In Unruly Words, Diana Raffman advances a new theory of vagueness which, unlike previous accounts, is genuinely semantic while preserving bivalence. According to this new approach, called the multiple range theory, vagueness consists essentially in a term's being applicable in multiple arbitrarily different, but equally competent, ways, even when contextual factors are fixed.

ABSTRACT A trivalent theory of indicative conditionals automatically enforces Stalnaker's thesis – the equation between probabilities of conditionals and conditional probabilities. This result holds because the trivalent semantics requires, for principled reasons, a modification of the ratio definition of conditional probability in order to accommodate the possibility of undefinedness. I explain how this modification is motivated and how it allows the trivalent semantics to avoid a number of well-known triviality results, in the process clarifying why these results hold for many (...) bivalent theories. In short, the slew of triviality results published in the last 40-odd years need not be viewed as an argument against Stalnaker's thesis: it can be construed instead as an argument for abandoning the bivalent requirement that indicative conditionals somehow be assigned a truth-value in worlds in which their antecedents are false. (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)

Though my title speaks of Kant’s mathematical realism, I want in this essay to explore Kant’s relation to a famous mathematical anti-realist. Specifically, I want to discuss Kant’s influence on L. E. J. Brouwer, the 20th-century Dutch mathematician who built a contemporary philosophy of mathematics on constructivist themes which were quite explicitly Kantian. Brouwer’s theory is perhaps most notable for its belief that constructivism requires us to abandon the traditional logic of mathematical reasoning in favor of different canon of reasoning, (...) called intuitionistic logic. Brouwer thought that classical logic is intrinsically bound up with a nonconstructive view of mathematics. This means that, according to Brouwer, when we do mathematics we must give up bivalence, we must no longer use such familiar logical laws as excluded middle, and we must sometimes forebear from the classic method of reductio ad absurdum. All of these are intuitionistically invalid classical principles. (shrink)

Karttunen's seminal 1973 article Presuppositions of compound sentences, lays the groundwork for the elegant and fruitful theory of this subject which he subsequently presented in (1974). In (1973, pp. 185–8), however, he fallaciously argued that the regularities he discovered concerning the behavior of and, or, and if ... then in English cannot be embodied in any three-valued logic giving a truth-functional interpretation to these connectives. The present paper refutes Karttunen's argument by exhibiting an interpretation with the desired properties, and shows (...) further how the full articulation of his 1974 system can be developed naturally within a truth-conditional semantics for English if bivalence is abandoned — contrary to what Karttunen expected. (shrink)

The term "fuzzy logic," as it is understood in this book, stands for all aspects of representing and manipulating knowledge based on the rejection of the most fundamental principle of classical logic---the principle of bivalence. According to this principle, each declarative sentence is required to be either true or false. In fuzzy logic, these classical truth values are not abandoned. However, additional, intermediate truth values between true and false are allowed, which are interpreted as degrees of truth. This opens (...) a new way of thinking---thinking in terms of degrees rather than absolutes. For example, it leads to the definition of a new kind of sets, referred to as fuzzy sets, in which membership is a matter of degree. The book examines the genesis and development of fuzzy logic. It surveys the prehistory of fuzzy logic and inspects circumstances that eventually lead to the emergence of fuzzy logic. The book explores in detail the development of propositional, predicate, and other calculi that admit degrees of truth, which are known as fuzzy logic in the narrow sense. Fuzzy logic in the broad sense, whose primary aim is to utilize degrees of truth for emulating common-sense human reasoning in natural language, is scrutinized as well. The book also examines principles for developing mathematics based on fuzzy logic and provides overviews of areas in which this has been done most effectively. It also presents a detailed survey of established and prospective applications of fuzzy logic in various areas of human affairs, and provides an assessment of the significance of fuzzy logic as a new paradigm. (shrink)

