Spinoza ’s letter of June 2, 1674 to his friend Jarig Jelles addresses several distinct and important issues in Spinoza ’s philosophy. It explains briefly the core of Spinoza ’s disagreement with Hobbes’ political theory, develops his innovative understanding of numbers, and elaborates on Spinoza ’s refusal to describe God as one or single. Then, toward the end of the letter, Spinoza writes: With regard to the statement that figure is a negation and not anything positive, it is obvious (...) that matter in its totality, considered without limitation [indefinitè consideratam], can have no figure, and that figure applies only to finite and determinate bodies. For he who says that he apprehends a figure, thereby means to indicate simply this, that he apprehends a determinate thing and the manner of its determination. This determination therefore does not pertain to the thing in regard to its being [esse]; on the contrary, it is its non-being [non-esse]. So since figure is nothing but determination, and determination is negation [Quia ergo figura non aliud, quam determinatio, et determinatio negatio est], figure can be nothing other than negation, as has been said. Arguably, what is most notable about this letter is the fate of a single subordinate clause which appears in the last sentence of this passage: et determinatio negatio est. That clause was to be adopted by Hegel and transformed into the slogan of his own dialectical method: Omnis determinatio est negatio. Of further significance is the fact that, while Hegel does credit Spinoza with the discovery of this most fundamental insight, he believes Spinoza failed to appreciate the importance of his discovery. The issue of negation and the possibility of self-negation stand at the very center of the philosophical dialogue between the systems of Spinoza and Hegel, and in this paper I will attempt to provide a preliminary explication of this foundational debate between the two systems. In the first part of the paper I will argue that the “determination is negation” formula has been understood in at least three distinct senses among the German Idealists, and as a result many of the participants in the discussion of this formula were actually talking past each other. The clarification of the three distinct senses of the formula will lead, in the second part of the paper, to a more precise evaluation of the fundamental debate between Spinoza and Hegel regarding the possibility of self-negation. In this part I will evaluate the validity of each interpretation of the determination formula, and motivate the positions of the various participants in the debate. (shrink)

Hegel’s theory of dialectic has long been a source of both endless confusion and bitter debate. It has, for instance, been oversimplified and characterized as the mechanical movement from thesis to antithesis to synthesis. In a similar vein, some philosophers in the analytic tradition have reproached Hegel’s notion of dialectic, claiming that it amounts to an outright and absurd denial of the law of contradiction. The dialectic has, moreover, been co-opted and developed by some of Hegel’s most impassioned critics such (...) as Marx and Kierkegaard. One of the most controversial aspects of Hegel’s theory of dialectic has been his perplexing doctrine of determinate negation, which has proven to be difficult for even the most sympathetic interpreters to make sense of. Determinate negation is Hegel’s way of referring to the positive aspect of the dialectic which makes the conceptual movement a constructive one and not a purely destructive or negative one. Although Hegel discusses his notion of determinate negation at some length in the Phenomenology, interpreters have traditionally gone to the Science of Logic in order to illustrate this doctrine. In this essay by contrast, I would like to discuss this difficult and disputed concept of determinate negation by means of an example drawn from the “Consciousness” chapter of the Phenomenology of Spirit. At the transition from “Sense-Certainty” to “Perception,” the first two sections of that chapter, Hegel describes explicitly the aspect of determinate negation that the dialectic evinces at that point, and it is this transition and this uncharacteristic passage that I wish to examine in some detail. In the first section of this essay, I will discuss Hegel’s notion of dialectic generally in order to pinpoint the role of determinate negation in relation to the dialectic’s other aspects. Then in the second section, I will proceed to analyze his account of determinate negation specifically, drawing on various passages throughout the Hegelian corpus. I will give a brief account of “Sense-Certainty” in my third section in order to set the context for the transition to “Perception.” Finally, in the fourth section, I will analyze this transition to “Perception” in terms of the doctrine of determinate negation. (shrink)

In this paper, I contend that Brandom’s interpretive oversights leave his inferentialist program vulnerable to Hegelian critique. My target is Brandom’s notion of “conceptual realism,” or the thesis that the structure of mind-independent reality mimics the structure of thought. I show, first, that the conceptual realism at the heart of Brandom’s empiricism finds root in his interpretation of Hegel. I then argue that conceptual realism is incompatible with Hegel’s thought, since the Jena Phenomenology, understood as a “way of despair,” includes (...) a critique of the philosophical framework upon which conceptual realism relies. Finally, I offer the Hegelian critique of Brandom that results from these textual infidelities. (shrink)

In Hegel’s Conception of the Determinate Negation , Terje Sparby develops a comprehensive account of the three forms of the determinate negation in Hegel’s philosophy.

