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)

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)

Since Peirce defined the first operators for three-valued logic, it is usually assumed that he rejected the principle of bivalence. However, I argue that, because bivalence is a principle, the strategy used by Peirce to defend logical principles can be used to defend bivalence. Construing logic as the study of substitutions of equivalent representations, Peirce showed that some patterns of substitution get realized in the very act of questioning them. While I recognize that we can devise non-classical (...) notations, I argue that, when we make claims about those notations, we inevitably get saddled with bivalent commitments. I present several simple inferences to show this. The argument that results from those examples is ‘pragmatic’, because the inevitability of the principle is revealed in use (not mention); and it is ‘semiotic’, because this revelation happens in the use of signs. (shrink)

Fregean theories of descriptions as terms have to deal with improper descriptions. To save bivalence various proposals have been made that involve assigning referents to improper descriptions. While bivalence is indeed saved, there is a price to be paid. Instantiations of the same general scheme, viz. the one and only individual that is F and G is G, are not only allowed but even required to have different truth values.

The aim of this paper is, firstly to explain Crispin Wright’s quandary view of vagueness, his intuitionistic response to sorites and the relation of borderline cases and bivalence, and, secondly assess the objections to his ideas.

In this fragment of Opuscula Logica it is displayed an arithmetical treatment of the aristotelic syllogisms upon the previous interpretations of Christine Ladd-Franklin and Jean Piaget. For the first time, the whole deductive corpus for each syllogism is presented in the two innovative modalities first proposed by Hugo Padilla Chacón. A. The Projection method (all the possible expressions that can be deduced through the conditional from a logical expression) and B. The Retrojection method (all the possible valid antecedents or premises (...) conjunction for an expression proposed as a conclusion). The results are numerically expressed, with their equivalents in the propositional language of bivalent logic. (shrink)

There is a familiar debate between Russell and Strawson concerning bivalence and ‘the present King of France’. According to the Strawsonian view, ‘The present King of France is bald’ is neither true nor false, whereas, on the Russellian view, that proposition is simply false. In this paper, I develop what I take to be a crucial connection between this debate and a different domain where bivalence has been at stake: future contingents. On the familiar ‘Aristotelian’ view, future contingent (...) propositions are neither true nor false. However, I argue that, just as there is a Russellian alternative to the Strawsonian view concerning ‘the present King of France’, according to which the relevant class of propositions all turn out false, so there is a Russellian alternative to the Aristotelian view, according to which future contingents all turn out false, not neither true nor false. The result: contrary to millennia of philosophical tradition, we can be open futurists without denying bivalence. (shrink)

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

Roman Suszko said that “Obviously, any multiplication of logical values is a mad idea and, in fact, Łukasiewicz did not actualize it.” The aim of the present paper is to qualify this ‘obvious’ statement through a number of logical and philosophical writings by Professor Jan Woleński, all focusing on the nature of truth-values and their multiple uses in philosophy. It results in a reconstruction of such an abstract object, doing justice to what Suszko held a ‘mad’ project within a generalized (...) logic of judgments. Four main issues raised by Woleński will be considered to test the insightfulness of such generalized truth-values, namely: the principle of bivalence, the logic of scepticism, the coherence theory of truth, and nothingness. (shrink)

In De interpretatione 9, Aristotle argues against the fatalist view that if statements about future contingent singular events (e.g. ‘There will be a sea battle tomorrow,’ ‘There will not be a sea battle tomorrow’) are already true or false, then the events to which those statements refer will necessarily occur or necessarily not occur. Scholars have generally held that, to refute this argument, Aristotle allows that future contingent statements are exempt from either the principle of bivalence, or the law (...) of excluded middle. This paper offers a new interpretation of Aristotle’s refutation of fatalism. According to this interpretation, each member of a pair of contradictory future contingent statements, in virtue of expressing modal necessity, is simply false. (shrink)

The aim of this paper is to argue that the adoption of an unrestricted principle of bivalence is compatible with a metaphysics that (i) denies that the future is real, (ii) adopts nomological indeterminism, and (iii) exploits a branching structure to provide a semantics for future contingent claims. To this end, we elaborate what we call Flow Fragmentalism, a view inspired by Kit Fine (2005)’s non-standard tense realism, according to which reality is divided up into maximally coherent collections of (...) tensed facts. In this way, we show how to reconcile a genuinely A-theoretic branching-time model with the idea that there is a branch corresponding to the thin red line, that is, the branch that will turn out to be the actual future history of the world. (shrink)

