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)

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.

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)

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)

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)

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)

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)

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)

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)

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 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 investigate the family LS0.5 of many-valued modal logics LS0.5's. We prove that the modalities of necessity and possibility of the logics LS0.5's capture well-defined bivalent concepts of logical validity and logical consistency. We also show that these modalities can be used as recovery operators.

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.

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)

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)

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)

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)

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.

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.

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)

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)

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)

I consider two related objections to the claim that the law of excluded middle does not imply bivalence. One objection claims that the truth predicate captured by supervaluation semantics is not properly motivated. The second objection says that even if it is, LEM still implies bivalence. I show that LEM does not imply bivalence in a supervaluational language. I also argue that considering supertruth as truth can be reasonably motivated.

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)

In several of his philosophical works, Cicero gives reports of the Epicurean views on bivalence and the excluded middle that are not always consistent. I attempt to establish a coherent account that fits the texts as well as possible and can reasonably be attributed to the Epicureans. I argue that they distinguish between a semantic and a syntactic version of the law of the excluded middle, and that whilst they reject bivalence and the semantic law for fear of (...) certain fatalistic consequences, they endorse the syntactic law. Subsequently, I show that certain principles that they seem to endorse in the context of Cicero’s discussion of Chrysippus’ argument for fate, when modified by the addition of the atomic swerve, suggest that the Epicureans have in mind something like a supervaluationist model of truth at times, which has the desired results with respect to the three logical principles. (shrink)

Evaluative processes have their roots in early evolutionary history, as survival is dependent on an organism’s ability to identify and respond appropriately to positive, rewarding or otherwise salubrious stimuli as well as to negative, noxious, or injurious stimuli. Consequently, evaluative processes are ubiquitous in the animal kingdom and are represented at multiple levels of the nervous system, including the lowest levels of the neuraxis. While evolution has sculpted higher level evaluative systems into complex and sophisticated information-processing networks, they do not (...) come to replace, but rather to interact with more primitive lower level representations. Indeed, there are basic features of the underlying neuroarchitectural plan for evaluative processes that are common across levels of organization—including that of evaluative bivalence. (shrink)

The passage 18a34-⁠b5 of Aristotle’s famous sea-battle chapter has often been misunderstood. My aim is to show, firstly, that Aristotle in this passage attempts to prove that the unrestricted validity of the Principle of Bivalence entails, in the case of singular statements, the validity of the Principle of Truth-value Distribution for the contradictory pairs they are members of. According to the latter principle either the affirmative member of a contradictory pair of statements must be true and the negative false (...) or vice versa. Secondly, I want to show what consequences the correct understanding of the passage in question has for the understanding of the introductory passage of the chapter (18a28-⁠33) and for the dispute over whether Aristotle exempts singular statements about contingent future events from the domain of the Principle of Bivalence. The thesis, advanced by some modern interpreters, that Aristotle refrains from doing so even though he exempts the contradictory pairs such statements are members of from the domain of the Principle of Truth-value Distribution will be rebutted as resulting from a fallacious line of reasoning. (shrink)

The principle of bivalence (PB) states that every declarative sentence is either true or false, and the principle of excluded middle (PEM) states that one member of any contradictory pair must be true. According to the standard interpretation of Int. 9, PB fails for future contingents. Moreover, some standardists believe that PEM fails for pairs of contradictory future contingents, whereas other standardists attempt to rescue PEM by applying the method of supervaluations. I argue that PB and PEM are not (...) suspended and that the method of supervaluations fails to explain how PEM holds. I propose an interpretation of Int. 9 according to which neither PB nor PEM fails. (shrink)

B. J. Copeland; Discussions: Vagueness and Bivalence: A Discussion of Williamson and Simons, Proceedings of the Aristotelian Society, Volume 95, Issue 1, 1 June.

My purpose in this paper is to argue that the classical notion of entailment is not suitable for non-bivalent logics, to propose an appropriate alternative and to suggest a generalized entailment notion suitable to bivalent and non-bivalent logics alike. In classical two valued logic, one can not infer a false statement from one that is not false, any more than one can infer from a true statement a statement that is not true. In classical logic in fact preserving truth and (...) preserving non-falsity are one and the same thing. They are not the same in non-bivalent logics however and I will argue that the classical notion of entailment that preserves only truth is not strong enough for such a logic. I will show that if we retain the classical notion of entailment in a logic that has three values, true, false and a third value in between, an inconsistency can be derived that can be resolved only by measures that seriously disable the logic. I will show this for a logic designed to allow for semantic presuppositions, then I will show that we get the same result in any three valued logic with the same value ordering. I will finally suggest how the notion of entailment should be generalized so that this problem may be avoided. The strengthened notion of entailment I am proposing is a conservative extension of the classical notion that preserves not only truth but the order of all values in a logic, so that the value of an entailed statement must alway be at least as great as the value of the sequence of statements entailing it. A notion of entailment this strong or stronger will, I believe, be found to be applicable to non-classical logics generally. In the opinion of Dana Scott, no really workable three valued logic has yet been developed. It is hard to disagree with this. A workable three valued logic however could perhaps be developed however, if we had a notion of entailment suitable to non-bivalent logics. (shrink)