My analysis of political debate in the United Kingdom during the summer of 2016 unpacks the compression of two highly complex issues within an unprecedented moment in British politics: reinvestment in nuclear arms and nuclear energy during the EU referendum crisis. I recover unities and discontinuities across events in this period and throughout history both to examine the non-identity between the particular and the universal as a major trope in parliamentary rhetoric, which construed the universal sentiment of world peace and (...) denied this in terms of security, and to seek out the use of determinate negation, especially when it has bearing on the advancing of climate-related policies. From nuanced speeches in the House of Commons and House of Lords, I move tentatively outwards by gesturing to the moment of truth in reified concepts, seeking to pry them open in their non-identity with art objects of the Anthropocene that are understood within a context of ecological poetics. (shrink)

Jason Lagapa’s Negative Theology and Utopian Thought in Contemporary American Poetry tackles a question that has been a difficult one to address for critics attempting to discuss contemporary experimental poetry in the line of “ Language writing.” This is a tradition that claims to be politically engaged but which nevertheless does not tend explicitly to exhort its readers to take concrete political actions. How can we thus judge this poetry’s political efficacy when there are no clear or obvious political actions (...) associated with it? Lagapa’s recourse to negative theology and utopian thought tells us that we should pay close attention to matters... (shrink)

Quantification, Negation, and Focus: Challenges at the Conceptual-Intentional Semantic Interface Tista Bagchi National Institute of Science, Technology, and Development Studies (NISTADS) and the University of Delhi Since the proposal of Logical Form (LF) was put forward by Robert May in his 1977 MIT doctoral dissertation and was subsequently adopted into the overall architecture of language as conceived under Government-Binding Theory (Chomsky 1981), there has been a steady research effort to determine the nature of LF in language in light of (...) structurally diverse languages around the world, which has ultimately contributed to the reinterpretation of LF as a Conceptual-Intentional (C-I) interface level between the computational syntactic component of the faculty of language and one or more interpretive faculties of the human mind. While this has opened up further possibilities of research in phenomena such as quantifier scope and scope interactions between negation, quantification, and focus, it has also given rise to a few real challenges to linguistic theory as well. Some of these are: (i) the split between lexical meaning – a matter supposedly belonging to the phase-wise selection of lexical arrays – and issues of semantic interpretation that arise purely from binding and scope phenomena (Mukherji 2010); (ii) partially relatedly, the level at which theta role assignment can be argued to take place, an issue that is taken up by me in Bagchi (2007); and (iii) how supposedly “pure” scopal phenomena relating to quantifiers, negation, and emphasizing expressions such as only and even (comparable to, e.g., Urdu/Hindi hii and bhii, Bangla –i and –o) also have dimensions of both focus and discourse reference. While recognizing all of these challenges, this talk aims to highlight particularly challenge (iii), both in terms of scholarship in the past and for the rich prospects for research on languages of south Asia with the semantics of quantification, negation, and focus in view. The scholarship of the past that I seek to relate this issue to is where, parallel to (and largely independently of) the research on LF that had been happening, Barwise and Cooper were developing their influential view of noun phrases as generalized quantifiers, culminating in their key 1981 article “Generalized Quantifiers and Natural Language” while, independently, McCawley, in his 1981 book Everything that Linguists have Always Wanted to Know about Logic, established through argumentation that all noun phrases semantically behave like generalized quantified expressions (further elaborated by him in the second – 1994 – revised edition of his book). I seek to demonstrate, based on limited data analysis from selected languages of south Asia, that our current understanding of quantification, negation, and focus under the Minimalist view owes something significant to the two major, but now largely marginalized, works of scholarship, and that for the way forward it is essential to adopt a more formal-semantic approach as adopted by them and also by later works such as Denis Bouchard’s (1995) The Semantics of Syntax, Mats Rooth’s work on focus (e.g., Rooth 1996, “Focus” in Shalom Lappin’s Handbook of Contemporary Semantic Theory), Heim and Kratzer’s Semantics in Generative Grammar (1998), and Yoad Winter’s (2002) Linguistic Inquiry article on semantic number, to cite just a few instances. (shrink)

We investigate the notion of classical negation from a non-classical perspective. In particular, one aim is to determine what classical negation amounts to in a paracomplete and paraconsistent four-valued setting. We first give a general semantic characterization of classical negation and then consider an axiomatic expansion BD+ of four-valued Belnap–Dunn logic by classical negation. We show the expansion complete and maximal. Finally, we compare BD+ to some related systems found in the literature, specifically a four-valued modal (...) logic of Béziau and the logic of classical implication and a paraconsistent de Morgan negation of Zaitsev. (shrink)

