calls spectrum arguments yield strong reason to reject Transitivity, according to which the ‘all-things-considered better than’ relation is transitive. Spectrum arguments do reveal that the conjunctions of independently plausible claims are inconsistent with Transitivity. But I argue that there is very strong independent reason to reject such conjunctions of claims, and thus that the fact that they are inconsistent with Transitivity does not yield strong reason to reject Transitivity.

Ben-Yami presents Wittgenstein’s explicit criticism of the Platonic identification of an explanation with a definition and the alternative forms of explanation he employed. He then discusses a few predecessors of Wittgenstein’s criticisms and the Fregean background against which he wrote. Next, the idea of family resemblance is introduced, and objections answered. Wittgenstein’s endorsement of vagueness and the indeterminacy of sense are presented, as well as the open texture of concepts. Common misunderstandings are addressed along the way. Wittgenstein’s ideas, as is (...) then shown, have far-reaching implications for knowledge of meaning and the nature of logic, and with them to the nature of the philosophical project and its possible achievements. (shrink)

What kind of semantics should someone who accepts the epistemicist theory of vagueness defended in Timothy Williamson’s Vagueness (1994) give a definiteness 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 find to be more plausible than the other. (shrink)

In Kripke’s classic paper on truth it is argued that by adding a new semantic category different from truth and falsity it is possible to have a language with its own truth predicate. A substantial problem with this approach is that it lacks the expressive resources to characterize those sentences which fall under the new category. The main goal of this paper is to offer a refinement of Kripke’s approach in which this difficulty does not arise. We tackle this characterization (...) problem by letting certain sentences belong to more than one semantic category. We also consider the prospect of generalizing this framework to deal with languages containing vague predicates. (shrink)

Agent relativists about vagueness (henceforth ‘agent relativists’) hold that whether or not an object x falls in the extension of a vague predicate ‘P’ at a time t depends on the judgemental dispositions of a particular competent agent at t. My aim in this paper is to critically examine arguments that purport to support agent relativism by appealing to data from forced-march Sorites experiments. The most simple and direct versions of such forced-march Sorites arguments rest on the following (implicit) premise: (...) If competent speakers' judgements vary in a certain way, then the extensions of ‘P’ as used by these speakers must vary in the same way. This premise is in need of independent support, since otherwise opponents of agent relativism can simply reject it. In this paper, I focus on the idea that one cannot plausibly reject this premise, as that would commit one to implausible claims about linguistic competence. Against this, I argue that one can accommodate the data from forced-march Sorites experiments in a way that is compatible with a plausible picture of linguistic competence, without going agent relativist. Thus, there is reason to be sceptical of the idea that such data paired with considerations about linguistic competence can be invoked in order to lend any solid support to agent relativism. Forced-march Sorites arguments of this kind can, and should be, resisted. (shrink)

Both the traditional Aristotelian and modern symbolic approaches to logic have seen logic in terms of discrete symbol processing. Yet there are several kinds of argument whose validity depends on some topological notion of continuous variation, which is not well captured by discrete symbols. Examples include extrapolation and slippery slope arguments, sorites, fuzzy logic, and those involving closeness of possible worlds. It is argued that the natural first attempts to analyze these notions and explain their relation to reasoning fail, so (...) that ignorance of their nature is profound. (shrink)

A logic is called 'paraconsistent' if it rejects the rule called 'ex contradictione quodlibet', according to which any conclusion follows from inconsistent premises. While logicians have proposed many technically developed paraconsistent logical systems and contemporary philosophers like Graham Priest have advanced the view that some contradictions can be true, and advocated a paraconsistent logic to deal with them, until recent times these systems have been little understood by philosophers. This book presents a comprehensive overview on paraconsistent logical systems to change (...) this situation. The book includes almost every major author currently working in the field. The papers are on the cutting edge of the literature some of which discuss current debates and others present important new ideas. The editors have avoided papers about technical details of paraconsistent logic, but instead concentrated upon works that discuss more 'big picture' ideas. Different treatments of paradoxes takes centre stage in many of the papers, but also there are several papers on how to interpret paraconistent logic and some on how it can be applied to philosophy of mathematics, the philosophy of language, and metaphysics. (shrink)