In this paper, I present a formal reconstruction of the classical argument for fatalism set forth by Aristotle in On Interpretation 9. From there, I expose two different formal solutions for avoiding the unwanted conclusion based on the traditional interpretation of Aristotle’s rejection of the Principle of Bivalence: On the one hand, Łukasiewicz's three-valued logic and, on the other hand, supervaluation semantics. I also address some criticisms made against these two proposals. To finish, I remark on some alternative interpretations (...) of Aristotle’s intentions maintaining that the Stagirite philosopher rejected fatalism without abandoning the Principle of Bivalence. (shrink)

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)

A three dimensional hypercube representing all of the 4,096 dyadic computations in a standard bivalent system has been created. It has been constructed from the 16 functions arrayed in a table of functional completeness that can compute a dyadic relationship. Each component of the dyad is an operator as well as a function, such as “implication” being a result, as well as an operation. Every function in the hypercube has been color keyed to enhance the display of emerging patterns. At (...) the minimum, the hypercube is a “multiplication table” of dyadic computations and produces values in a way that shortens the time to do operations that normally would take longer using conventional truth table methods. It also can serve as a theorem prover and creator. With the hypercube comes a deductive system without the need for axioms. The main significance of the 3-D hypercube at this point is that it is the most fundamental way of displaying all dyadic computations in binary space, thus serving as a way of normalizing the rendition of uninterpreted, or raw, binary space. The hypercube is a dimensionless entity, a standard by which in binary spaces can be measured and classified, analogous to a meter stick. (shrink)

This chapter contains sections titled: 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9.

This paper outlines an approach to the principle of bivalence based on very general, but still elementary, semantic considerations. The principle of bivalence states that “every sentence is either true or false”. Clearly, some logics are bivalent while others are not. A more general formulation of uses the concept of designated and non-designated logical values and is captured by “every sentence is either designated or non-designated”. Yet this formulation seems trivial, because the concept of non-designated value is negative. (...) In order to refine the analysis, the class of anti-designated values has been distinguished. The non-trivial version of the principle of bivalence is expressed by “every sentence is either designated or anti-designated”. The last part of the paper mentions some extralogical reasons for considering the principle of bivalence with truth being a designated value as intimately connected to human thinking and behavior. (shrink)

Scott Soames proposes in his book Understanding Truth (1999) a motivation to reject bivalence. It is his claim that if bivalence is assumed to apply to liar sentences, contradictions will follow. However, contradictions will equally follow if bivalence is denied of liar sentences (in fact, of any truth-bearers). Soames avoids contradictions by treating truth as a partially defined predicate: for certain sentences, truth is not defined to apply or not to apply. Liar sentences are some of such (...) cases. However, there are several problems with Soames’s proposal. Liar sentences also pose a problem for the disquotational schema (T). Soames only avoids this problem by changing the meaning of the biconditional in schema (T), and this will be shown to be an unsatisfactory move. Furthermore, we should only regard liar sentences as possible counterexamples to bivalence provided they are regarded as proper truth-bearers. Soames takes truth-bearers to be sentences that express propositions. However, it is questionable that liar sentences, such as the example considered, can count as proper truth-bearers. Soames imposes on truth a condition I will call the priority requirement: that the status of the claim that some sentence S is true (or not true) depends on the prior status of S itself, hence on the status of sentences not containing the truth-predicate. Soames regards the cases where the truth-predicate cannot be eliminated as cases of truth-value gaps. I will argue that the priority requirement means that liar sentences fail to express propositions: for liar sentences, there is no prior sentence S on whose truth or falsity the liar case would turn. Nothing is said constraining a possible way for things to be, so nothing said to call true or false. Sentences that do not say anything are neither true nor false without contradiction. (shrink)

In De int. 9 Aristotle argues that some declarative sentences are neither true nor false. This raises the problem of how we should understand the words of ch. 4, which introduces the declarative sentence as ‘that in which being true or being false holds’. In this paper I remove the contradiction by arguing that in ch. 4 Aristotle does not intend to claim that *all* declarative sentences are either true or false, but rather that *only* they are either true or (...) false, unlike other kinds of sentence; I defend the soundness of this interpretation on the linguistic and textual level; and I show how we can make good philosophical sense of it. (shrink)