The problems of the meaning and function of negation are disentangled from ontological issues with which they have been long entangled. The question of the function of negation is the crucial issue separating relevant and paraconsistent logics from classical theories. The function is illuminated by considering the inferential role of contradictions, contradiction being parasitic on negation. Three basic modelings emerge: a cancellation model, which leads towards connexivism, an explosion model, appropriate to classical and intuitionistic theories, and a (...) constraint model, which includes relevant theories. These three modelings have been seriously confused in the modern literature: untangling them helps motivate the main themes advanced concerning traditional negation and natural negation. Firstly, the dominant traditional view, except around scholastic times when the explosion view was in ascendency, has been the cancellation view, so that the mainstream negation of much of traditional logic is distinctively nonclassical. Secondly, the primary negation determinable of natural negation is ·relevant negation. In order to picture relevant negation the traditional idea of negation as otherthanness is progressive) refined, to nonexclusive restricted otherthanness. Several pictures result, a reversal picture, a debate model, a record cabinet (or files of the universe) model which help explain relevant negation. Two appendices are attached, one on negation in Hegel and the Marxist tradition, the other on Wittgenstein's treatment of negation and contradiction. (shrink)

Four experiments and 2 corpus-based studies demonstrate that negation is a determinant factor affecting novel nonliteral utterance-interpretation by default. For a nonliteral utterance-interpretation to be favored by default, utterances should be potentially ambiguous between literal and nonliteral interpretations. They should therefore be (a) unfamiliar, (b) free of semantic anomaly or any kind of internal incongruity, and (c) unbiased by contextual information. Experiments 1–3 demonstrate that negative utterances, meeting these 3 conditions, were interpreted metaphorically (This is not a safe) or (...) sarcastically (Ambitious she is not) when presented in isolation and were therefore processed faster in contexts strongly biasing them toward their nonliteral than toward their (equally biased) literal interpretation. Experiment 4 reduces the possibility that it is structural markedness on its own that induces nonliteralness. Two corpus-based studies provide corroborating evidence, supporting the view of negation as an operator generating nonliteral interpretations by default. (shrink)

L’auteur discerne trois intuitions majeures dans la théorie éliminativiste de la négation développée par les idéalistes anglais, d’après laquelle une négation est l’élimination d’une alternative au sein d’un ensemble complet d’alternatives disjonctivement affirmées du sujet de la négation : premièrement, la détermination du sens d’une proposition est l’assignation à une proposition de coordonnées logiques dans un espace logique ; deuxièmement, le sens d’une proposition entretient une relation interne avec le sens de sa négation ; troisièmement, l’espace logique dans lequel une (...) négation acquiert un sens dépend du contexte. Il est montré que la théorie éliminativiste de la négation échoue à composer adéquatement les deux premiers aperçus en essayant de faire droit au troisième. A l’inverse, le Tractatus de Wittgenstein les combine adéquatement mais passe à côté du troisième. Une théorie adéquate de la négation devrait, minimalement, satisfaire simultanément les trois contraintes. (shrink)

By 1929, after the full acknowledgment of the colour–exclusion problem, Wittgenstein had to admit that material incompatibilities presented in conceptual systems could not be reduced to formal tautologies and contradictions. Wittgenstein then, in his middle period, had to examine the kind of negation which, for instance, colour systems should render, which expose not just one but many or, in some cases, infinite inferentially articulated alternatives. Here, inspired by Brandom’s inferentialism, I explore the idea that Wittgenstein, in his middle period, (...) advocated a form of inferentialism based on the inferentially articulated content of propositions in Satzsysteme. At that time, Wittgenstein suggested that every sentence should be logically connected to many others. I call this feature inferential thickness. Therefore, I use Löf’s normative read of verificationism to explain Middle Wittgenstein’s holist solution to problems concerning the use of negation related to material incompatibilities and determination of propositional sense. I also investigate the distinction between contrariety and contradiction and some possible connections to a mandatory restriction of the principle of excluded middle in Satzsysteme. (shrink)

Thinkers have never been able to deny the centrality of negation and contradiction in everything human, despite all their efforts to banish both from the domains of truth, right, and beauty. Unless we properly understand the fundamental significance of negation and contradiction, we cannot free ourselves from bondage to opinion, arbitrary convention, and subjective taste. Of all philosophers, Hegel has most resolutely confronted the role of negation and contradiction in the most essential strivings of humanity, and it (...) is high time to put his comprehension to work. The essay does just this, showing how negation and contradiction operate in the different levels of logical determinacy and how their role in self-determination allows for truth, right, and beauty to be and to be conceived. (shrink)

Das Konzept der Negation ist der zentrale Operator bei der Unterscheidung zwischen historischem Wandel und natürlicher Evolution, welche grundlegend für das moderne Denken ist. Die Krise dieser Abgrenzung ist somit auch eine »Krise der Negation« (AlainBadiou). Der vorliegende Text untersucht die Krise, indem er zuerst Hegels Konzept der »bestimmten Negation« und deren Auswirkungen auf das moderne Verständnis von Revolution beleuchtet und erörtert im Anschluss zwei mögliche Alternativen, wie Negation noch verstanden werden kann: als abstrakte Negation (...) (Luhmann) und als endlose Negation (Agamben). The concept of negation is the central operator in distinguishing between historical change and natural evolution, which is constitutive of modern thinking. The crisis of this distinction is therefore the »crisis of negation « (Alain Badiou). The text examines this crisis by first considering Hegel’s concept of »determinate negation« and its impact on the modern understanding of revolution and then discusses two possible alternative understandings of negation: abstract negation (Luhmann) and infinite negation (Agamben). (shrink)

