The negation constitutes one of the main troubles for attempts to naturalise the semantics of the logical vocabulary, as shown by the problems related to the interpretation of disjunction in the treatment of error (Fodor) or to the definition of contraries in the analysis of reidentification abilities (Millikan). There seems to be no way out between “no (naturalized) negation, no grip of logic on the world” and “no (truth-functional) negation, no logic”. Unexpected help may come from the (...) cognitive phenomenology of negation. For three reasons: • Firstly, because it allows a distinction to be made between semantic analysis, identification of the possession conditions and psychology of the acquisition of the notion of negation, a distinction whose absence produces many misunderstandings between those who support a naturalization of semantics and those who oppose it. • Then because it places negation at the level of the contrast between beliefs, a terrain that is hostile to some naturalizers. • Finally, because it makes it possible to define a peculiar type of liberalised naturalization of semantics and, consequently, to present a hypothesis for the naturalization of cognitive phenomenology. (shrink)

Etude de l'extension par la negation semi-intuitionniste de la logique positive des propositions appelee logique C, developpee par A. Urquhart afin de definir une semantique relationnelle valable pour la logique des valeurs infinies de Lukasiewicz (Lw). Evitant les axiomes de contraction et de reduction propres a la logique classique de Dummett, l'A. propose une semantique de type Routley-Meyer pour le systeme d'Urquhart (CI) en tant que celle-la ne fournit que des theories consistantes pour la completude de celui-ci.

This book offers a unique synthesis of past and current work on the structure, meaning, and use of negation and negative expressions, a topic that has engaged thinkers from Aristotle and the Buddha to Freud and Chomsky. Horn's masterful study melds a review of scholarship in philosophy, psychology, and linguistics with original research, providing a full picture of negation in natural language and thought; this new edition adds a comprehensive preface and bibliography, surveying research since the book's (...) original publication. (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)

Let MK3 I and MK3 II be Kleene's strong 3-valued matrix with only one and two designated values, respectively. Next, let MK3 G be defined exactly as MK3 I, except th...

In game-theoretical semantics, perfectlyclassical rules yield a strong negation thatviolates tertium non datur when informationalindependence is allowed. Contradictorynegation can be introduced only by a metalogicalstipulation, not by game rules. Accordingly, it mayoccur (without further stipulations) onlysentence-initially. The resulting logic (extendedindependence-friendly logic) explains several regularitiesin natural languages, e.g., why contradictory negation is abarrier to anaphase. In natural language, contradictory negationsometimes occurs nevertheless witin the scope of aquantifier. Such sentences require a secondary interpretationresembling the so-called substitutionalinterpretation of quantifiers.This (...) interpretation is sometimes impossible,and it means a step beyond thenormal first-order semantics, not an alternative to it. (shrink)

The cognitive anthropological approach has provided a powerful means of beginning to understand religious representations. I suggest that two extant approaches, despite their general plausibility, may not accurately characterise the detailed nature of those representations. A major source of this inaccuracy lies in the characterisation of negation of ontological properties, which gives rise to broader questions about their ontological determinacy and counter-intuitiveness. I suggest that a more plausible account may be forthcoming by allowing a more complex approach to the (...) representations, deriving from understanding their nature as concept combinations. Such an account also suggests an alternative approach to the role of deference in religious representations. In sum, the empirical and theoretical implications of a more fine-grained analysis of religious representations suggest a vindication of the cognitive anthropology approach to integrating culture and cognition. (shrink)

The present paper is an attempt at the investigation of the nature of polarity contrast in natural languages. Truth conditions for natural language sentences are incomplete unless they include a proper definition of the conditions under which they are false. It is argued that the tertium non datur principle of classical bivalent logical systems is empirically invalid for natural languages: falsity cannot be equated with non-truth. Lacking a direct intuition about the conditions under which a sentence is (...) false, we need an independent foundation of the concept of falsity. The solution I offer is a definition of falsity in terms of the truth of a syntactic negation of the sentence. A definition of syntactic negation is proposed for English. (shrink)

Since antiquity two different negations in natural languages have been noted: predicate negation (not honest) and predicate term negation (dishonest). The extensive literature offers no models. We propose category-theoretic models with two distinct negation operators, neither of them in general Boolean. We study combinations of the two (not dishonest) and sentential counterparts of each. We emphasize the relevance of our work for the theory of cognition.