While the past century of neuroscientific research has brought considerable progress in defining the boundaries of the human cerebral cortex, there are cases in which the demarcation of one area from another remains fuzzy. Despite the existence of clearly demarcated areas, examples of gradual transitions between areas are known since early cytoarchitectonic studies. Since multi-modal anatomical approaches and functional connectivity studies brought renewed attention to the topic, a better understanding of the theoretical and methodological implications of fuzzy boundaries in brain (...) science can be conceptually useful. This article provides a preliminary conceptual framework to understand this problem by applying philosophical theories of vagueness to three levels of neuroanatomical research. For the first two levels (cytoarchitectonics and fMRI studies), vagueness will be distinguished from other forms of uncertainty, such as imprecise measurement or ambiguous causal sources of activation. The article proceeds to discuss the implications of these levels for the anatomical study of connectivity between cortical areas. There, vagueness gets imported into connectivity studies since the network structure is dependent on the parcellation scheme and thresholds have to be used to delineate functional boundaries. Functional connectivity may introduce an additional form of vagueness, as it is an organizational principle of the brain. The article concludes by discussing what steps are appropriate to define areal boundaries more precisely. (shrink)

In this paper I offer a counterexample to the so called vagueness argument against restricted composition. This will be done in the lines of a recent suggestion by Trenton Merricks, namely by challenging the claim that there cannot be a sharp cut-off point in a composition sequence. It will be suggested that causal powers which emerge when composition occurs can serve as an indicator of such sharp cut-off points. The main example will be the case of a heap. It seems (...) that heaps might provide a very plausible counterexample to the vagueness argument if we accept the idea that four grains of sand is the least number required to compose a heap—the case has been supported by W. D. Hart. My purpose here is not to put forward a new theory of composition, I only wish to refute the vagueness argument and point out that we should be wary of arguments of its form. (shrink)

The dissertation has two parts, each dealing with a problem, namely: 1) What is the most adequate account of fuzziness -the so-called phenomenon of vagueness?, and 2) what is the most plausible solution to the sorites, or heap paradox? I will try to show that fuzzy properties are those which are gradual, amenable to be possessed in a greater or smaller extent. Acknowledgement of degrees in the instantiation of a property allows for a gradual transition from one opposite to the (...) other, each intermediate stage constituting an overlap in certain proportion of both contraries. Hence, degrees in the possession of a property give rise to simple contradictions. The reason why I have chosen those two questions is that they provide the main philosophical motivation for a particular brand of an infinite valued and paraconsistent logic. I will claim that Classical logic (CL) is not adequate to handle fuzzy situations, and, being deficient, is in need of being expanded to make room for degrees of truth and weak contradictions. One can hardly deny the importance of the debate, since what is ultimately at stake is what the limits of truth, rationality, intelligibility and possibility are. The main disciplines within which the research moves are the philosophy of language, philosophy of logic, and ontology. (shrink)

Doris Olin's Paradox is a very helpful book for those who want to be introduced to the philosophical treatment of paradoxes, or for those who already have knowledge of the general area and would like to have a helpful resource book.

Contrary to the great bulk of philosophical work on vagueness, the core of vagueness is not to be found in vague monadic predicates such as ‘bald’, ‘tall’, or ‘old’. The true source of vagueness – at least vagueness of the type that typically appears in the sorites – lies beneath these, in a mechanism using a buried quantifier operative over the comparatives ‘balder’, ‘taller’ and ‘older’.

