We present and defend the Australian Plan semantics for negation. This is a comprehensive account, suitable for a variety of different logics. It is based on two ideas. The first is that negation is an exclusion-expressing device: we utter negations to express incompatibilities. The second is that, because incompatibility is modal, negation is a modal operator as well. It can, then, be modelled as a quantifier over points in frames, restricted by accessibility relations representing compatibilities and incompatibilities (...) between such points. We defuse a number of objections to this Plan, raised by supporters of the American Plan for negation, in which negation is handled via a many-valued semantics. We show that the Australian Plan has substantial advantages over the American Plan. (shrink)

In this paper, new evidence is presented for the assumption that the reason-relation reading of indicative conditionals ('if A, then C') reflects a conventional implicature. In four experiments, it is investigated whether relevance effects found for the probability assessment of indicative conditionals (Skovgaard-Olsen, Singmann, and Klauer, 2016a) can be classified as being produced by a) a conversational implicature, b) a (probabilistic) presupposition failure, or c) a conventional implicature. After considering several alternative hypotheses and the accumulating evidence from other studies as (...) well, we conclude that the evidence is most consistent with the Relevance Effect being the outcome of a conventional implicature. This finding indicates that the reason-relation reading is part of the semantic content of indicative conditionals, albeit not part of their primary truth-conditional content. (shrink)

Here is one argument against realism. (1) Realists are committed to the classical rules for negation. But (2) legitimate rules of inference must conserve evidence. And (3) the classical rules for negation do not conserve evidence. So (4) realism is wrong. Most realists reject 2. But it has recently been argued that if we allow denied sentences as premisses and conclusions in inferences we will be able to reject 3. And this new argument against 3 generates a new (...) response to the antirealist argument: keep 1 and 2, avoiding 4 by rejecting 3. My aim in this paper is to see how much work in the fight against anti-realism this new response can really do. I argue that there is a powerful objection to the response: 2 is in tension with the claim that denied sentences can be premisses and conclusions in inferences. But I show that, even given this objection, the new response has an important role to play. (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)

No detailed description available for "Negation".

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)

Many relevant logics are conservatively extended by Boolean negation. Not all, however. This paper shows an acute form of non-conservativeness, namely that the Boolean-free fragment of the Boolean extension of a relevant logic need not always satisfy the variable-sharing property. In fact, it is shown that such an extension can in fact yield classical logic. For a vast range of relevant logic, however, it is shown that the variable-sharing property, restricted to the Boolean-free fragment, still holds for the Boolean (...) extended logic. (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.

An extension of intuitionism to empirical discourse, a project most seriously taken up by Dummett and Tennant, requires an empirical negation whose strength lies somewhere between classical negation (‘It is unwarranted that... ’) and intuitionistic negation (‘It is refutable that... ’). I put forward one plausible candidate that compares favorably to some others that have been propounded in the literature. A tableau calculus is presented and shown to be strongly complete.

Minimal Negation is defined within the basic positive relevance logic in the relational ternary semantics: B+. Thus, by defining a number of subminimal negations in the B+ context, principles of weak negation are shown to be isolable. Complete ternary semantics are offered for minimal negation in B+. Certain forms of reductio are conjectured to be undefinable (in ternary frames) without extending the positive logic. Complete semantics for such kinds of reductio in a properly extended positive logic are (...) offered. (shrink)

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)

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)

Aristotle draws what are, by our lights, two unusual relationships between predication and existence. First, true universal affirmations carry existential import. If ‘All humans are mortal’ is true, for example, then at least one human exists. And secondly, although affirmations with empty terms in subject position are all false, empty negations are all true: if ‘Socrates’ lacks a referent, then both ‘Socrates is well’ and ‘Socrates is ill’ are false but both ‘Socrates is not well’ and ‘Socrates is not ill’ (...) are true. In this paper, I conjecture that for Aristotle predications have mereological truth conditions: for example, ‘Socrates is pale’ is true just in case Socrates is a part of the mereological sum of pale things. The existential import of universal affirmations and the semantic profile of empty negations follow from this mereological semantics. (shrink)

Many relevant logics can be conservatively extended by Boolean negation. Mares showed, however, that E is a notable exception. Mares’ proof is by and large a rather involved model-theoretic one. This paper presents a much easier proof-theoretic proof which not only covers E but also generalizes so as to also cover relevant logics with a primitive modal operator added. It is shown that from even very weak relevant logics augmented by a weak K-ish modal operator, and up to the (...) strong relevant logic R with a S5 modal operator, all fail to be conservatively extended by Boolean negation. The proof, therefore, also covers Meyer and Mares’ proof that NR—R with a primitive S4-modality added—also fails to be conservatively extended by Boolean negation. (shrink)

This chapter is concerned with logical aspects of negation, i.e. with the role of negation in valid inferences and hence with the contribution negation makes to the truth and falsity conditions of declarative expressions. Negation is an important philosophical and logical concept. Often differences between logical systems can ‐ at least partially ‐ be described as differences between the notions of negation used in these logics.