A cognitive pragmatic approach is taken to some long-standing problem cases of negation, the so-called presupposition denial cases. It is argued that a full account of the processes and levels of representation involved in their interpretation typically requires the sequential pragmatic derivation of two different propositions expressed. The first is one in which the presupposition is preserved and, following the rejection of this, the second involves the echoic (metalinguistic) use of material falling in the scope of the negation. (...) The semantic base for these processes is the standard anti-presuppositionalist wide-scope negation. A different view, developed by Burton-Roberts (1989a, 1989b), takes presupposition to be a semantic relation encoded in natural language and so argues for a negation operator that does not cancel presuppositions. This view is shown to be flawed, in that it makes the false prediction that presupposition denial cases are semantic contradictions and it is based on too narrow a view of the role of pragmatic inferencing. (shrink)

The operators of first-order logic, including negation, operate on whole formulae. This makes it unsuitable as a tool for the formal analysis of reasoning with non-sentential forms of negation such as predicate term negation. We extend its language with negation operators whose scope is more narrow than an atomic formula. Exploiting the usefulness of subatomic proof-theoretic considerations for the study of subatomic inferential structure, we define intuitionistic subatomic natural deduction systems which have several subatomic operators (...) and an additional operator for formula negation at their disposal. We establish normalization and subexpression property results for the systems. The normalization results allow us to formulate a proof-theoretic semantics for formulae composed of the subatomic operators. We illustrate the systems with applications to reasoning with combinations of sentential negation, predicate term negation, subject term negation, and antonymy. (shrink)

In this paper, a family of paraconsistent propositional logics with constructive negation, constructive implication, and constructive co-implication is introduced. Although some fragments of these logics are known from the literature and although these logics emerge quite naturally, it seems that none of them has been considered so far. A relational possible worlds semantics as well as sound and complete display sequent calculi for the logics under consideration are presented.

This paper introduces and explores a conservative extension of inquisitive logic. In particular, weak negation is added to the standard propositional language of inquisitive semantics, and it is shown that, although we lose some general semantic properties of the original framework, such an enrichment enables us to model some previously inexpressible speech acts such as weak denial and ‘might’-assertions. As a result, a new modal logic emerges. For this logic, a Fitch-style system of natural deduction is formulated. The (...) main result of this paper is a theorem establishing the completeness of the system with respect to inquisitive semantics with weak negation. At the conclusion of the paper, the possibility of extending the framework to the level of first order logic is briefly discussed. (shrink)

We have become estranged from the cosmic movements, according to Bataille. We are confined by the error linked to the representation of ‘the stationary earth’. We have negated the immersive immanence of the whole and made nature into a fixed world of tools and things. How then do we recognise ourselves as part of the ‘rapture of the heavens’? Bataille urges us to consider life as a solar phenomenon, the free play of solar energy on the earth. This paper argues (...) how Bataille’s cosmic vision anticipates both planetary and ecological thinking. This entails a base materialism of difference based on a radical Nietzschean and Heraclitean theory of becoming and flux. This thought is developed along Bataillean concepts such as the movement of the whole, the labyrinthine, and compound beings, ipseity, as well as the concept of a communifying movement. This materialism paired with Vladimir Vernadsky’s biosphere lays the foundation for the laws of the general economy, which also constitutes an economy inclusive of a general ecology: a tragic vision of Gaia and Dionysos. Finally, poetry is considered, as a way we might reconnect with the universe and stimulate ecological imagination. (shrink)