This paper deals with the problem of future contingents, and focuses on two classical logical principles, excluded middle and bivalence. One may think that different attitudes are to be adopted towards these two principles in order to solve the problem. According to what seems to be a widely held hypothesis, excluded middle must be accepted while bivalence must be rejected. The paper goes against that line of thought. In the first place, it shows how the rejection of (...) class='Hi'>bivalence leads to implausible consequences if excluded middle is accepted. In the second place, it addresses the question of why one should reject bivalence, and finds no satisfactory answer. /// Este artículo trata el problema de los futuros contingentes, y se enfoca en dos principios lógicos clásicos: el tercero excluido y la bivalencia. Se podría pensar que una solución del problema requiere actitudes diferentes hacia estos dos principios. Según una hipótesis que parece ampliamente compartida, el tercero excluido debe ser aceptado, mientras que la bivalencia debe ser rechazada. Este artículo argumenta en contra de esta línea de pensamiento. En primer lugar, se aborda cómo el rechazo de la bivalencia lleva a consecuencias poco plausibles si el tercero excluido es aceptado. En segundo lugar, se enfrenta la cuestión de por qué se debería rechazar la bivalencia, sin encontrar una respuesta satisfactoria. (shrink)

A general framework for translating various logical systems is presented, including a set of partial unary operators of affirmation and negation. Despite its usual reading, affirmation is not redundant in any domain of values and whenever it does not behave like a full mapping. After depicting the process of partial functions, a number of logics are translated through a variety of affirmations and a unique pair of negations. This relies upon two preconditions: a deconstruction of truth-values as ordered and structured (...) objects, unlike its mainstream presentation as a simple object; a redefinition of the Principle of Bivalence as a set of four independent properties, such that its definition does not equate with normality. (shrink)

I offer a new theory of faultless disagreement, according to which truth is absolute (non-relative) but can still be non-objective. What's relative is truth-aptness: a sentence like ‘Vegemite is tasty’ (V) can be truth-accessible and bivalent in one context but not in another. Within a context in which V fails to be bivalent, we can affirm that there is no issue of truth or falsity about V, still disputants, affirming and denying V, were not at fault, since, in their context (...) of assertion V was bivalent. This theory requires a theory of assertion that is a form of cognitive expressivism. (shrink)

The incompleteness of set theory ZF C leads one to look for natural nonconservative extensions of ZF C in which one can prove statements independent of ZF C which appear to be “true”. One approach has been to add large cardinal axioms.Or, one can investigate second-order expansions like Kelley-Morse class theory, KM or Tarski-Grothendieck set theory T G or It is a nonconservative extension of ZF C and is obtained from other axiomatic set theories by the inclusion of Tarski’s axiom (...) which implies the existence of inaccessible cardinals. See also related set theory with a filter quantifier ZF (aa). In this paper we look at a set theory NC# ∞# , based on bivalent gyper infinitary logic with restricted Modus Ponens Rule In this paper we deal with set theory NC# ∞# based on bivalent gyper infinitary logic with Restricted Modus Ponens Rule. Nonconservative extensions of the canonical internal set theories IST and HST are proposed. (shrink)

The present contribution might be regarded as a kind of defense of the common sense in logic. It is demonstrated that if the classical negation is interpreted as the minimal negation with n = 2 truth values, then deviant logics can be conceived as extension of the classical bivalent frame. Such classical apprehension of negation is possible in non- classical logics as well, if truth value is internalized and bivalence is replaced by bipartition.

Is logic empirical? Is logic to be found in the world? Or is logic rather a convention, a product of conventions, part of the many rules that regulate the language game? Answers fall in either camp. We like the linguistic answer. In this paper, we want to analyze how a linguistic community would tackle the problem of developing a logic and show how the linguistic conventions adopted by the community determine the properties of the local logic. Then show how to (...) move from a notion of logic that varies from community to community to a notion of logic that is in a sense universal. The framework is conventional up to a point: we have sentences, atomic and composite, the connectives are interpreted, values are computed, and the value of a composite sentence is a function of the values of its subsentences. Less conventional is the use of a plurality of truth values, and the sharp distinction we draw between sentences and statements, in the spirit of the distinction between proposition and judgment that one may find in proof theory. The linguistic community will face many choices. What are the good ones, the ones to avoid? Are there, in some sense, optimal choices? These are the kind of issues we are addressing. Where do we end up? With some kind of universal bivalent logic, ironically enough. We start from an arbitrarily large number of truth values, atomic sentences and connectives, construct a generic many-valued logic, recover more or less the usual results and issues, and in the end it all comes down to a positive bivalent logic with two connectives, `and' and `or', as if logic is nothing more than a mere accounting of possibilities. (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)