The concept of negation is the central operator in distinguishing between historical change and natural evolution, which is constitutive of modern thinking. The crisis of this distinction is therefore the »crisis of negation « (Alain Badiou). The text examines this crisis by first considering Hegel’s concept of »determinate negation« and its impact on the modern understanding of revolution and then discusses two possible alternative understandings of negation: abstract negation (Luhmann) and infinite negation (Agamben).

We consider the variety of Dynamic Relation Algebras V(DRA). We show that the monoid of an algebra in this variety determines dynamic negation uniquely.

Some authors have claimed that Hegel's ‘determinate negation’ should be distinguished from ‘logical’ or ‘formal’ negation, that is, from a view of negation as a contradictory forming operator. In contrast, I argue that dialectical determinate negation involves a view of negation as a contradictory forming operator, and can therefore count as formal negation in every respect. However, as it is clear in contemporary glutty semantics of negation, one may distinguish between different (...) accounts of the relationship between negation, contradiction and content. I suggest that, differently from classical negation, and as glutty negation, dialectical negation has partial content. (shrink)

In one of their papers, Michael De and Hitoshi Omori observed that the notion of classical negation is not uniquely determined in the context of so-called Belnap-Dunn logic, and in fact there are 16 unary operations that qualify to be called classical negation. These varieties are due to different falsity conditions one may assume for classical negation. The aim of this paper is to observe that there is an interesting way to make sense of classical negation (...) independent of falsity conditions. We discuss two equivalent semantics, and offer a Hilbert-style system that is sound and complete with respect to the semantics. (shrink)

We explored the influence of negation on cognitive dynamics, measured using mouse‐movement trajectories, to test the classic notion that negation acts as an operator on linguistic processing. In three experiments, participants verified the truth or falsity of simple statements, and we tracked the computer‐mouse trajectories of their responses. Sentences expressing these facts sometimes contained a negation. Such negated statements could be true (e.g., “elephants are not small”) or false (e.g., “elephants are not large”). In the first experiment, (...) as predicted by the classic notion of negation, we found that negation caused more discreteness in the mouse trajectory of a response. The second experiment induced a simple context for these statements, yet negation still increased discreteness in trajectories. A third experiment enhanced the pragmatic context of sentences, and the discreteness was substantially diminished, with one primary measure no longer significantly showing increased discreteness at all. Traditional linguistic theories predict rapid shifts in cognitive dynamics occur due to the nature of negation: It is an operator that reverses the truth or falsity of an interpretation. We argue that these results support both propositional and contextual accounts of negation present in the literature, suggesting that contextual factors are crucial for determining the kind of cognitive dynamics displayed. We conclude by drawing broader lessons about theories of cognition from the case of negation. (shrink)

For some while we have been witnessing a series of destructive phenomena which seem to indicate a full-fledged return to the negative on the world stage – from terrorism and armed conflict to the threat of environmental catastrophe. At the same time, politics seems increasingly impotent in the face of these threats. In this book, the leading Italian philosopher Roberto Esposito reconstructs the genealogy of the reciprocal intertwining of politics and negation. He retraces the intensification of negation in (...) the thought of various thinkers, from Schmitt and Freud to Heidegger, and examines the negative slant of some of our fundamental political categories, such as sovereignty, property and freedom. Against the centrality of negation, Esposito proposes an affirmative philosophy that does not negate or repress negation but radically rethinks it in the positive cipher of difference, determination and opposition. The result is a rigorous and original pathway which, in the tension between affirmation and negation, recognizes the disturbing traumas of our time, as well as the harbingers of what awaits at its limits. This highly original and timely book will be of great value to students and scholars in philosophy, cultural theory and the humanities more generally, and to anyone interested in contemporary European thought. (shrink)