Many think that expressivists have a special problem with negation. I disagree. For if there is a problem with negation, I argue, it is a problem shared by those who accept some plausible claims about the nature of intentionality. Whether there is any special problem for expressivists turns, I will argue, on whether facts about what truth-conditions beliefs have can explain facts about basic inferential relations among those beliefs. And I will suggest that the answer to this last (...) question is, on most plausible attempts at solving the problem of intentionality, ‘no’. (shrink)

In this paper, I will first draw attention to the central fact that Freud describes in his article on “Negation”; i.e., “recognition of the unconscious on the part of the ego is expressed in a negative formula.” Technically, such a negative formula is called denial. Second, I will ask whether such denials are accidental or necessary in the life of consciousness. To answer this question I will use the philosophical system of Hegel. I will make use of his ideas (...) at two levels of generality. At the first level of generality, I will analyze one of Hegel’s most famous and influential passages: “Lordship and Bondage.” Hegel presents this passage as a necessary moment in the development of consciousness. If we find in that passage a necessary denial, then we can argue that denial is not just a fact but is, at least in any philosophical anthropology akin to Hegel’s, a necessity. At the second level of generality, we will then quote two statements by Hegel, one about the nature of the will and one about the path to truth for consciousness. Both statements will be clarified by Hegel’s analysis of “Lordship and Bondage.” Both statements will be used to demonstrate that Hegel affirmed explicitly the necessity of negativity in the life of consciousness. (shrink)

This essay proposes a procedural interpretation of negative information in terms of split negation as procedural prohibition. Information frames and models are introduced, with negation defined as the implication of bottom, 0. A method for extracting the procedures prohibited by complex formulas is outlined, and the relationship between types of prohibited procedures is identified. Definitions of negation types in terms of the implication of 0 on an informational interpretation have been criticized. This criticism turns on the definitions (...) creating a purportedly unnatural asymmetry between positive and negative information. It is demonstrated below that a strong asymmetry between positive and negative information is in fact the case. As such, an asymmetry between positive and negative information is natural, and something that we should want an informational interpretation of negation to preserve. (shrink)

In a series of articles, Kit Fine presents some highly compelling objections to monism, the doctrine that spatially coincident objects are identical. His objections rely on Leibniz’s Law and linguistic environments that appear to be immune to the standard charge of non-transparency and substitution failure. In this paper, I respond to Fine’s objections on behalf of the monist. Following Benjamin Schnieder, I observe that arguments from Leibniz’s Law are valid only if they involve descriptive, rather than metalinguistic, negation. Then (...) I show that the monist is justified in treating the negation in Fine’s objections as metalinguistic in nature. Along the way I make a few methodological remarks about the interaction between the study of natural language and metaphysics. I also present evidence that some of the linguistic environments which Fine relies on are, contrary to appearances, non-transparent. (shrink)

Frame semantics for negation on the Australian Plan accommodates many different negations, but it falls short on accommodating subminimal negation when the language contains conjunction and disjunction. In this paper, I will present a multi-relational frame semantics –multi-incompatibility frame semantics– that can accommodate subminimal negation. I will first argue that multi-incompatibility frames are in accordance with the philosophical motivations behind negation on the Australian Plan, namely its modal and exclusion-expressing nature. Then, I will prove the soundness (...) and completeness results of a subminimal logic that consists of the multi-incompatibility semantics and a proof system with operational rules that characterize subminimal negation, conjunction and disjunction. Lastly, I will prove some key correspondence theorems that relate frame conditions to certain principles that are associated with stronger negations, which will give rise to a new kite of negations that includes subminimal negation. (shrink)

In this article, I present a semantically natural conservative extension of Urquhart’s positive semilattice logic with a sort of constructive negation. A subscripted sequent calculus is given for this logic and proofs of its soundness and completeness are sketched. It is shown that the logic lacks the finite model property. I discuss certain questions Urquhart has raised concerning the decision problem for the positive semilattice logic in the context of this logic and pose some problems for further research.