[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)

In Vagueness and Contradiction (2001), Roy Sorensen defends and extends his epistemic account of vagueness. In the process, he appeals to connections between vagueness and semantic paradox. These appeals come mainly in Chapter 11, where Sorensen offers a solution to what he calls the no-no paradox—a “neglected cousin” of the more famous liar—and attempts to use this solution as a precedent for an epistemic account of the sorites paradox. This strategy is problematic for Sorensen’s project, however, since, as we establish, (...) he fails to resolve the semantic pathology of the no-no paradox. (shrink)

A sorites argument is a symptom of the vagueness of the predicate with which it is constructed. A vague predicate admits of at least one dimension of variation (and typically more than one) in its intended range along which we are at a loss when to say the predicate ceases to apply, though we start out confident that it does. It is this feature of them that the sorites arguments exploit. Exactly how is part of the subject of this paper. (...) The majority of philosophers writing on vagueness take it to be a kind of semantic phenomenon. If we are right, they are correct in this assumption, which is surely the default position, but they have not so far provided a satisfactory account of the implications of this or a satisfactory diagnosis of the sorites arguments. Other philosophers have urged more exotic responses, which range from the view that the fault lies not in our language, but in the world, which they propose to be populated with vague objects which our semantics precisely reflects, to the view that the world and language are both perfectly in order, but that the fault lies with our knowledge of the properties of the words we use (epistemicism). In contrast to the exotica to which some philosophers have found themselves driven in an attempt to respond to the sorites puzzles, we undertake a defense of the commonsense view that vague terms are semantically vague. Our strategy is to take fresh look at the phenomenon of vagueness. Rather than attempting to adjudicate between different extant theories, we begin with certain pre-theoretic intuitions about vague terms, and a default position on classical logic. The aim is to see whether (i) a natural story can be told which will explain the vagueness phenomenon and the puzzling nature of soritical arguments, and, in the course of this, to see whether (ii) there arises any compelling pressure to give up the natural stance. We conclude that there is a simple and natural story to be told, and we tell it, and that there is no good reason to abandon our intuitively compelling starting point. The importance of the strategy lies in its dialectical structure. Not all positions on vagueness are on a par. Some are so incredible that even their defenders think of them as positions of last resort, positions to which we must be driven by the power of philosophical argument. We aim to show that there is no pressure to adopt these incredible positions, obviating the need to respond to them directly. If we are right, semantic vagueness is neither surprising, nor threatening. It provides no reason to suppose that the logic of natural languages is not classical or to give up any independently plausible principle of bivalence. Properly understood, it provides us with a satisfying diagnosis of the sorites argumentation. It would be rash to claim to have any completely novel view about a topic so well worked as vagueness. But we believe that the subject, though ancient, still retains its power to inform and challenge us. In particular, we will argue that taking seriously the central phenomenon of predicate vagueness—the “boundarylessness” of vague predicates—on the commonsense assumption that vagueness is semantic, leads ineluctably to the view that no sentences containing vague expressions (henceforth ‘vague sentences’) are truth-evaluable. This runs counter to much of the literature on vagueness, which commonly assumes that, though some applications of vague predicates to objects fail to be truth-evaluable, in clear positive and negative cases vague sentences are unproblematically true or false. It is clarity on this, and related points, that removes the puzzles associated with vagueness, and helps us to a satisfying diagnosis of why the sorites arguments both seem compelling and yet so obviously a bit of trickery. We give a proof that semantically vague predicates neither apply nor fail-to-apply to anything, and that consequently it is a mistake to diagnose sorites arguments, as is commonly done, by attempting to locate in them a false premise. Sorites arguments are not sound, but not unsound either. We offer an explanation of their appeal, and defend our position against a variety of worries that might arise about it. The plan of the paper is as follows. We first introduce an important distinction in terms of which we characterize what has gone wrong with vague predicates. We characterize what we believe to be our natural starting point in thinking about the phenomenon of vagueness, from which only a powerful argument should move us, and then trace out the consequences of accepting this starting point. We consider the charge that among the consequences of semantic vagueness are that we must give up classical logic and the principle of bivalence, which has figured prominently in arguments for epistemicism. We argue there are no such consequences of our view: neither the view that the logic of natural languages is classical, nor any plausible principle of bivalence, need be given up. Next, we offer a diagnosis of what has gone wrong in sorites arguments on the basis of our account. We then present an argument to show that our account must be accepted on pain of embracing (in one way or another) the epistemic view of “vagueness”, i.e., of denying that there are any semantically vague terms at all. Next, we discuss some worries that may arise about the intelligibility of our linguistic practices if our account is correct. We argue none of these worries should force us from our intuitive starting point. Finally, we cast a quick glance at other forms of semantic incompleteness. (shrink)