Todd (2016) proposes an analysis of future-directed sentences, in particular sentences of the form 'will(φ)', that is based on the classic Russellian analysis of definite descriptions. Todd's analysis is supposed to vindicate the claim that the future is metaphysically open while retaining a simple Ockhamist semantics of future contingents and the principles of classical logic, i.e. bivalence and the law of excluded middle. Consequently, an open futurist can straightforwardly retain classical logic without appeal to supervaluations, determinacy operators, or any (...) further controversial semantical or metaphysical complication. In this paper, we will show that this quasi-Russellian analysis of 'will' both lacks linguistic motivation and faces a variety of significant problems. In particular, we show that the standard arguments for Russell's treatment of definite descriptions fail to apply to statements of the form 'will(φ)'. (shrink)

It is a widely held principle that no one is able to do something that would require the past to have been different from how it actually is. This principle of the fixity of the past has been presented in numerous ways, playing a crucial role in arguments for logical and theological fatalism, and for the incompatibility of causal determinism and the ability to do otherwise. I will argue that, assuming bivalence, this principle is in conflict with standard views (...) about knowledge and the semantics for ‘actually’. I also consider many possible responses to the argument. (shrink)

I propose an approach to liar and Curry paradoxes inspired by the work of Roger Swyneshed in his treatise on insolubles (1330-1335). The keystone of the account is the idea that liar sentences and their ilk are false (and only false) and that the so-called ''capture'' direction of the T-schema should be restricted. The proposed account retains what I take to be the attractive features of Swyneshed's approach without leading to some worrying consequences Swyneshed accepts. The approach and the resulting (...) logic (called ''Swynish Logic'') are non-classical, but are consistent and compatible with many elements of the classical picture including modus ponens, modus tollens, and double-negation elimination and introduction. It is also compatible with bivalence and contravalence. My approach to these paradoxes is also immune to an important kind of revenge challenge that plagues some of its rivals. (shrink)

There is a long-standing disagreement among Branching-Time theorists. Even though they all believe that the branching representation accurately grasps the idea that the future, contrary to the past, is open, they argue whether this representation is compatible with the claim that one among many possible futures is distinguished—the single future that will come to be. This disagreement is paralleled in an argument about the bivalence of future contingents. The single, privileged future is often called the Thin Red Line. I (...) reconstruct the history of the arguments for and against this idea. Then, I propose my own version of the Thin Red Line theory which is immune to the major objections found in the literature. I argue that the semantic disagreement is grounded in distinct metaphysical presuppositions. My solution is expressed in a conceptual framework proposed by John MacFarlane, who distinguishes semantics from postsemantics. I extend his distinction and introduce a new notion of presemantics to elucidate my idea. (shrink)

In this paper we consider the theory of predicate logics in which the principle of Bivalence or the principle of Non-Contradiction or both fail. Such logics are partial or paraconsistent or both. We consider sequent calculi for these logics and prove Model Existence. For L4, the most general logic under consideration, we also prove a version of the Craig-Lyndon Interpolation Theorem. The paper shows that many techniques used for classical predicate logic generalise to partial and paraconsistent logics once the (...) right set-up is chosen. Our logic L4 has a semantics that also underlies Belnap’s [4] and is related to the logic of bilattices. L4 is in focus most of the time, but it is also shown how results obtained for L4 can be transferred to several variants. (shrink)

There are three main traditional accounts of vagueness : one takes it as a genuinely metaphysical phenomenon, one takes it as a phenomenon of ignorance, and one takes it as a linguistic or conceptual phenomenon. In this paper I first very briefly present these views, especially the epistemicist and supervaluationist strategies, and shortly point to some well-known problems that the views carry. I then examine a 'statistical epistemicist' account of vagueness that is designed to avoid precisely these problems – it (...) will be a view that provides an account of the phenomenon of vagueness as coming from our linguistic practices, while insisting that meaning supervenes on use, and that our use of vague terms does yield sharp and precise meanings, which we ignore, thus allowing bivalence to hold. (shrink)