The present essay includes six thematically connected papers on negation in the areas of the philosophy of logic, philosophical logic and metaphysics. Each of the chapters besides the first, which puts each the chapters to follow into context, highlights a central problem negation poses to a certain area of philosophy. Chapter 2 discusses the problem of logical revisionism and whether there is any room for genuine disagreement, and hence shared meaning, between the classicist and deviant's respective uses of (...) 'not'. If there is not, revision is impossible. I argue that revision is indeed possible and provide an account of negation as contradictoriness according to which a number of alleged negations are declared genuine. Among them are the negations of FDE and a wide family of other relevant logics, LP, Kleene weak and strong 3-valued logics with either "exclusion" or "choice" negation, and intuitionistic logic. Chapter 3 discusses the problem of furnishing intuitionistic logic with an empirical negation for adequately expressing claims of the form 'A is undecided at present' or 'A may never be decided' the latter of which has been argued to be intuitionistically inconsistent. Chapter 4 highlights the importance of various notions of consequence-as-s-preservation where s may be falsity, indeterminacy or some other semantic value, in formulating rationality constraints on speech acts and propositional attitudes such as rejection, denial and dubitability. Chapter 5 provides an account of the nature of truth values regarded as objects. It is argued that only truth exists as the maximal truthmaker. The consequences this has for semantics representationally construed are considered and it is argued that every logic, from classical to non-classical, is gappy. Moreover, a truthmaker theory is developed whereby only positive truths, an account of which is also developed therein, have truthmakers. Chapter 6 investigates the definability of negation as "absolute" impossibility, i.e. where the notion of necessity or possibility in question corresponds to the global modality. The modality is not readily definable in the usual Kripkean languages and so neither is impossibility taken in the broadest sense. The languages considered here include one with counterfactual operators and propositional quantification and another bimodal language with a modality and its complementary. Among the definability results we give some preservation and translation results as well. (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)

In the context of a distributive lattice we specify the sort of mappings that could be generally called ''negations'' and study their behavior under iteration. We show that there are periodic and nonperiodic ones. Natural periodic negations exist with periods 2, 3, and 4 and pace 2, as well as natural nonperiodic ones, arising from the interaction of interior and quasi interior mappings with the pseudocomplement. For any n and any even , negations of period n and pace (...) s can also be constructed, but in a rather ad hoc and trivial way. (shrink)

There are two natural ways of thinking about negation: (i) as a form of complementation and (ii) as an operation of reversal, or inversion (to deny that p is to say that things are “the other way around”). A variety of techniques exist to model conception (i), from Euler and Venn diagrams to Boolean algebras. Conception (ii), by contrast, has not been given comparable attention. In this note we outline a twofold geometric proposal, where the inversion metaphor is (...) understoood as involving a rotation o a reflection, respectively. These two options are equivalent in classical two-valued logic but they differ significantly in many-valued logics. Here we show that they correspond to two basic sorts of negation operators—Post’s and Kleene’s—and we provide a simple group-theoretic argument demonstrating their generative power. (shrink)

The influential Italian thinker offers three essays in the political philosophy of language. Multitude between Innovation and Negation by Paolo Virno translated by James Cascaito. The publication of Paolo Virno's first book in English, Grammar of the Multitude, by Semiotext in 2004 was an event within the field of radical political thought and introduced post-'68 currents in Italy to American readers. Multitude between Innovation and Negation, written several years later, offers three essays that take the reader on a (...) journey through the political philosophy of language. “Wit and Innovative Action” explores the ambivalence inevitably arising when the semiotic and the semantic, grammar and experience, rule and regularity, and right and fact intersect. Virno unravels the infinite potential and wonders of everyday linguistic praxis and ambiguity. Wit, he argues, is a public performance, and its modus operandi characterizes human action in a state of emergency; it is a reaction, an articulate response, and a possible solution to a state of crisis. “Mirror Neurons, Linguistic Negation, and Mutual Recognition” examines the relationship of language and intersubjective empathy: without language, would human beings be able to recognize other members of their species? And finally, in “Multitude and Evil,” Virno challenges the distinction between the state of nature and civil society and argues for a political institution that resembles language in its ability to be at once nature and history. Few thinkers take the risks required by innovation. Like a philosophical entrepreneur, Virno is engaged in no less than rewriting the dictionary of political theory, an urgent and ambitious project when language, caught in a permanent state of emergency impossible to sustain, desperately needs to articulate and enact new practices of freedom for the multitude. Paolo Virno is the author of several books, including A Grammar of the Multitude, 2004). (shrink)