In this paper, I give a systematic account of the core features of Jürgen Habermas’s revised approach to truth that comprises both realist and epistemic components. While agents in the lifeworld are pragmatic realists and work on the basic assumption that their beliefs about the world are true, beliefs that have become problematic can be scrutinized only in the form of validity-claims in rational discourses. Thus Habermas introduces a discursive truth predicate that involves a procedural idealization of the conditions of (...) discourse in terms of ‘justified acceptability under conditions that are currently ideal’. He argues that this truth predicate has to be understood in connection with agents’ pragmatic realism to give a comprehensive account of truth. This paper is concerned with the tension between truth-in-the-lifeworld and discursive truth. I argue that Habermas’s account is troubled by two problems: firstly an unclear conception of the role of the truth predicate; and secondly, the suggestion that agents in discourse temporarily abandon their lifeworldly realism. While the first problem can be remedied by exploring the distinction between a truth predicate and a truth criterion, the second difficulty requires a modification of Habermas’s account. I show that the consistency of his approach depends on the acceptance of the principle of bivalence throughout, and hence on the subscription to a certain kind of realism that extends beyond the lifeworld. Key Words: bivalence • discourse • Habermas • lifeworld • pragmatism • realism • truth. (shrink)

Richard Robinson has argued that “no consistent and useful and desirable meaning” can be given to the philosophical terms “corrigible” and “incorrigible” so long as one espouses a bivalent theory of truth with the law of excluded middle operative. The crux of his argument is that the corrigibility-incorrigibility distinction can be shown to be redundant since, in effect, incorrigibility is materially equivalent to truth and corrigibility materially equivalent to falsehood. Robinson understands the correcting of a proposition to consist in “ (...) class='Hi'>abandoning one's belief in a false proposition and adopting its true contradictory instead. “ But given that it makes no sense to speak of correcting a true proposition, all true propositions are incapable of emendation simply by virtue of their being true, and all incorrigible propositions are by definition true. (shrink)

In this introduction to Part 1 of the Common Knowledge symposium, “Fuzzy Studies,” the journal's editor discusses four essays from the 1980s by Richard Rorty, in which Rorty chose to associate himself with various neopragmatists, Continental thinkers, and “left-wing Kuhnians” under the rubric of the “new fuzziness.” The term had been introduced as an insult by a philosopher of science with positivist leanings, but Rorty took it up as an “endearing” compliment, arguing that “to be less fuzzy” was also to (...) be “less genial, tolerant, open-minded, and fallibilist.” He defined the “new fuzziness” as “an attempt to blur just those distinctions between the objective and subjective and between fact and value which the critical conception of rationality has developed.” This introduction also examines W. V. Quine's essay “Speaking of Objects” (1957), which describes objects as fuzzy “half-entities”; Clifford Geertz's essay “Blurred Genres” (1980), which advises social scientists that being “taxonomically upstanding” is futile; and Lofti Zadeh's article “The Concept of a Linguistic Variable and Its Application to Approximate Reasoning” (1975), which abandons “Aristotelian, bivalent logic” in favor of a “fuzzy logic” based on Zadeh's “fuzzy set theory.” This introductory piece relates these theoretical works of the past half-century to the sorites paradox and to classical issues of vagueness raised and still unresolved in Western philosophy. Returning then to Rorty, the author questions how Rorty expected his endorsement of the “new fuzziness” to be applied, as proposed, to theology and politics. Suggesting that such applications are the natural work of historians, the author, having asked the historian Natalie Zemon Davis for comment, then quotes her response—which associates fuzzy studies, “common knowledge,” and peacemaking—at length. (shrink)

In this introduction to Part 1 of the Common Knowledge symposium, “Fuzzy Studies,” the journal's editor discusses four essays from the 1980s by Richard Rorty, in which Rorty chose to associate himself with various neopragmatists, Continental thinkers, and “left-wing Kuhnians” under the rubric of the “new fuzziness.” The term had been introduced as an insult by a philosopher of science with positivist leanings, but Rorty took it up as an “endearing” compliment, arguing that “to be less fuzzy” was also to (...) be “less genial, tolerant, open-minded, and fallibilist.” He defined the “new fuzziness” as “an attempt to blur just those distinctions between the objective and subjective and between fact and value which the critical conception of rationality has developed.” This introduction also examines W. V. Quine's essay “Speaking of Objects”, which describes objects as fuzzy “half-entities”; Clifford Geertz's essay “Blurred Genres”, which advises social scientists that being “taxonomically upstanding” is futile; and Lofti Zadeh's article “The Concept of a Linguistic Variable and Its Application to Approximate Reasoning”, which abandons “Aristotelian, bivalent logic” in favor of a “fuzzy logic” based on Zadeh's “fuzzy set theory.” This introductory piece relates these theoretical works of the past half-century to the sorites paradox and to classical issues of vagueness raised and still unresolved in Western philosophy. Returning then to Rorty, the author questions how Rorty expected his endorsement of the “new fuzziness” to be applied, as proposed, to theology and politics. Suggesting that such applications are the natural work of historians, the author, having asked the historian Natalie Zemon Davis for comment, then quotes her response—which associates fuzzy studies, “common knowledge,” and peacemaking—at length. (shrink)