What the world needs now is another theory of vagueness. Not because the old theories are useless. Quite the contrary, the old theories provide many of the materials we need to construct the truest theory of vagueness ever seen. The theory shall be similar in motivation to supervaluationism, but more akin to many-valued theories in conceptualisation. What I take from the many-valued theories is the idea that some sentences can be truer than others. But I say very different things to (...) the ordering over sentences this relation generates. I say it is not a linear ordering, so it cannot be represented by the real numbers. I also argue that since there is higher-order vagueness, any mapping between sentences and mathematical objects is bound to be inappropriate. This is no cause for regret; we can say all we want to say by using the comparative truer than without mapping it onto some mathematical objects. From supervaluationism I take the idea that we can keep classical logic without keeping the familiar bivalent semantics for classical logic. But my preservation of classical logic is more comprehensive than is normally permitted by supervaluationism, for I preserve classical inference rules as well as classical sequents. And I do this without relying on the concept of acceptable precisifications as an unexplained explainer. The world does not need another guide to varieties of theories of vagueness, especially since Timothy Williamson (1994) and Rosanna Keefe (2000) have already provided quite good guides. I assume throughout familiarity with popular theories of vagueness. (shrink)

This article tries to present a possible way of proliferating bivalent, trivalent, etc. calculation. Beyond the controversies as to the effects that involve the presence versus the absence of the temporal factor, no significant attempts of adapting a bivalent logic to other pairs of values have been mentioned. Should the transformation of the alethiologic modalities into logic values be possible, five more bivalent calculations may be added to the classic bivalent calculation. All these new calculations are compatible with the traditional (...) one, to the extent that they submit to the conditions of non-contradictoriness and exhaustibility, and the connectors receive, even if partially, the corresponding value interpretations. (shrink)

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

ABSTRACT: This chapter offers a revenge-free solution to the liar paradox (at the centre of which is the notion of Gestalt shift) and presents a formal representation of truth in, or for, a natural language like English, which proposes to show both why -- and how -- truth is coherent and how it appears to be incoherent, while preserving classical logic and most principles that some philosophers have taken to be central to the concept of truth and our use of (...) that notion. The chapter argues that, by using a truth operator rather than truth predicate, it is possible to provide a coherent, model-theoretic representation of truth with various desirable features. After investigating what features of liar sentences are responsible for their paradoxicality, the chapter identifies the logic as the normal modal logic KT4M (= S4M). Drawing on the structure of KT4M (=S4M), the author proposes that, pace deflationism, truth has content, that the content of truth is bivalence, and that the notions of both truth and bivalence are semideterminable. (shrink)

In matters of personal taste, faultless disagreement occurs between people who disagree over what is tasty, fun, etc., in those cases when each of these people seems equally far from the objective truth. Faultless disagreement is often taken as evidence that truth is relative. This article aims to help us avoid the truth-relativist conclusion. The article, however, does not argue directly against relativism; instead, the article defends non-relative truth constructively, aiming to explain faultless disagreement with the resources of semantic contextualism. (...) To this end the article describes and advocates a contextualist solution inspired by supervaluationist truth-value gap approaches. The solution presented here, however, does not require truth value gaps; it preserves both logical bivalence and non-relative truth, even while it acknowledges and explains the possibility of faultless disagreement. The solution is motivated by the correlation between assertions’ being true and their being useful. This correlation, furthermore, is used not only to tell which assertions are true, but also to determine which linguistic intuitions are reliable. (shrink)

T In [Rea 2006], Michael Rea presents an argument that presentism is incompatible with a libertarian view of human freedom and the unrestricted principle of bivalence. I aim to show that Rea’s argument fails. The outline of my paper is as follows. In Part I, I briefly explain the above three views and I present Rea’sargument. In Part II, I argue that one of the premises of the argument is unjustified.