There is a relatively recent trend in treating negation as a modal operator. One such reason is that doing so provides a uniform semantics for the negations of a wide variety of logics and arguably speaks to a longstanding challenge of Quine put to non-classical logics. One might be tempted to draw the conclusion that negation is a modal operator, a claim Francesco Berto, 761–793, 2015) defends at length in a recent paper. According to one such modal account, (...) the negation of a sentence is true at a world x just in case all the worlds at which the sentence is true are incompatible with x. Incompatibility is taken to be the key notion in the account, and what minimal properties a negation has comes down to which minimal conditions incompatibility satisfies. Our aims in this paper are twofold. First, we wish to point out problems for the modal account that make us question its tenability on a fundamental level. Second, in its place we propose an alternative, non-modal, account of negation as a contradictory-forming operator that we argue is superior to, and more natural than, the modal account. (shrink)

In this paper, we axiomatize the negatable consequences in dependence and independence logic by extending the systems of natural deduction of the logics given in [22] and [11]. We prove a characterization theorem for negatable formulas in independence logic and negatable sentences in dependence logic, and identify an interesting class of formulas that are negatable in independence logic. Dependence and independence atoms, first-order formulas belong to this class. We also demonstrate our extended system of independence logic by giving explicit (...) derivations for Armstrong's Axioms and the Geiger-Paz-Pearl axioms of dependence and independence atoms. (shrink)

Ever since Kant, the possibility of having objects of knowledge has been one of the most basic anthropological questions (“what can I know?”). For the logician, the linguist, or the semiologist who studies natural language, negation is one of these objects. However, as an operation and as a symbol, it has the paradoxical property of not being able to be objectivized in the discourse that treats it without being used in this construction. Of course, it is an entirely (...) general problem that it is not possible to have a “distant view” of symbolism without using a symbolism of some sort or another. Anthropology is essentially concerned with such objects. The descriptions of ethnologists are thus fertile ground for the epistemologist who believes, like Aristotle, that non-being is not and therefore that, even though in fact we speak only of things (otherwise, what good is speaking?) our discourse cannot be a copy of them. (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)

This short paper has two loosely connected parts. In the first part, I discuss the difference between classical and intuitionist logic in relation to different the role of hypotheses play in each logic. Harmony is normally understood as a relation between two ways of manipulating formulas in systems of natural deduction: their introduction and elimination. I argue, however, that there is at least a third way of manipulating formulas, namely the discharge of assumption, and that the difference between classical (...) and intuitionist logic can be characterised as a difference of the conditions under which discharge is allowed. Harmony, as ordinarily understood, has nothing to say about discharge. This raises the question whether the notion of harmony can be suitably extended. This requires there to be a suitable fourth way of manipulating formulas that discharge can stand in harmony to. The question is whether there is such a notion: what might it be that stands to discharge of formulas as introduction stands to elimination? One that immediately comes to mind is the making of assumptions. I leave it as an open question for further research whether the notion of harmony can be fruitfully extended in the way suggested here. In the second part, I discuss bilateralism, which proposes a wholesale revision of what it is that is assumed and manipulated by rules of inference in deductions: rules apply to speech acts – assertions and denials – rather than propositions. I point out two problems for bilateralism. First, bilaterlists cannot, contrary to what they claim to be able to do, draw a distinction between the truth and assertibility of a proposition. Secondly, it is not clear what it means to assume an expression such as '+ A' that is supposed to stand for an assertion. Worse than that, it is plausible that making an assumption is a particular speech act, as argued by Dummett (Frege: Philosophy of Language, p.309ff). Bilaterlists accept that speech acts cannot be embedded in other speech acts. But then it is meaningless to assume + A or − A. (shrink)