This work presents an overview of four different approaches to the problem of future contingency and determinism in temporal logics. All of them are bivalent, viz. they share the assumption that propositions concerning future contingent facts have a determinate truth-value. We introduce Ockhamism, Peirceanism, Actualism and T x W semantics, the four most relevant bivalent alternatives in this area, and compare them from the point of view of their expressiveness and their underlying metaphysics of time.

Putative resolutions of the sorites paradox in which the major premise is declared false or illegitimate, Including max black's treatment in terms of the alleged illegitimacy of vague attributions to borderline cases, Are rejected on semantical grounds. The resort to a non-Bivalent logic of representational "accuracy" with a continuum of accuracy values is shown to resolve the paradox, And the identification of accuracy values as truth values is defended as compatible with the central insight of the correspondence theory of truth (...) and with the practical legitimacy of most applications of ordinary, Bivalent logic to statements involving vague predicates. (shrink)

This paper is an attack on the Dummett-Prawitz view that the principle of bivalence has a crucial double significance, metaphysical and meaning theoretical. On the one hand it is said that holding bivalence valid is what characterizes a realistic view, i.e. a view in metaphysics, and on the other hand it is said that there are meaning theoretical arguments against its acceptability. I argue that these two aspects are incompatible. If the failure of validity of bivalence depends (...) on properties of linguistic meaning, then there are no metaphysical consequences to be drawn. The case for this view is straightforward as long as we are discussing a language different from our own. But it seems that the distinction between failure because of meaning and failure because of reality cannot be applied to our own language, simply because our own language is just what we use to represent reality. I argue that this impression is illusory. In order to draw a conclusion about reality, meaning must be connected with truth in a non-trivial way, and precisely this cannot be done in the language for which the meaning theory itself is correct. (shrink)

Silencing of tumor suppressor genes (TSGs), by DNA methylation, is well known in adult cancers. However, based on the “stem cell” theory of tumorigenesis, the early epigenetic events arising in malignant precursors remain unknown. A recent report1 demonstrates that, while pluripotent embryonic stem cells lack DNA methylation and possess a “bivalent” pattern of activating and repressive histone marks in numerous TSGs, analogous multipotent malignant cells derived from germ cell tumors (embryonic carcinoma cells) gain additional silencing modifications to those same genes. (...) These results suggest a possible mechanism by which aberrant differentiation, mediated by histone and DNA methylation, instigates tumor progression. BioEssays 29:842–845, 2007. © 2007 Wiley Periodicals, Inc. (shrink)

Philosophers are approaching a consensus that biological individuality, including evolutionary individuality, comes in degrees. Graded evolutionary individuality presents a puzzle when juxtaposed with another widely embraced view: that evolutionary individuality follows from being a selectable member of a Darwinian population. Population membership is, on the orthodox view, a bivalent condition, so how can members of Darwinian populations vary in their degree of individuality? This article offers a solution to the puzzle, by locating difference in degree of evolutionary individuality at the (...) level of population lineages, some of which are more Darwinian than others. In doing so, it sheds light on graded individuality in overlapping and nested population lineages, such as those that arise in multilevel selection and symbiotic collectives. (shrink)

Dorothy Edgington; X*—Meaning, Bivalence and Realism, Proceedings of the Aristotelian Society, Volume 81, Issue 1, 1 June 1981, Pages 153–174, https://doi.org/1.