We report two studies on the effect of implicitly versus explicitly conveying affirmation and denial problems about conditionals. Recently Evans and Handley (1999) and Schroyens et al. (1999b, 2000b) showed that implicit referencing elicits matching bias: Fewer determinate inferences are made, when the categorical premise (e.g., B) mismatches the conditional's referred clause (e.g., A). Also, the effect of implicit affirmation (B affirms not-A) is larger than the effect of implicit denial (B denies A). Schroyens et al. hypothesised that this (...) interaction is due to uncertainty in the case-wise affirmation of the contrast class of negated elements involved in implicit affirmations. In Experiment 1 we tested this hypothesis by manipulating the set size of the conditional clauses. The results confirm that binary sets, where the contrast class is a singleton, eliminate the differential effect of implicit affirmation and denial. With non-binary sets the interaction is not modulated by the scope of the contrast class (3, 5, 9 elements). Experiment 2 further investigated the role of contrast classes by using class inclusion to construct implicit affirmations (Mammal vs Mammal or Monkey) and implicit denial (No-Mammal vs Mammal or Monkey), in addition to the standard implicit problems mediated by contrast-class inclusion [(No-)Mammal/No-Mammal; Reptile; Snake). Findings indicate that class inclusion (Mammal/Monkey; Reptile/ Snake) only marginally affects performance, and is independent of the type of problem. This would suggest that the implicitness problem-type interaction is dependent on constructing contrast classes. However, the experiment failed to replicate the interaction, even on the subset of problems repeating the abstract letter/number format of Experiment 1. Moreover, with the natural binary set-sizes (vowels/consonants) the implicitness effect was eliminated entirely. (shrink)

A series of five experiments investigated the extent of subliminal processing of negation. Participants were presented with a subliminal instruction to either pick or not pick an accompanying noun, followed by a choice of two nouns. By employing subjective measures to determine individual thresholds of subliminal priming, the results of these studies indicated that participants were able to identify the correct noun of the pair – even when the correct noun was specified by negation. Furthermore, using a grey-scale (...) contrast method of masking, Experiment 5 confirmed that these priming effects were evidenced in the absence of partial awareness, and without the effect being attributed to the retrieval of stimulus–response links established during conscious rehearsal. (shrink)

1 What are the features of a paraconsistent negation? Since paraconsistent logic was launched by da Costa in his seminal paper [4], one of the fundamental problems has been to determine what exactly are the theoretical or metatheoretical properties of classical negation that can have a unary operator not obeying the principle of noncontradiction, that is, a paraconsistent operator. What the result presented here shows is that some of these properties are not compatible with each other, so that (...) in constructing a paraconsistent negation as close as possible to classical negation, we have to make a choice among classical properties compatible with the idea of paraconsistency. In particular, there is no paraconsistent negation more classical than all the others. (shrink)

The present paper is inspired by Sylvan and Plumwood’s logicBM defined in “Non-normal relevant logics” and by their treatmentof negation with the ∗-operator in “The semantics of first-degree en-tailment”. Given a positive logic L including Routley and Meyer’sbasic positive logic and included in either the positive fragment of Eor in that of RW, we investigate the essential De Morgan negation ex-pansions of L and determine all the deductive relations they maintainto each other. A Routley-Meyer semantics is provided for (...) each logicdefined in the paper. (shrink)

This essay introduces a formal structure to model the indeterminacy of the future in Einstein-Minkowski space-time. We consider a first-order language, supplemented with an operator for predicate-term negation, and defend the claim that such an operation provides an appropriate model for the indeterminacy of future contingents. In the final section, it is proved that given a language otherwise adequate to represent a physical theory, at least some of the predicates of that language are indeterminate when the future is not (...) causally determined by the present and the past. (shrink)

The essay draws on the concept of ‘asymmetric counter-concepts’ as developed by Reinhart Koselleck starting with twin-formulas such as ‘the familiar and the unfamiliar’ which are generally used to establish collective des­ignations of the self and others and which institutionalize the axiological and the epistemological. These counter-concepts can have different semantic temperatures. The focus is on the underlying meaning-production schemes which produce value-asymmetries. The essay tries to show that a process of heating up these value-asymmetries is only one side of (...) the history of such asymmetric counter-concepts from medieval to modern times. Simultaneously a cooling down can be observed in written texts from different periods; examples include the 12th century Rolandslied and the 16th century Essais of Michel de Montaigne. Full negation eliminates uncertainties and value insecurities. But the complexities and contingencies that emerge since Early Modern times then lead to losses of negatability, which in turn render gains in unfamiliarity. The modern experience of the foreign is indeterminate otherness instead of determined negation that characterized pre-modern alterity. Modern societies therefore need to mediate between validity and contingency under the circumstances of plurality. Interpretational demands and uncertainty about the relevant interpretive frames increase. Foreignness is then experienced as unfamiliarity. This presupposes intellectual attitudes like irritability, curiosity, and willingness to learn. The modern concept of ‘culture’ then is proposed as a comparative pattern where only unavoidable structural asymmetry remains. It explains cultural differences and the experience of foreignness through heterogeneity. Using this specifically modern pattern, there is no longer a legitimate value slope between one’s own position and its negation. The distinction is then between the familiar and the unfamiliar. (shrink)