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)

Negation is a cornerstone of human language and one of the few universals found in all languages. Without negation, neither categorization nor efficient communication would be possible. Languages, however, differ remarkably in how they express negation. It is yet widely unknown how the way negation is marked influences the acquisition process of first language learners. Here, we investigate whether universal or language-specific cues are more relevant for the acquisition process. We test to what extent frequency and (...) salience features (morphosyntactic boundedness, position of the negation marker, allomorphy) influence the acquisition of negation. We use naturalistic longitudinal data from 17 children (age 24–36 months) learning nine typologically maximally diverse languages spoken in very diverse cultural contexts ranging from western urban to subsistence farming: Chintang, English, Indonesian, Japanese, Qaqet, Russian, Sesotho, Turkish, and Yucatec Mayan. Distributional analyses show that the amount and type of input of negation that children hear vary considerably across cultures. Despite significant differences in the input that children receive, we observe a universal pattern in the acquisition of negation: Children transition from relatively easy and salient free negators to less salient bound negator morphemes in their use of negation. Our results show that frequency and morphosyntactic boundedness explain the development of flexibility in the acquisition of negation across all of the nine languages. We further find that the developmental path is shaped by the interaction between frequency and language-specific parameters of salience that are contingent on grammatical features of negation marking in different languages, such as the position of the negation marker and allomorphic variation. Our language-specific findings highlight cross-linguistic variation, which reflects cross-cultural differences in the amount of input of negative utterances children receive. Taken together, this study provides cross-linguistic evidence for the acquisition of negation and emphasizes the interplay of universal and language-specific factors in the acquisition process. (shrink)

We present a case study for the debate between the American and the Australian plans, analyzing a crucial aspect of negation: expressivity within a theory. We discuss the case of non-classical set theories, presenting three different negations and testing their expressivity within algebra-valued structures for ZF-like set theories. We end by proposing a minimal definitional account of negation, inspired by the algebraic framework discussed.

This paper addresses the two interpretations that a combination ofnegative indefinites can get in concord languages like French:a concord reading, which amounts to a single negation, and a doublenegation reading. We develop an analysis within a polyadic framework,where a sequence of negative indefinites can be interpreted as aniteration of quantifiers or via resumption. The first option leadsto a scopal relation, interpreted as double negation. The secondoption leads to the construction of a polyadic negative quantifiercorresponding to the concord reading. (...) Given that sentential negationparticipates in negative concord, we develop an extension of thepolyadic approach which can deal with non-variable binding operators,treating the contribution of negation in a concord context assemantically empty. Our semantic analysis, incorporated into agrammatical analysis formulated in HPSG, crucially relies on theassumption that quantifiers can be combined in more than one wayupon retrieval from the quantifier store. We also considercross-linguistic variation regarding the participation ofsentential negation in negative concord. (shrink)

A recurring narrative in the literature is that the empirical facts about negated conditionals provide compelling evidence for the principle of Conditional Excluded Middle and sit uncomfortably with a large family of analyses of conditionals as universal quantifiers over possible worlds. I show that both parts of the narrative are in need of a rewrite by articulating an innovative conditional analysis in a bilateral semantic setting that takes inspiration from the Ramsey test for conditionals but distinguishes the classical Ramseyan question (...) of what it takes to _accept _a conditional from the one of what it takes to _reject_ a conditional. The resulting framework disentangles the empirical facts about negated conditionals from the validity of Conditional Excluded Middle but also shows how the principle can live happily in a strict analysis of conditionals, and how it can co-exist with other non-classical principles such as Simplification of Disjunctive Antecedents without negative side-effects. (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)