On standard versions of supervaluationism, truth is equated with supertruth, and does not satisfy bivalence: some truth-bearers are neither true nor false. In this paper I want to confront a well-known worry about this, recently put by Wright as follows: ‘The downside . . . rightly emphasized by Williamson . . . is the implicit surrender of the T-scheme’. I will argue that such a cost is not high: independently motivated philosophical distinctions support the surrender of the T- scheme, (...) and suggest acceptable approximations. (shrink)

Writers such as Stalnaker and Dummett have argued that specific features of subjunctive conditional statements undermine the principle of bivalence. This, paper is concerned with rebutting such claims. 1. It is shown how subjective conditionals pose a prima facie threat to bivalence, and how this threat can be dissolved by a distinction between the results of negating a subjective conditional and of negating its consequent. To make this distinction is to side with Lewis against Stalnaker in a dispute (...) about possible worlds semantics for such conditionals, and reasons are given for doing so. 2. These arguments are extended to answer Dummett''s claim that behaviourist and phenomenalist analyses in terms of subjunctive conditions violate bivalence. This answer is shown to be compatible with the principle that hypothetical statements are true only in virtue of categorical facts. (shrink)

Philosophers are approaching a consensus that biological individuality, including evolutionary individuality, comes in degrees. Graded evolutionary individuality presents a puzzle when juxtaposed with another widely embraced view: that evolutionary individuality follows from being a selectable member of a Darwinian population. Population membership is, on the orthodox view, a bivalent condition, so how can members of Darwinian populations vary in their degree of individuality? This article offers a solution to the puzzle, by locating difference in degree of evolutionary individuality at the (...) level of population lineages, some of which are more Darwinian than others. In doing so, it sheds light on graded individuality in overlapping and nested population lineages, such as those that arise in multilevel selection and symbiotic collectives. (shrink)

This thesis is concerned with the challenge truth-value gaps pose to the principle of bivalence. The central question addressed is: are truth-value gaps counterexamples to bivalence and is the supposition of counterexamples coherent? My aim is to examine putative cases of truth-value gaps against an argument by Timothy Williamson, which shows that the supposition of counterexamples to bivalence is contradictory. The upshot of his argument is that either problematic utterances say nothing, or they cannot be neither true (...) nor false. I start by identifying truth-bearers: an utterance, for instance, is a truth-bearer if it says that something is the case. Truth-bearers are evaluable items, with truth- and falsity-conditions statable in corresponding instances of schemas for truth and falsehood. A genuine case of a truth-value gap should be an utterance that is neither true nor false but says something to be the case. But it is inconsistent to accept the schemas for truth and falsehood and the existence of genuine cases of truth-value gaps. Secondly, I expound Williamson’s argument, which explores this inconsistency, and I identify two kinds of strategy to disarm his argument: those that preserve the schemas for truth and falsehood, and those that do not. Neither strategy is found to be persuasive. Thirdly, I argue that cases of reference failure causing truth-value gaps illustrate the upshot of Williamson’s argument. Fourthly, I examine Scott Soames’s account of liar sentences as counterexamples to bivalence. Soames adopts a strategy of the first kind to avoid contradictions. I argue that his solution allows some contradictions to be true, and that he fails to show that liar sentences are truth-bearers. Finally, I examine Charles Travis’s case for isostheneia: an equal balancing of reasons to evaluate a statement as true or as false, in which case a statement is neither. Travis avoids contradictions by adopting a strategy of the second kind. I argue that the schemas for truth and falsehood are immune to Travis’s objections, and that isostheneia fails to identify evaluable items. The cases examined confirm that utterances that are neither true nor false say nothing. My claim is thus that truth-value gaps are not counterexamples to bivalence. (shrink)

It is highly now intuitive that the future is open and the past is closed now—whereas it is unsettled whether there will be a fourth world war, it is settled that there was a first. Recently, it has become increasingly popular to claim that the intuitive openness of the future implies that contingent statements about the future, such as ‘There will be a sea battle tomorrow,’ are non-bivalent (neither true nor false). In this paper, we argue that the non-bivalence (...) of future contingents is at odds with our pre-theoretic intuitions about the openness of the future. These intuitions are revealed by our pragmatic judgments concerning the correctness and incorrectness of assertions of future contingents. We argue that the pragmatic data together with a plausible account of assertion shows that in many cases we take future contingents to be true (or to be false), though we take the future to be open in relevant respects. It follows that appeals to intuition to support the non-bivalence of future contingents are untenable. Intuition favours bivalence. (shrink)