One of Da Costa’s motives when he constructed the paraconsistent logic C! was to dualise the negation of intuitionistic logic. In this paper I explore a different way of going about this task. A logic is defined by taking the Kripke semantics for intuitionistic logic, and dualising the truth conditions for negation. Various properties of the logic are established, including its relation to C!. Tableau and natural deduction systems for the logic are produced, as are appropriate algebraic (...) structures. The paper then investigates dualising the intuitionistic conditional in the same way. This establishes various connections between the logic, and a logic called in the literature ‘Brouwerian logic’ or ‘closed-set logic’. (shrink)

It is argued that the "inner" negation $\mathord{\sim}$ familiar from 3-valued logic can be interpreted as a form of "conditional" negation: $\mathord{\sim}$ is read '$A$ is false if it has a truth value'. It is argued that this reading squares well with a particular 3-valued interpretation of a conditional that in the literature has been seen as a serious candidate for capturing the truth conditions of the natural language indicative conditional (e.g., "If Jim went to the party (...) he had a good time"). It is shown that the logic induced by the semantics shares many familiar properties with classical negation, but is orthogonal to both intuitionistic and classical negation: it differs from both in validating the inference from $A \rightarrow \nega B$ to $\nega(A\rightarrow B)$ to. (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)

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)

Negation can be considered a shared social action that develops since early infancy with very basic acts of refusals or rejection. Inspired by an approach to the embodiment of concepts known as Multiple Representation Theories, the present paper explores negation as an embodied action that relies on both sensorimotor and linguistic/social information. Despite the different variants, MRT accounts share the basic ideas that both linguistic/social and sensorimotor information concur to the processes of concepts formation and representation and that (...) the balance between these components depends on the kind of concept, the context, or the performed task. In the present research we will apply the MRT framework for exploring negation in Italian sign language. The nature of negation in LIS has been explored in continuity with the co-speech gesture where negative elements are encoded through differentiated prosodic and gestural strategies across languages. Data have been collected in naturalistic settings that may allow a much wider understanding of negation both in speech and in spoken language with a semi-structured interview. Five LIS participants with age range 30–80 were recruited and interviewed with the aim of understanding the continuity between gesture and sign in negation. Results highlight that negation utterances mirror the functions of rejection, non-existence and denial that have been described in language acquisition both in deaf and hearing children. These different steps of acquisition of negation show a different balance between sensorimotor, linguistic and social information in the construction of negative meaning that the MRT is able to enlighten. (shrink)

The concepts of choice, negation, and infinity are considered jointly. The link is the quantity of information interpreted as the quantity of choices measured in units of elementary choice: a bit is an elementary choice between two equally probable alternatives. “Negation” supposes a choice between it and confirmation. Thus quantity of information can be also interpreted as quantity of negations. The disjunctive choice between confirmation and negation as to infinity can be chosen or not in turn: This (...) corresponds to set-theory or intuitionist approach to the foundation of mathematics and to Peano or Heyting arithmetic. Quantum mechanics can be reformulated in terms of information introducing the concept and quantity of quantum information. A qubit can be equivalently interpreted as that generalization of “bit” where the choice is among an infinite set or series of alternatives. The complex Hilbert space can be represented as both series of qubits and value of quantum information. The complex Hilbert space is that generalization of Peano arithmetic where any natural number is substituted by a qubit. “Negation”, “choice”, and “infinity” can be inherently linked to each other both in the foundation of mathematics and quantum mechanics by the meditation of “information” and “quantum information”. (shrink)