There are many kinds of negation and denial. Perhaps the most common is the Boolean negation not that applies to propositions-in-extension, i.e. truth-values. The others are, inter alia, the property of propositions of not being true which applies to propositions; the complement function which applies to sets; privation which applies to properties; negation as failure applied in logic programming; negation as argumentation ad absurdum, and many others. The goal of this paper is neither to provide a (...) complete list, nor to analyse all of them. Rather, I am going to deal with negation of propositions that come attached with a presupposition that is entailed by the positive as well as negated form of a given proposition. However, there are two kinds of negation, namely internal and external negation. I am going to prove that while the former is presupposition-preserving, the latter is presupposition-denying. This issue has much in common with the difference between topic and focus articulation within a sentence. Whereas articulating the topic of a sentence activates a presupposition, articulating the focus frequently yields merely an entailment. The main contribution of this paper is the proof that the two kinds of negation are not equivalent. While the Russellian wide-scope (external) negation gets the truthconditions of a sentence right for a subject occurring as a focus, Strawsonian narrow-scope (internal) negation is validly applicable for a subject occurring as the topic. I also deal with other kinds of presupposition triggers, in particular factive attitudes and prerequisites of a given property. My background theory is Transparent Intensional Logic (TIL). TIL is an expressive logic apt for the analysis of sentences with presuppositions, because in TIL we work with partial functions, in particular with propositions with truth-value gaps. Moreover, the procedural semantics of TIL make it possible to uncover the hidden semantic features of sentences, make them explicit and logically tractable. (shrink)

It is known that many relevant logics can be conservatively extended by the truth constant known as the Ackermann constant. It is also known that many relevant logics can be conservatively extended by Boolean negation. This essay, however, shows that a range of relevant logics with the Ackermann constant cannot be conservatively extended by a Boolean negation.

A theory of substructural negations as impossibility and as unnecessity based on bi-intuitionistic logic, also known as Heyting-Brouwer logic, has been developed by Takuro Onishi. He notes two problems for that theory and offers the identification of the two negations as a solution to both problems. The first problem is the lack of a structural rule corresponding with double negation elimination for negation as impossibility, DNE, and the second problem is a lack of correspondence between certain sequents and (...) a characterizing frame property. While the identification of negation as impossibility and negation as unnecessity does solve Onishi’s problems, in general it nevertheless seems desirable to keep the two notions separate. The present paper addresses the first problem by introducing a reformulation of Onishi’s display sequent calculus in a language of sequents that incorporates Boolean negation. Instead of identifying negation as impossibility and negation as unnecessity, a notion of weak frame correspondence is defined. It is observed that DNE weakly corresponds with a certain frame property, and two structural sequent rules are presented that together also weakly correspond with that frame property and allow one to derive DNE. Moreover, the reformulated display calculus has an independent motivation by considerations on proof-theoretic semantics. (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)

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)

It has been said that, when some paraconsistent logicians supporting dialetheism assert: "For some sentence α, both α and not-α are true", therefore claiming that the Law of Non-Contradiction (LNC) fails, we should wonder what "true" and "not" mean here. After surveying two classical paraconsistent approaches to negation (provided by da Costa's positive-plus systems and Graham Priest's Logic of Paradox), I describe a negation with the following features: (1) its definition does not make reference to the controversial concept (...) truth; (2) it has strong pre-theoretical appeal and motivation, because it performs an indispensable expressive function in language and communication; and (3) it is accepted by dialetheists, too, since it is based on a very deep metaphysical intuition they also show to fully share: this intuition I call the one of material exclusion. If my characterization is sufficient to confer a determinate meaning to the negation in question, we can conveniently formulate via this negation a version of the LNC which I take, therefore, to be indisputable also from the dialetheist's point of view. Such a result, however, does not constitute a quick and easy success against supporters of true contradictions. It may simply show that the versions of the LNC dialetheists most convincingly attack are those that were not to be defended, and that supporters of consistency have been historically confused in assimilating them to the indisputable one. Quine's famous argument, that to change the logic is to change the subject, may be right to this extent: classical negation and non-classical negations have different meanings. But the substantial issue that Quine never addressed is why we should suppose that the meaning of the vernacular negation is classical. © 2011 Elsevier B.V., All rights reserved. (shrink)

Here is an account of recent investigations into the two main concepts of negation developed in the constructive logic: the negation as reduction to absurdity, and the strong negation. These concepts are studied in the setting of paraconsistent logic.

Four weak positional calculi are constructed and examined. They refer to the use of the connective of negation within the scope of the positional connective “R” of realization. The connective of negation may be fully classical, partially analogical or independent from the classical, truth-functional negation. It has been also proved that the strongest system, containing fully classical connective of negation, is deductively equivalent to the system MR from Jarmużek and Pietruszczak.