هو أول كتاب باللغة العربية يعرض لمراحل وآليات تطور المنطق الرمزي المعاصر متعدد القيم بأنساقه المختلفة، مركزًا على مشكلة الغموض المعرفي للإنسان بأبعادها اللغوية والإبستمولوجية والأنطولوجية، والتي تتجلى – على سبيل المثال – فيما تحفل به الدراسات الفلسفية والمنطقية والعلمية من مفارقات تمثل تحديًا قويًا لثنائية الصدق والكذب الكلاسيكية، وكذلك في اكتشاف «هيزنبرج» لمبدأ اللايقين، وتأكيده وعلماء الكمّ على ضرورة التفسيرات الإحصائية في المجال دون الذري، الأمر الذي يؤكد عدم فعالية قانون الثالث المرفوع في التعامل مع معطيات الواقع الفعلي، واستحالة (...) مشروع إقامة لغة مثالية أو اصطناعية أو كاملة منطقيًا تتجاوز عيوب ونقائص اللغة العادية التي نفكر ونتعامل بها. وينتهي الكتاب إلى نتيجة مؤداها أن ما واجهه المنطق الكلاسيكي – ثنائي القيم – من مشكلات أدت إلى تطويره، لاسيما مشكلة الغموض، لابد وأن يواجهه بالمثل المنطق متعدد القيم، ذلك أن الغموض ظاهرة إبستمولوجية في المحل الأول، مردودها إلى الذات العارفة وقصور إمكاناتها الإدراكية والقياسية، لا إلى الوجود ذاته. وأننا حتى لو سمحنا لأية قضية منطقية بقيمة صدق ثالثة، أو بأكثر من قيمة تتوسط بين الصدق التام والكذب التام، فسوف تظل القضية – كتمثيل لغوي لإحدى وقائع العالم – صادقة أو كاذبة، سواء أدركنا ذلك أو لم ندركه، وذلك تأكيد لنزعة أفلاطون الواقعية القائلة بوجود أزلي وثابت للحقائق في عالم المثل، تحول دون معرفتنا الظنية بظلالها في عالم الحس المتغير. ويعني ذلك بعبارة أخرى أن الشك في قانون الثالث المرفوع هو إسقاط من الذات على الموضوع، مبعثه عدم اكتمال العملية المعرفية ومحدوديتها، وأن ظهور الأنساق المنطقية ذات القيم المتعددة ما هو إلا حلقة من حلقات العلاقة الجدلية اللامنتهية بين الإنسان والطبيعة، أو بيم ما هو مُدرك وما هو موجود بالفعل. والكتاب بصفة عامة هو أول كتاب باللغة العربية يعرض لأنساق المنطق متعدد القيم بعد مرحلة البرينكيبيا ماثيماتيكا. (shrink)

This article provides an intuitive semantic account of a new logic for comparisons (CL), in which atomic statements are assigned both a classical truth-value and a “how much” value or extension in the range [0, 1]. The truth-value of each comparison is determined by the extensions of its component sentences; the truth-value of each atomic depends on whether its extension matches a separate standard for its predicate; everything else is computed classically. CL is less radical than Casari’s comparative logics, in (...) that it does not allow for the formation of comparative statements out of truth-functional molecules. It is argued that CL provides a better analysis of predicate vagueness than classical logic, fuzzy logic or supervaluation theory. (shrink)