Multitude between Innovation and NegationPaolo Virnotranslated by James CascaitoThe publication of Paolo Virno's first book in English, Grammar of the Multitude, by Semiotext in 2004 was an event within the field of radical political thought and introduced post-'68 currents in Italy to American readers. Multitude between Innovation and Negation, written several years later, offers three essays that take the reader on a journey through the political philosophy of language."Wit and Innovative Action" explores the ambivalence inevitably arising when the semiotic (...) and the semantic, grammar and experience, rule and regularity, and right and fact intersect. Virno unravels the infinite potential and wonders of everyday linguistic praxis and ambiguity. Wit, he argues, is a public performance, and its modus operandi characterizes human action in a state of emergency; it is a reaction, an articulate response, and a possible solution to a state of crisis. "Mirror Neurons, Linguistic Negation, and Mutual Recognition" examines the relationship of language and intersubjective empathy: without language, would human beings be able to recognize other members of their species? And finally, in "Multitude and Evil," Virno challenges the distinction between the state of nature and civil society and argues for a political institution that resembles language in its ability to be at once nature and history.Few thinkers take the risks required by innovation. Like a philosophical entrepreneur, Virno is engaged in no less than rewriting the dictionary of political theory, an urgent and ambitious project when language, caught in a permanent state of emergency impossible to sustain, desperately needs to articulate and enact new practices of freedom for the multitude.Paolo Virno is the author of several books, including A Grammar of the Multitude, 2004). (shrink)

In the present paper we analyse how intuitionist negation relates to the concept of triviality. A proposition is trivial if it implies every other proposition. We claim that for natural deduction intuitionist logic such concept affords us a negation introduction rule schema that corresponds by the inversion principle to the usual negation elimination shcema, ex contradictionerm quodlibet.1.

Constructive logic with Nelson negation is an extension of the intuitionistic logic with a special type of negation expressing some features of constructive falsity and refutation by counterexample. In this paper we generalize this logic weakening maximally the underlying intuitionistic negation. The resulting system, called subminimal logic with Nelson negation, is studied by means of a kind of algebras called generalized N-lattices. We show that generalized N-lattices admit representation formalizing the intuitive idea of refutation by means (...) of counterexamples giving in this way a counterexample semantics of the logic in question and some of its natural extensions. Among the extensions which are near to the intuitionistic logic are the minimal logic with Nelson negation which is an extension of the Johansson's minimal logic with Nelson negation and its in a sense dual version — the co-minimal logic with Nelson negation. Among the extensions near to the classical logic are the well known 3-valued logic of Lukasiewicz, two 12-valued logics and one 48-valued logic. Standard questions for all these logics — decidability, Kripke-style semantics, complete axiomatizability, conservativeness are studied. At the end of the paper extensions based on a new connective of self-dual conjunction and an analog of the Lukasiewicz middle value ½ have also been considered. (shrink)

The paper is concerned with negation in artificial and natural languages. "Negation" is an ambiguous word. It can mean three different things: An operation(negating), an operator (a sign of negation), the result of an operation. The threethings, however, are intimately linked. An operation such as negation, is realizedthrough an operator of negation, i.e. consists in adding a symbol of negation to an entity to obtain an entity of the same type; and which operation (...) it is dependson what it applies to and on what results from its application.I argue that negation is not an operation on linguistic acts but rather anoperation on the objects of linguistic acts, namely sentences. And I assume that the negation of a sentence is a sentence that contradicts it. If so, the negation of a sentence may be obtained, in case the sentence is molecular, by applying the operation of negation not to the sentence itself but to a constituent sentence. To put it in a succinct and paradoxically sounding way we could say that in order to negate a sentence it is sufficient but not necessary to negate it.However that negation applies to sentences is true only for artificial languages, in which the sign of negation is a monadic sentential connective. In natural language, negation applies to expressions other than sentences, namely word sand non-sentential phrases. Still words and not sentential phrases are interesting and valuable only as ultimate or immediate constituents of sentences, as a means of saying (something that can be true or false) and the concern with negation is ultimately the concern with the negation of sentences. So the problem is what sub-sentential and non sentential expressions negation should apply to in order to obtain the negation of the containing sentence. The standard answer is that the negation of a natural language sentence is equivalent to the negation of its predicate. Yet, I argue, predicate negation is necessary but not sufficient, due to the existence of molecular sentences.Finally I notice that if to apply negation to an artificial sentence is to put the negation sign in front of it, to negate the predicate of a natural language sentencemay or may not be to put the negation sign in front of it. (shrink)