In this work, we propose a variant of so-called informational semantics, a technique elaborated by Voishvillo, for two infectious logics, Deutsch’s |${\mathbf{S}_{\mathbf{fde}}}$| and Szmuc’s |$\mathbf{dS}_{\mathbf{fde}}$|⁠. We show how the machinery of informational semantics can be effectively used to analyse truth and falsity conditions of disjunction and conjunction. Using this technique, it is possible to claim that disjunction and conjunction can be rightfully regarded as such, a claim which was disputed in the recent literature. Both |${\mathbf{S}_{\mathbf{fde}}}$| and |$\mathbf{dS}_{\mathbf{fde}}$| are formalized in (...) terms of natural deduction. This allows us to solve several problems: to develop a natural deduction calculus for |${\mathbf{S}_{\mathbf{fde}}}$| containing the standard form of disjunction elimination (in contrast to the calculus by Petrukhin), to introduce the first natural deduction calculus for |$\mathbf{dS}_{\mathbf{fde}}$| and to reflect the fundamental symmetry between |${\mathbf{S}_{\mathbf{fde}}}$| and |$\mathbf{dS}_{\mathbf{fde}}$| on proof-theoretical level forming a convenient basis for obtaining their well-known extensions |$\mathbf{K}^{\mathbf{w}}_{\mathbf{3}}$| and |$\mathbf{PWK}$|⁠. (shrink)

In this paper we aim to disentangle the thesis that the future is open from theses that often get associated or even conflated with it. In particular, we argue that the open future thesis is compatible with both the unrestricted principle of bivalence and determinism with respect to the laws of nature. We also argue that whether or not the future (and indeed the past) is open has no consequences as to the existence of (past and) future ontology.

This work presents an overview of four different approaches to the problem of future contingency and determinism in temporal logics. All of them are bivalent, viz. they share the assumption that propositions concerning future contingent facts have a determinate truth-value. We introduce Ockhamism, Peirceanism, Actualism and T × W semantics, the four most relevant bivalent alternatives in this area, and compare them from the point of view of their expressiveness and their underlying metaphysics of time.

Timothy Williamson famously offered an argument from these Tarskian principles in favor of bivalence. I show, dwelling on (Andjelkovic & Williamson, 2000), that the argument depends on a contentious formulation of the Tarskian principles about truth (and falsity), which the supervaluationist can reject without jeopardizing the Tarskian insight. In the mentioned paper, Adjelkovic and Williamson argue that, even if the appropriate formulation seems to make room for failure of bivalence in borderline cases, this appearance is illusory, once one (...) grants an independent further principle involving biconditionals. I finally argue that the formulation of this is, however, contentious in a similar manner. (shrink)

In his paper ‘Some Comments on Fatalism’, The Philosophical Quartery, 46 (1996), pp. 1–11, James Cargile offers an argument against the view that the correct response to fatalism is to restrict the principle of bivalence with respect to statements about future contingencies. His argument fails because it is question‐begging. Further, he fails to give due weight to the reason behind this view, which is the desire to give an adequate account of the past/future asymmetry. He supposes that mere appeal (...) to the direction of causation will suffice to explain this asymmetry, whereas in fact the causal asymmetry is the same as the temporal asymmetry, and so cannot ground it. I finish by drawing a connection between the power asymmetry (our ability to affect the future but not the past) and the memory/intention asymmetry. (shrink)

In this paper, I focus on some intuitionistic solutions to the Paradox of Knowability. I first consider the relatively little discussed idea that, on an intuitionistic interpretation of the conditional, there is no paradox to start with. I show that this proposal only works if proofs are thought of as tokens, and suggest that anti-realists themselves have good reasons for thinking of proofs as types. In then turn to more standard intuitionistic treatments, as proposed by Timothy Williamson and, most recently, (...) Michael Dummett. Intuitionists can either point out the intuitionistc invalidity of the inference from the claim that all truths are knowable to the insane conclusion that all truths are known, or they can outright demur from asserting the existence of forever-unknown truths, perhaps questioning—as Dummett now suggests—the applicability of the Principle of Bivalence to a certain class of empirical statements. I argue that if intuitionists reject strict finitism—the view that all truths are knowable by beings just like us—the prospects for either proposal look bleak. (shrink)

Timothy Williamson, in various places, has put forward an argument that is supposed to show that denying bivalence is absurd. This paper is an examination of the logical force of this argument, which is found wanting.