Children are the most sensitive part of a sciety, therefore the violence against them is considered a serious violation of their personal rights and their higher interests. In most of cases, children in Republic of Macedonia are very little or not at all informed concerning the possibilities of reporting the cases of violence against them by their parents or relatives. The issue of domestic violence is still considered a private problem which occurs within the home. Thus, in most of cases, (...) this problem remains unsolved in silence, without any alert for the state institutions or SOS phone lines for reporting domestic violence. Some children who are more aware for being subjected to violation of their rights are afraid to report the case because of further consequences. In this article some facts will be given about observing the symptoms and signs of violence against children, forms and types of violence against children, determination factors of the violence as well as the consequences and the impact of the domestic violence on the physical and psychological development of children. Based on the legal provisions and international conventions, parents, responsible institutions and the society in general are morally and legally obliged to respect and preserve the interest of the child. This article will show that in practice, these legal obligations are violated by parents and in specific cases also by the competent institutions because the lack of intervention. (shrink)

We investigate two large families of logics, differing from each other by the treatment of negation. The logics in one of them are obtained from the positive fragment of classical logic (with or without a propositional constant ff for “the false”) by adding various standard Gentzen-type rules for negation. The logics in the other family are similarly obtained from LJ+, the positive fragment of intuitionistic logic (again, with or without ff). For all the systems, we provide simple semantics (...) which is based on non-deterministic four-valued or three-valued structures, and prove soundness and completeness for all of them. We show that the role of each rule is to reduce the degree of non-determinism in the corresponding systems. We also show that all the systems considered are decidable, and our semantics can be used for the corresponding decision procedures. Most of the extensions of LJ+ (with or without ff) are shown to be conservative over the underlying logic, and it is determined which of them are not. (shrink)

In this note we introduce and study algebras of type such that is a bounded distributive lattice and ⌝ is an operator that satisfies the condition ⌝ = a ⌝ b and ⌝ 0 = 1. We develop the topological duality between these algebras and Priestley spaces with a relation. In addition, we characterize the congruences and the subalgebras of such an algebra. As an application, we will determine the Priestley spaces of quasi-Stone algebras.

In the Transcendental Ideal Kant discusses the principle of complete determination: for every object and every predicate A, the object is either determinately A or not-A. He claims this principle is synthetic, but it appears to follow from the principle of excluded middle, which is analytic. He also makes a puzzling claim in support of its syntheticity: that it represents individual objects as deriving their possibility from the whole of possibility. This raises a puzzle about why Kant regarded it as (...) synthetic, and what his explanatory claim means. I argue that the principle of complete determination does not follow from the principle of excluded middle because the externally negated or ?negative? judgement ?Not (S is P)? does not entail the internally negated or ?infinite? judgement ?S is not-P.? Kant's puzzling explanatory claim means that empirical objects are determined by the content of the totality of experience. This entails that empirical objects are completely determinate if and only if the totality of experience has a completely determinate content. I argue that it is not a priori whether experience has such a completely determinate content and thus not analytic that objects obey the principle of complete determination. (shrink)

In this paper we will study the properties of the least extension n(Λ) of a given intermediate logic Λ by a strong negation. It is shown that the mapping from Λ to n(Λ) is a homomorphism of complete lattices, preserving and reflecting finite model property, frame-completeness, interpolation and decidability. A general characterization of those constructive logics is given which are of the form n(Λ). This summarizes results that can be found already in [13, 14] and [4]. Furthermore, we determine (...) the structure of the lattice of extensions of n(LC). (shrink)

Vann McGee has recently argued that Belnap’s criteria constrain the formal rules of classical natural deduction to uniquely determine the semantic values of the propositional logical connectives and quantifiers if the rules are taken to be open-ended, i.e., if they are truth-preserving within any mathematically possible extension of the original language. The main assumption of his argument is that for any class of models there is a mathematically possible language in which there is a sentence true in just those models. (...) I show that this assumption does not hold for the class of models of classical propositional logic. In particular, I show that the existence of non-normal models for negation undermines McGee’s argument. (shrink)

In the context of a weak formal theory called Basic Intuitionistic Mathematics $$\textsf{BIM}$$ BIM, we study Brouwer’s Fan Theorem and a strong negation of the Fan Theorem, Kleene’s Alternative (to the Fan Theorem). We prove that the Fan Theorem is equivalent to contrapositions of a number of intuitionistically accepted axioms of countable choice and that Kleene’s Alternative is equivalent to strong negations of these statements. We discuss finite and infinite games and introduce a constructively useful notion of determinacy. We (...) prove that the Fan Theorem is equivalent to the Intuitionistic Determinacy Theorem. This theorem says that every subset of Cantor space $$2^\omega $$ 2 ω is, in our constructively meaningful sense, determinate. Kleene’s Alternative is equivalent to a strong negation of a special case of this theorem. We also consider a uniform intermediate value theorem and a compactness theorem for classical propositional logic. The Fan Theorem is equivalent to each of these theorems and Kleene’s Alternative is equivalent to strong negations of them. We end with a note on ‘stronger’ Fan Theorems. The paper is a sequel to Veldman (Arch Math Logic 53:621–693, 2014). (shrink)