One well known approach to the soritical paradoxes is epistemicism, the view that propositions involving vague notions have definite truth values, though it is impossible in principle to know what they are. Recently, Paul Horwich has extended this approach to the liar paradox, arguing that the liar proposition has a truth value, though it is impossible to know which one it is. The main virtue of the epistemicist approach is that it need not reject classical logic, and in particular the (...) unrestricted acceptance of the principle of bivalence and law of excluded middle. Regardless of its success in solving the soritical paradoxes, the epistemicist approach faces a number of independent objections when it is applied to the liar paradox. I argue that the approach does not offer a satisfying, stable response to the paradoxes—not in general, and not for a minimalist about truth like Horwich. (shrink)

This paper considers logics which are formally dual to intuitionistic logic in order to investigate a co-constructive logic for proofs and refutations. This is philosophically motivated by a set of problems regarding the nature of constructive truth, and its relation to falsity. It is well known both that intuitionism can not deal constructively with negative information, and that defining falsity by means of intuitionistic negation leads, under widely-held assumptions, to a justification of bivalence. For example, we do not want (...) to equate falsity with the non-existence of a proof since this would render a statement such as “pi is transcendental” false prior to 1882. In addition, the intuitionist account of negation as shorthand for the derivation of absurdity is inadequate, particularly outside of purely mathematical contexts. To deal with these issues, I investigate the dual of intuitionistic logic, co-intuitionistic logic, as a logic of refutation, alongside intuitionistic logic of proofs. Direct proof and refutation are dual to each other, and are constructive, whilst there also exist syntactic, weak, negations within both logics. In this respect, the logic of refutation is weakly paraconsistent in the sense that it allows for statements for which, neither they, nor their negation, are refuted. I provide a proof theory for the co-constructive logic, a formal dualizing map between the logics, and a Kripke-style semantics. This is given an intuitive philosophical rendering in a re-interpretation of Kolmogorov’s logic of problems. (shrink)

FDE, LP and K3 are closely related to each other and admit of an intuitive informational interpretation. However, all these logics are co-NP complete, and so idealized models of how an agent can think. We address this issue by shifting to signed formulae, where the signs express imprecise values associated with two bipartitions of the corresponding set of standard values. We present proof systems whose operational rules are all linear and have only two structural branching rules that express a generalized (...) Principle of Bivalence. Each of these systems leads to defining an infinite hierarchy of tractable approximations to the respective logic, in terms of the maximum number of allowed nested applications of the two branching rules. Further, each resulting hierarchy admits of an intuitive 5-valued non-deterministic semantics. (shrink)

There is no interesting entailment either way between theism and various forms of realism. Taking its cue from Dummett’s characterisation of realism and his discussion of it with respect to theistic belief, this paper argues both that theism does not follow from realism, and that God cannot be appealed to in order to secure bivalence for an otherwise indeterminate subject matter. In both cases, significant appeal is made to the position that God is not a language user, which in (...) turn is motivated by an account of understanding as aptitude possession. The resulting picture sits comfortably with the apophatism common within living religious traditions and with the view that the philosophy of religion ought to reorientate itself away from metaphysics towards more practical questions. (shrink)

At the heart of the Stoic theory of modality is a strict commitment to bivalence, even for future contingents. A commitment to both future truth and contingency has often been thought paradoxical. This paper argues that the Stoic retreat from necessity is successful. it maintains that the Stoics recognized three distinct senses of necessity and possibility: logical, metaphysical and providential. Logical necessity consists of truths that are knowable a priori. Metaphysical necessity consists of truths that are knowable a posteriori, (...) a world order according to certain metaphysical principles and natures that god crafts within the constraints of matter. Finally, what is providentially necessary is what occurs according to the chain of fate, but only once it is in process or past. -/- The method of the paper is a close reading of Diogenes Laertius 7.75, adducing broad textual evidence along the way, to show that the Stoic theory of modality embraces Philonian possibility, both that which is capable of being true as a matter of logical consistency, and that which is possible according to the bare fitness of the entity. What differentiates the Stoics from Philo is their additional commitment to possibility as opportunity, resisting the collapse of determinism into necessity. (shrink)