Semantic justifications of the classical rules of logical inference typically make use of a notion of bivalent truth, understood as a property guaranteed to attach to a sentence or its negation regardless of the prospects for speakers to determine it as so doing. For want of a convincing alternative account of classical logic, some philosophers suspicious of such recognition-transcending bivalence have seen no choice but to declare classical deduction unwarranted and settle for a weaker system; intuitionistic logic in particular, (...) buttressed by assertion-conditional semantics, is often considered to enjoy a degree of meaning-theoretical respectability unattainable by classical logic.The decision to forgo the classical inference rules is not always made lightly. Thus, Dummett: " In the resolution of the conflict between [the view that generally accepted classical modes of inference ought to be theoretically accommodated, and the demand that any such accommodation be achieved without recourse to bivalence] lies, as I see it, one of the most fundamental and intractable problems in the theory of meaning; indeed, in all philosophy. "The present article ventures to suggest that the conflict need not be irresoluble. Helping ourselves only to such conceptual resources as are standardly invoked by proponents of intuitionistic logic, we shall formulate a rudimentary meaning theory for a selection of logical constants, define a relation of valid inferability, and prove that the latter includes all of classical first-order logic. The result is essentially that reported in Sandqvist as Theorem 2.20.2. TheoryWe will be considering a standard-syntax first-order language with ⊃, ⊥ and ∀ as its only primitive logical constants. We assume countable supplies of individual constants, individual functors , and predicates , as well as a denumerably infinite set of individual variables. By a basic 1 formula we will mean one that …[Full Text of this Article]. (shrink)

Non-bivalent languages (languages containing sentences that can be true, false or neither) are given a probabilitistic interpretation in terms of betting quotients. Necessary and sufficient conditions for avoiding Dutch books—the laws of non-bivalent probability—in such a setting are provided.

Regarding the famous Sea Battle Argument, which Aristotle presents in De Interpretatione 9, there has never been a general agreement not only about its correctness but also, and mainly, about what the argument really is. According to the most natural reading of the chapter, the argument appeals to a temporal concept of truth and concludes that not every statement is always either true or false. However, many of Aristotle’s followers and commentators have not adopted this reading. I believe that it (...) has faced so much resistance for reasons of hermeneutic charity: denying the law of universal bivalence seems to be overly disruptive to logical orthodoxy – the kind of logical orthodoxy represented by what we now call classical propositional logic, much of which Aristotle clearly supports in many texts. I intend to show that the logical-semantic theses that the traditional reading finds in De Interpretatione 9 are much more conservative than they may seem to be at first glance. First, I will show that they complement, and do not contradict in any way, the orthodox definitions of the concepts of truth and statement that Aristotle advances in other texts. Second, by resorting in an anachronistic vein to concepts and methods peculiar to contemporary logic, I will show that a trivalent modal semantics conforming to those theses can be built for a standard formal language of the classical propositional calculus. It is remarkable that reasonable concepts of logical truth and logical consequence that may be defined on the basis of this trivalent modal semantics are coextensive with their orthodox counterparts, the concepts of tautology and tautological consequence of classical bivalent and extensional semantics. (shrink)

Classical logic rests on the assumption that there are two mutually exclusive and jointly exhaustive truth values. This assumption has always been surrounded by philosophical controversy. Doubts have been raised about its legitimacy, and hence about the legitimacy of classical logic. Usually, the assumption is stated in the form of a general principle, namely the principle that every proposition is either true or false. Then, the philosophical controversy is often framed in terms of the question whether every proposition is either (...) true or false. The main purpose of the paper is to show that there is something wrong in this way of putting things. The point is that the common way of understanding the controversial assumption is misconceived, as it rests on a wrong picture of propositions. In the first part of the paper I outline this picture and I argue against it. In the second part I sketch a different picture of propositions and I suggest how this leads to conceive the issue of classical logic in different terms. (shrink)

Bivalent chromatin with active H3K4me3 and repressive H3K27me3 was initially identified in embryonic stem cells (ESCs) to poise expression of developmental genes upon lineage commitment. Since then, many more different bivalent modifications have been demonstrated in both ESCs and fully differentiated cells. Bivalency not only spatiotemporally controls gene transcription but also acts to fine‐tune the level of transcription during development. Although increasing number of studies demonstrated the functional significance of bivalent chromatin, the molecular connection of bivalent chromatin and transcriptional regulation (...) remains largely elusive. Recently, we showed Spindlin1/C11orf84 complex prefers to recognize the non‐canonical histone H3K4me3K9me3 bivalent mark, which is required for timely ribosomal RNA transcription. Here, we hypothesize the recognition of K4me3 and K9me3 at the same histone tail by Spindlin1/C11orf84 complex may serve as a general mechanism of conversion from a repressed to an active chromatin structure for transcriptional activation. (shrink)