Jc Beall argues that if FDE is logic proper, then there is no logical negation. This claim is largely based on the fact that, in standard proof systems for FDE, there are no stand-alone negation rules that suffice to capture the behavior of negation. In this paper, I show that by adopting a bilateral proof system for FDE, one can maintain that there is a logical negation, it is the very same logical negation that belongs (...) to classical logic, and its basic function is to flip-flop between assertion and denial. After laying out the bilateral proof systems on which this claim is technically based, I develop the conception of assertion and denial on which this claim is philosophically based, responding to a number of objections. I conclude by considering the possibility of a different, so-called "Boolean" negation, in the context of this bilateral framework. (shrink)

Negation, denial and falsity lie at the heart of debates about logic. We set out the classical account of the relationship between negation and denial, owing to Frege and Geach. We then challenge this on the basis that it does not permit an adequate account of falsity. A dialetheic alternative is minuted and criticised before a novel rejectivist account is proposed according to which falsity is the aim of the speech‐act of denial, whilst negation embeds deniability into (...) assertoric contexts. (shrink)

There is a widely held view that visual representations (images) do not depict negation, for example, as expressed by the sentence, “the train is not coming.” The present study focuses on the real-world visual representations of photographs and comic (manga) illustrations and empirically challenges the question of whether humans and machines, that is, modern deep neural networks, can recognize visual representations as expressing negation. By collecting data on the captions humans gave to images and analyzing the occurrences of (...) negation phrases, we show some evidence that humans recognize certain images as expressing negation. Furthermore, based on this finding, we examined whether or not humans and machines can classify novel images as expressing negation. The humans were able to correctly classify images to some extent, as expected from the analysis of the image captions. On the other hand, the machine learning model of image processing was only able to perform this classification at about the chance level, not at the same level of performance as the human. Based on these results, we discuss what makes humans capable of recognizing negation in visual representations, highlighting the role of the background commonsense knowledge that humans can exploit. Comparing human and machine learning performances suggests new ways to understand human cognitive abilities and to build artificial intelligence systems with more human-like abilities to understand logical concepts. (shrink)

In this paper, we shall consider the so-called cancellation view of negation and the inferential role of contradictions. We will discuss some of the problematic aspects of negation as cancellation, such as its original presentation by Richard and Valery Routley and its role in motivating connexive logic. Furthermore, we will show that the idea of inferential ineffectiveness of contradictions can be conceptually separated from the cancellation model of negation by developing a system we call qLPm, a combination (...) of Graham Priest’s minimally inconsistent Logic of Paradox with q-entailment as introduced by Grzegorz Malinowski. (shrink)

Of the various accounts of negation that have been offered by logicians in the history of Western logic, that of negation as cancellation is a very distinctive one, quite different from the explosive accounts of modern "classical" and intuitionist logics, and from the accounts offered in standard relevant and paraconsistent logics. Despite its ancient origin, however, a precise understanding of the notion is still wanting. The first half of this paper offers one. Both conceptually and historically, the account (...) of negation as cancellation is intimately connected with connexivist principles such as ¬( ¬). Despite this, standard connexivist logics incorporate quite different accounts of negation. The second half of the paper shows how the cancellation account of negation of the first part gives rise to a semantics for a simple connexivist logic. (shrink)

At least since [Frege, 1960] and [Geach, 1965], there has been some consensus about the relation between negation, the speech act of denial, and the attitude of rejection: a denial, the consensus has had it, is the assertion of a negation, and a rejection is a belief in a negation. Recently, though, there have been notable deviations from this orthodox view. Rejectivists have maintained that negation is to be explained in terms of denial or rejection, rather (...) than vice versa. Some other theorists have maintained that negation is a separate phenomenon from denial, and that neither is to be explained in terms of the other. In this paper, I present and consider these heterodox theories of the relation between negation, denial, and rejection. (shrink)

Infinity and negation are in various relations and interdependencies one to another. The analysis of negation and infinity aims to better understanding them. Semantical, syntactical, and pragmatic issues will be considered.

This is only a very short essay on negation and harmony in philosophical logic. If you buy it anyway, you'll help me pay the bills and I'll be able to write longer things.

In this article, I discuss Alain Badiou’s 2008 address titled “The Three Negations.” Though the text was originally presented in a symposium concerning the relationship of law to Badiou’s theory of the event, I discuss the way this brief address offers an introduction to the broad sweep of Badiou’s metaphysics, outlining his accounts of being, appearing, and transformation. To do so, Badiou calls on the resources of three paradigms of negation: from classical Aristotelian logic, from Brouwer’s intuitionist logic, and (...) in paraconsistent logics developed by DaCosta. I explain Badiou’s use of negation in the three primary areas of his metaphysics, as well as to diagnose the degrees of transformation that may have occurred in a situation. My analysis of Badiou’s use of negation in this text is aided by examples from his broader ontological oeuvre. I also explain the underlying requirement in Badiou’s work that formal considerations - mathematical or logical - get their sense by being tethered to readily-identifiable political, aesthetic, scientific, or interpersonal concerns. I conclude by addressing the foundation Badiou’s work establishes for pursuing a new metaphysics, and by discussing certain of the liabilities that remain in the wake of his account. (shrink)