Intuitionistic logic formalises the foundational ideas of L.E.J. Brouwer’s mathematical programme of intuitionism. It is one of the earliest non-classical logics, and the difference between classical and intuitionistic logic may be interpreted to lie in the law of the excluded middle, which asserts that either a proposition is true or its negation is true. This principle is deemed unacceptable from the constructive point of view, in whose understanding the law means that there is an effective procedure to determine the (...) truth of all propositions. This understanding of the distinction between the two logics supports the view that negation plays a vital role in the formulation of intuitionistic logic.Nonetheless, the formalisation of negation in intuitionistic logic has not been universally accepted, and many alternative accounts of negation have been proposed. Some seek to weaken or strengthen the negation, and others actively supporting negative inferences that are impossible with it.This thesis follows this tradition and investigates various aspects of negation in intuitionistic logic. Firstly, we look at a problem proposed by H. Ishihara, which asks how effectively one can conserve the deducibility of classical theorems into intuitionistic logic, by assuming atomic classes of non-constructive principles. The classes given in this section improve a previous class given by K. Ishii in two respects: (a) instead of a single class for the law of the excluded middle, two classes are given in terms of weaker principles, allowing a finer analysis and (b) the conservation now extends to a subsystem of intuitionistic logic called Glivenko’s logic. This section also discusses the extension of Ishihara’s problem to minimal logic.Secondly, we study the relationship between two frameworks for weak constructive negation, the approach of D. Vakarelov on one hand and the framework of subminimal negation by A. Colacito, D. de Jongh, and A. L. Vargas on the other hand. We capture a version of Vakarelov’s logic with the semantics of the latter framework, and clarify the relationship between the two semantics. This also provides proof-theoretic insights, which results in the formulation of a cut-free sequent calculus for the aforementioned system.Thirdly, we investigate the ways to unify the formalisations of some logics with contra-intuitionistic inferences. The enquiry concerns paraconsistent logics by R. Sylvan and A. B. Gordienko, as well as the logic of co-negation by G. Priest and of empirical negation by M. De and H. Omori. We take Sylvan’s system as basic, and formulate the frame conditions of the defining axioms of the other systems. The conditions are then used to obtain cut-free labelled sequent calculi for the systems.Finally, we consider L. Humberstone’s actuality operator for intuitionistic logic, which can be seen as the dualisation of a contra-intuitionistic negation. A compete axiomatisation of intuitionistic logic with actuality operator is given, and comparisons are made for some related operators.Abstract prepared by Satoru Niki.E-mail:[email protected]. (shrink)

Marcuse and Sartre take up the problem of alienating otherness from a Marxist perspective, Marcuse in One-Dimensional Man and Sartre in his Critique of Dialectical Reason. For Sartre, the “series” is a social relation that places individuals in competition, mediated by the materialized result of past praxis. For Marcuse, the loss of agency results from the productive apparatus determining the needs and aspirations of individuals. The question is how to convert alienating negativity into a negation of the society that (...) negates individuals. For Sartre, this “negation of the whole” can come only from a mortal threat facing all members of the serialized group. For Marcuse, it comes from the individual becoming aware of her alienation, especially through works of art. For both, revolt must be a historically constituted, collective “living contradiction.”Marcuse et Sartre abordent le problème de l’altérité aliénante à partir d’une perspective marxiste, Marcuse dans L’homme unidimensionnel et Sartre dans sa Critique de la raison dialectique. Pour ce dernier, la « série » est une relation sociale qui met les individus en compétition, médiatisée par le résultat matérialisé de la praxis passée. Pour Marcuse, la perte de pouvoir est causée par le dispositif de production qui régule les besoins et les aspirations des individus. La question est ainsi comment transformer la négativité aliénante en une négation de la société qui nie les individus : d’après Sartre, cette « négation du tout » ne peut venir que d’une menace mortelle subie par tous les membres du groupe sérialisé ; si l’on en croit Marcuse, elle vient plutôt du fait qu’un individu prenne conscience de son aliénation, notamment à travers l’oeuvre d’art. Pour les deux, la révolte doit être, dans tous les cas, une « contradiction vivante » collective et historiquement constituée. (shrink)

In 2004, C. Sanza, with the purpose of legitimizing the study of \-valued Łukasiewicz algebras with negation -algebras) introduced \-valued Łukasiewicz algebras with negation. Despite the various results obtained about \-algebras, the structure of the free algebras for this variety has not been determined yet. She only obtained a bound for their cardinal number with a finite number of free generators. In this note we describe the structure of the free finitely generated \-algebras and we determine a formula (...) to calculate its cardinal number in terms of the number of free generators. Moreover, we obtain the lattice \\) of all subvarieties of \ and we show that the varieties of Boolean algebras, three-valued Łukasiewicz algebras and four-valued Łukasiewicz algebras are proper subvarieties of \. (shrink)