The general fact of the impossibility of a bivalent, truth-functional semantics for the propositional structures determined by quantum mechanics should be more subtly demarcated according to whether the structures are taken to be orthomodular latticesP L or partial-Boolean algebrasP A; according to whether the semantic mappings are required to be truth-functional or truth-functional ; and according to whether two-or-higher dimensional Hilbert spaceP structures or three-or-higher dimensional Hilbert spaceP structures are being considered. If the quantumP structures are taken to be orthomodular (...) latticesP L, then bivalent mappings which preserve the operations and relations of aP L must be truth-functional . Then as suggested by von Neumann and Jauch-Piron and as proven in this paper, the mere presence of incompatible elements in aP L is sufficient to rule out any semantical or hidden-variable proposal which imposes this strong condition, for anytwo-or-higher dimensional Hilbert spaceP L structure. Thus from the orthomodular lattice perspective, the peculiarly non-classical feature of quantum mechanics and the peculiarly non-Boolean feature of the quantum propositional structures is the existence of incompatible magnitudes and propositions. However, the weaker truth-functionality condition can instead be imposed upon the semantic or hidden-variable mappings on theP L structures, although such mappings ignore the lattice meets and joins of incompatibles and preserve only the partial-Boolean algebra structural features of theP L structures. Or alternatively, the quantum propositional structures can be taken to be partial-Boolean algebrasP A, where bivalent mappings which preserve the operations and relations of aP A need only be truth-functional (c). In either case, the Gleason, Kochen-Specker proofs show that any semantical or hidden variable proposal which imposes this truth-functionality (c) condition is impossible for anythree-or-higher dimensional Hilbert spaceP A orP L structures. But such semantical or hidden-variable proposals are possible for any two dimensional Hilbert spaceP A orP L structures, in spite of the presence of incompatibles in these structures, in spite of the fact that Heisenberg's Uncertainty Principle applies to the incompatible elements in these structures, and in spite of the fact that these structures are non-Boolean in the Piron sense. (shrink)

It is generally agreed that vague predicates like ‘red’, ‘rich’, ‘tall’, and ‘bald’, have borderline cases of application. For instance, a cloth patch whose color lies midway between a definite red and a definite orange is a borderline case for ‘red’, and an American man five feet eleven inches in height is (arguably) a borderline case for ‘tall’. The proper analysis of borderline cases is a matter of dispute, but most theorists of vagueness agree at least in the thought that (...) borderline cases for vague predicate ‘ ’ are items whose satisfaction of ‘ ’ is in some sense unclear or problematic: it is unclear whether or not the patch is red, unclear whether or not the man is tall.1 For example, Lynda Burns cites a widespread view as holding that borderline cases “are not definitely within the positive or negative extension of the predicate. … Border- line cases are seen as falling within a gap between the cases of definite application of the predicate and cases of definite application of its negation” (1995, 30). Michael Tye writes that the “concept of a border- line case is the concept of a case that is neither definitely in nor defi- nitely out” (1994b, 18). (shrink)

Name der Zeitschrift: Archiv für Geschichte der Philosophie Jahrgang: 98 Heft: 3 Seiten: 239-271.

Timothy Williamson, in various places, has put forward an argument that is supposed to show that denying bivalence is absurd. This paper is an examination of the logical force of this argument, which is found wanting.

In this paper it is argued that the buck-passing analysis (BPA) of final value is not a plausible analysis of value and should be abandoned. While considering the influential wrong kind of reason problem and other more recent technical objections, this paper contends that there are broader reasons for giving up on buck-passing. It is argued that the BPA, even if it can respond to the various technical objections, is not an attractive analysis of final value. It is not attractive (...) for two reasons: the first being that the BPA lacks the features typical of successful conceptual analyses and the second being that it is unable to deliver on the advantages that its proponents claim for it. While not offering a knock-down technical refutation of the BPA, this paper aims to show that there is little reason to think that the BPA is correct, and that it should therefore be given up as an analysis of final value. (shrink)