In this paper we critically review Correia’s and Rosenkranz’s Nothing to Come. A Defence of the Growing Block Theory of Time, published by Springer in 2018. By taking into account the essential reliance of the book on tense logic, we bring out the existence of a conflict between their logical axioms, that presuppose truth bivalence even for statements concerning future contingents, and the principle of groundedness that they also advocate. According to this principle, a proposition Q is now groundedly (...) true as long as sometimes in the future it will be the case that there exists something that makes Q true. However, if the events that occur in 2060 do not exist unrestrictedly (for us, here in 2021), what, in reality, can possibly ground the truth (or falsity) of Q? In the second part of the paper, we concentrate on some conceptual difficulties raised by their brilliant attempt to adapt the growing block view of reality to a relativistic setting, based on what they call bow-tie presentism, according to which for each spacetime point s only s plus the spacelike-related region with respect to s exists. By extending temporal logic into a relativity-friendly spatiotemporal logic, we conclude by noting that Correia and Rosenkranz’s Nothing to Come makes very important strides in showing that there are technical ways to develop the growing block of reality into an ontology that coheres with relativity theories perfectly well, while retaining most of the distinctive content of classical growing block theory. (shrink)

This paper investigates Ockhamism from a metaphysical point of view. Its main point is that the claim that future contingents are true or false is less demanding than usually expected, as it does not require particularly contentious assumptions about the future. First it will be argued that Ockhamism is consistent with a wide range of metaphysical views. Then it will be shown that each of these views leaves room for the claim that the future is open, at least on some (...) plausible interpretations of that claim. (shrink)

This paper explores what the epistemic account of vagueness means for theories of legal interpretation. The thesis of epistemicism is that vague statements are true or false even though it is impossible to know which. I argue that if epistemicism is accepted within the domain of the law, then the following three conditions must be satisfied: Interpretative reasoning within the law must adhere to the principle of bivalence and the law of excluded middle, interpretative reasoning within the law must (...) construe vague statements as an epistemic phenomenon, and epistemicism must be expanded to include normative considerations in order to account for legal theories that are consistent with the first two conditions. The first two conditions are internal to a particular theory of legal interpretation, while the third condition is external to a particular theory of legal interpretation. My conclusion shows that there are legal theories that are internally consistent with the fundamental features of epistemicism. However, within the domain of law—and specifically in the case of legal theories that are internally consistent with epistemicism—I show that vagueness cannot be explained simply by our ignorance of the meaning and use of vague expressions. Rather, epistemicism must also account for ignorance of the requisite normative considerations in legal theories with which it is otherwise consistent. (shrink)

Various philosophers have long since been attracted to the doctrine that future contingent propositions systematically fail to be true—what is sometimes called the doctrine of the open future. However, open futurists have always struggled to articulate how their view interacts with standard principles of classical logic—most notably, with the Law of Excluded Middle. For consider the following two claims: Trump will be impeached tomorrow; Trump will not be impeached tomorrow. According to the kind of open futurist at issue, both of (...) these claims may well fail to be true. According to many, however, the disjunction of these claims can be represented as p ∨ ~p—that is, as an instance of LEM. In this essay, however, I wish to defend the view that the disjunction these claims cannot be represented as an instance of p ∨ ~p. And this is for the following reason: the latter claim is not, in fact, the strict negation of the former. More particularly, there is an important semantic distinction between the strict negation of the first claim [~] and the latter claim. However, the viability of this approach has been denied by Thomason, and more recently by MacFarlane and Cariani and Santorio, the latter of whom call the denial of the given semantic distinction “scopelessness”. According to these authors, that is, will is “scopeless” with respect to negation; whereas there is perhaps a syntactic distinction between ‘~Will p’ and ‘Will ~p’, there is no corresponding semantic distinction. And if this is so, the approach in question fails. In this paper, then, I criticize the claim that will is “scopeless” with respect to negation. I argue that will is a so-called “neg-raising” predicate—and that, in this light, we can see that the requisite scope distinctions aren’t missing, but are simply being masked. The result: a under-appreciated solution to the problem of future contingents that sees and as contraries, not contradictories. (shrink)

Note: The author holds the copyright, and there was no agreement, express or implied, not to use a facsimile PDF. -/- Using erotetic logic, the paper defines the "the whole truth" in a manner consistent with U.S. Supreme Court precedent. It cannot mean "the whole story," as witnesses in an adversary system are permitted /only/ to answer the questions put to them, nor are they permitted to speculate, add irrelevant material, etc. Nor can it mean not to add an admixture (...) of falsity, as that is already included in "nothing but the truth," and, strictly speaking, in "the truth," as any such as admixture renders the whole thing false (given bivalence is presupposed) by &-introduction. (shrink)