The distinction between the "permanent" (will not reverse) and "irreversible" (cannot reverse) cessation of functions is critical to understand the meaning of a determination of death using circulatory–respiratory tests. Physicians determining death test only for the permanent cessation of circulation and respiration because they know that irreversible cessation follows rapidly and inevitably once circulation no longer will restore itself spontaneously and will not be restored medically. Although most statutes of death stipulate irreversible cessation of circulatory and respiratory functions, the accepted (...) medical standard is their permanent cessation because permanence is a perfect surrogate indicator for irreversibility, and using it permits a more timely declaration. Therefore, patients properly declared dead in donation after circulatory death (DCD) protocols satisfy the requirements of death statutes and do not violate the dead donor rule. The acronym DCD should represent organ "donation after circulatory death" to clarify that the death standard is the permanent cessation of circulation, not heartbeat. Heart donation in DCD does not retroactively negate the donor's death determination because circulation has ceased permanently. (shrink)

It is well-known that intuitionistic logics can be formalized by means of Heyting algebras, i.e. relatively pseudocomplemented semilattices. Within such algebras the logical connectives implication and conjunction are formalized as the relative pseudocomplement and the semilattice operation meet, respectively. If the Heyting algebra has a bottom element |$0$|⁠, then the relative pseudocomplement with respect to |$0$| is called the pseudocomplement and it is considered as the connective negation in this logic. Our idea is to consider an arbitrary meet-semilattice with (...) |$0$| satisfying only the Ascending Chain Condition (these assumptions are trivially satisfied in finite meet-semilattices) and introduce the operators formalizing the connectives negation |$x^{0}$| and implication |$x\rightarrow y$| as the set of all maximal elements |$z$| satisfying |$x\wedge z=0$| and as the set of all maximal elements |$z$| satisfying |$x\wedge z\leq y$|⁠, respectively. Such a negation and implication is ‘unsharp’ since it assigns to one entry |$x$| or to two entries |$x$| and |$y$| belonging to the semilattice, respectively, a subset instead of an element of the semilattice. Surprisingly, this kind of negation and implication still shares a number of properties of these connectives in intuitionistic logic, in particular the derivation rule Modus Ponens. Moreover, unsharp negation and unsharp implication can be characterized by means of five, respectively seven simple axioms. We present several examples. The concepts of a deductive system and of a filter are introduced as well as the congruence determined by such a filter. We finally describe certain relationships between these concepts. (shrink)

In this paper we shall give a characterization of D-algebras in terms of lattice ordered abelian groups. To make this paper self-contained we shall recall some notations from [4]. The symbols !; ^; _; serve as implication, conjunction, disjunction, and negation, respectively. By D we mean a set of connectives from the list above containing the implication connective !. By a D-formula we mean a formula built up in a usual way from an innite set of the propositional variables (...) and connectives from D. (shrink)

Many arguments that show p to be enthymematic (in an argument for q) rely on claims like “if one did not believe that p, one would not have a reason for believing that q.” Such arguments are susceptible to the neg-raising fallacy. We tend to interpret claims like “X does not believe that p” as statements of disbelief (X's belief that not-p) rather than as statements of withholding the belief that p. This article argues that there is a tendency to (...) equivocate in arguments for the enthymematicity of arguments (e.g., Lewis Carroll's paradox, Hume's problem) as well as in arguments for the enthymematicity of action explanations (e.g., arguments for psychologism and for explanatory individualism). The article concludes with a warning, because the equivocation is often helpful in teaching and because neg-raising verbs include philosophically vital verbs: desire, want, intend, think, suppose, imagine, expect, feel, seem, appear. (shrink)

In this paper I rely on recent literature that emphasises the importance of recognition in Hegel's philosophy in order to apply the recognition-theoretic approach to the notion of sacrifice in the Phenomenology of Spirit. Firstly, I conduct a preliminary analysis by examining the general meaning of sacrifice as a form of determinate negation. Secondly, I focus on two phenomenological moments (the struggle between ?faith? and ?pure insight?, and the cult) in order to answer the question, ?Is a real (...) (effective and unselfish) sacrifice possible?? Finally, I argue that sacrifice should be considered as a Darstellung, and I explain the twofold connection between sacrifice and recognition. I conclude that there is no sacrifice without recognition, and the process of recognition is intrinsically sacrificial. (shrink)

The aim of the paper is to analyse Brandom’s account of Hegel’s conception of determinate negation and the role this structure plays in the dialectical process with respect to the problem of contradiction. After having shown both the merits and the limits of Brandom’s account, I will refer to Priest’s dialetheistic approach to contradiction as an alternative contemporary perspective from which it is possible to capture essential features of Hegel’s notion of contradiction, and I will test the equation (...) of Hegel’s dialectic with Priest dialetheism. (shrink)