The incompleteness of set theory ZFC leads one to look for natural extensions of ZFC in which one can prove statements independent of ZFC which appear to be "true". One approach has been to add large cardinal axioms. Or, one can investigate second-order expansions like Kelley-Morse class theory, KM or Tarski- Grothendieck set theory TG [1]-[3] It is a non-conservative extension of ZFC and is obtaineed from other axiomatic set theories by the inclusion of Tarski's axiom which implies the existence (...) of inaccessible cardinals [1].Non-conservative extension of ZFC based on an generalized quantifiers considered in [4]. In this paper we look at a set theory NC_{∞^{#}}^{#},based on bivalent gyper infinitary logic with restricted Modus Ponens Rule [5]-[8]. In this paper we deal with set theory NC_{∞^{#}}^{#} based on gyper infinitary logic with Restricted Modus Ponens Rule. Set theory NC_{∞^{}}^{} contains Aczel's anti-foundation axiom [9]. We present a new approach to the invariant subspace problem for Hilbert spaces. Our main result will be that: if T is a bounded linear operator on an infinite-dimensional complex separable Hilbert space H,it follow that T has a non-trivial closed invariant subspace. Non-conservative extension based on set theory NC_{∞}^{#} of the model theoretical nonstandard analysis [10]-[12] also is considered. (shrink)

Anaxagoras’ theory of the nous constitutes one aspect of his philosophy particularly interesting for Aristotle. However, he maintains a somewhat bivalent position about it: on the one hand, he praises the Presocratic philosopher for putting the nous as the first principle, while on the other, he shows his disappointment. According to him, Anaxagoras’ nous works insufficiently in the universe, but it is also the cause of goodness, indeed it is the Good capitalized. Our goal is to explain how Aristotle was (...) able to develop both types of seemingly inconsistent discourse concerning Anaxagorean nous. Two solutions to this dilemma are offered: even although some teleology exists in Anaxagoras’ philosophy, it is insufficient for Plato and Aristotle; the reception of the thought of Anaxagoras in later authors still before Aristotle furtively associates goodness with nous. (shrink)

Chrysippus often talks as if there is a third option when we might expect that two options in response to a question are exhaustive. Things are true, false or neither; equal, unequal, or neither; the same, different, or neither.. and so on. There seems to be a general pattern here that calls for a general explanation. This paper offers a general explanation of this pattern, preserving Stoic commitments to excluded middle and bivalence, arguing that Chrysippus employs this trichotomy move (...) when he wishes to argue that apparent contradictories are only contraries, and wishes to endorse a third option. This general explanation of the pattern of trichotomies sheds light on a number of interpretive puzzles, including on Chrysippus's response to the paradox of the cone. The purpose of these trichotomies is also discussed, and it is suggested that they originate from the dialectical context in which philosophical problems were posed. (shrink)

It is the aim of this paper to show that [the theological argument from Divine omniscience] is not more than a needlessly (and confusingly) elaborate version of the argument for fatalism discussed by Aristotle in de Interpretatione 9, which, since its sole premise is the Principle of Bivalence, may conveniently be called the logical argument for fatalism. If this is right, if the theological premisses of the theological argument can be shown to be strictly irrelevant to the fatalist conclusion, (...) then it follows that it is pointless to try to avoid fatalism by modification of those theological premisses. (shrink)

The paper presents Alfred Tarski’s debate with the semantic antinomies: the basic Liar Paradox, and its more sophisticated versions, which are currently discussed in philosophy: Strengthen Liar Paradox, Cyclical Liar Paradox, Contingent Liar Paradox, Correct Liar Paradox, Card Paradox, Yablo’s Paradox and a few others. Since Tarski, himself did not addressed these paradoxes—neither in his famous work published in 1933, nor in later papers in which he developed the Semantic Theory of Truth—therefore, We try to defend his concept of truth (...) against these antinomies. It is demonstrated that Tarskian theory of truth is resistant to the paradoxes and it is still the best solution to avoid the antinomies and remain within a classical logic, that is, accepting the laws of noncontradiction, excluded middle, and the principle of bivalence. Thus, the goal of the paper is double—firstly, to show that none of the versions of the Liar Paradox’s is a serious threat to Tarski’s concept of truth, and secondly, that Semantic Theory of Truth allows to remain within classical logic, and at the same time, avoid antinomies—which makes it the most attractive among classical theories of truth. (shrink)

It is disputed what norm, if any, governs assertion. We address this question by looking at assertions of future contingents: statements about the future that are neither metaphysically necessary nor metaphysically impossible. Many philosophers think that future contingents are not truth apt, which together with a Truth Norm or a Knowledge Norm of assertion implies that assertions of these future contingents are systematically infelicitous. In this article, we argue that our practice of asserting future contingents is incompatible with the view (...) that they are not truth apt. We consider a range of norms of assertion and argue that the best explanation of the data is provided by the view that assertion is governed by the Knowledge Norm. (shrink)

This dialogue between Achilles and the Tortoise – in the spirit of those of Carroll and Hofstadter – argues against the idea, identified with the “traditional” interpretation of Aristotle’s “sea battle argument”, that future contingents are an exception to the Principle of Bivalence. It presents examples of correct everyday predictions, without which one would not be able to decide and to act; however, doing this is incompatible with the belief that the content of these predictions lacks a truth-value. The (...) cost of using a non-classical logic to cope with that may be too high for Stagirite’s defenders, and they would still need to explain why our ordinary predictions seem to have a binary truth-value. In the end, the paper suggests that the problem of future contingents – and of free will – is not a logical problem at all, but rather a limit on what an agent can believe before taking a decision. (shrink)

There is no interesting entailment either way between theism and various forms of realism. Taking its cue from Dummett's characterisation of realism and his discussion of it with respect to theistic belief, this paper argues both that theism does not follow from realism, and that God cannot be appealed to in order to secure bivalence for an otherwise indeterminate subject matter. In both cases, significant appeal is made to the position that God is not a language user, which in (...) turn is motivated by an account of understanding as aptitude possession. The resulting picture sits comfortably with the apophatism common within living religious traditions and with the view that the philosophy of religion ought to reorientate itself away from metaphysics towards more practical questions. (shrink)

Study represents an application of the neutrosophic method, for solving the contradiction between communication and information. In addition, it recourse to an appropriate method of approaching the contradictions: Extensics, as the method and the science of solving the contradictions. The research core is the reality that the scientific research of communication-information relationship has reached a dead end. The bivalent relationship communicationinformation, information-communication has come to be contradictory, and the two concepts to block each other. After the critical examination of conflicting (...) positions expressed by many experts in the field, the extensic and inclusive hypothesis is issued that information is a form of communication. The object of communication is the sending of a message. The message may consist of thoughts, ideas, opinions, feelings, beliefs, facts, information, intelligence or other significational elements. When the message content is primarily informational, communication will become information or intelligence. The arguments of supporting the hypothesis are: a) linguistic (the most important being that there is "communication of information" but not "information of communication"; also, it is clarified and reinforced the over situated referent, that of the communication as a process), b) systemic-procedural (in the communication system is developing an information system; the informing actant is a type of communicator, the information process is a communication process), c) practical (the delimitation eliminates the efforts of disparate and inconsistent understanding of the two concepts), d) epistemological arguments (the possibility of intersubjective thinking of reality is created), linguistic arguments, e) logical and realistic arguments (it is noted the situation that allows to think coherently in a system of concepts - derivative series or integrative groups) f) and arguments from historical experience (the concept of communication has temporal priority, it appears 13 times in Julius Caesar’s writings ). In an axiomatic conclusion, the main arguments are summarized in four axioms: three are based on the pertinent observations of specialists, and the fourth is a relevant application of Florentin Smarandache’s neutrosophic theory. (shrink)

Some have argued that the possibility of faultless disagreement gives relativist semantic theories an important explanatory advantage over their absolutist and contextualist rivals. Here I combat this argument, focusing on the specific case of aesthetic discourse. My argument has two stages. First, I argue that while relativists may be able to account for the possibility of faultless aesthetic disagreement, they nevertheless face difficulty in accounting for the intuitive limits of faultless disagreement. Second, I develop a new non-relativist theory which can (...) account for the full range of data regarding faultless disagreement. This view—‘Humean Absolutism’—integrates two of Hume’s central principles from Of the Standard of Taste into a truth-conditional framework, resulting in a non-bivalent theory of aesthetic truth. I argue that Humean Absolutism can underwrite the possibility of faultless disagreement whilst retaining reasonable limits around the phenomenon. I close by relating this positive account of faultless disagreement to broader issues concerning the cognitive role of truth-value gaps. -/- *NB - this is an unpublished paper and is no longer in progress.*. (shrink)