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).

This book is the first in the field of paraconsistency to offer a comprehensive overview of the subject, including connections to other logics and applications in information processing, linguistics, reasoning and argumentation, and philosophy of science. It is recommended reading for anyone interested in the question of reasoning and argumentation in the presence of contradictions, in semantics, in the paradoxes of set theory and in the puzzling properties of negation in logic programming. Paraconsistent logic comprises a major (...) logical theory and offers the broadest possible perspective on the debate of negation in logic and philosophy. It is a powerful tool for reasoning under contradictoriness as it investigates logic systems in which contradictory information does not lead to arbitrary conclusions. Reasoning under contradictions constitutes one of most important and creative achievements in contemporary logic, with deep roots in philosophical questions involving negation and consistency This book offers an invaluable introduction to a topic of central importance in logic and philosophy. It discusses the history of paraconsistent logic; language, negation, contradiction, consistency and inconsistency; logics of formal inconsistency and the main paraconsistent propositional systems; many-valued companions, possible-translations semantics and non-deterministic semantics; paraconsistent modal logics; first-order paraconsistent logics; applications to information processing, databases and quantum computation; and applications to deontic paradoxes, connections to Eastern thought and to dialogical reasoning. (shrink)

The paper discusses properties of 4QL, a DATALOG⌉⌉-like query language, originally outlined by Maluszyński and Szalas (Maluszyński & Szalas, 2011). 4QL allows one to use rules with negation in heads and bodies of rules. It is based on a simple and intuitive semantics and provides uniform tools for “lightweight” versions of known forms of nonmonotonic reasoning. Negated literals in heads of rules may naturally lead to inconsistencies. On the other hand, rules do not have to attach meaning to some (...) literals. Therefore 4QL is founded on a four-valued semantics, employing the logic introduced in (Maluszyński et al., 2008; Vitória et al., 2009) with truth values: ‘true’, ‘false’, ‘inconsistent’ and ‘unknown’. In addition, 4QL is tractable w.r.t. data complexity and captures PTIME queries. Even though DATALOG⌉⌉ is known as a concept for the last 30 years, to our best knowledge no existing approach enjoys these properties.In the current paper we:investigate properties of well-supported models of 4QLprove the correctness of the algorithm for computing well-supported modelsshow that 4QL has PTIME data complexity and captures PTIME. (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 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 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)

Review of the book "Paraconsistent Logic: Consistency, Contradiction, and Negation by Water Carnielli and Marcelo Coniglio.

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)

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)

This paper is a reaction to the following remark by grzegorczyk: "the compound sentences are not a product of experiment. they arise from reasoning. this concerns also negations; we see that the lemon is yellow, we do not see that it is not blue." generally, in science the truth is ascertained as indirectly as falsehood. an example: a litmus-paper is used to verify the sentence "the solution is acid." this approach gives rise to a (very intuitionistic indeed) conservative extension of (...) the heyting logic satisfying natural duality laws. (shrink)

The goal of this two-part series of papers is to show that constructive logic with strong negation N is definitionally equivalent to a certain axiomatic extension NFL ew of the substructural logic FL ew . In this paper, it is shown that the equivalent variety semantics of N (namely, the variety of Nelson algebras) and the equivalent variety semantics of NFL ew (namely, a certain variety of FL ew -algebras) are term equivalent. This answers a longstanding question (...) of Nelson [30]. Extensive use is made of the automated theorem-prover Prover9 in order to establish the result. The main result of this paper is exploited in Part II of this series [40] to show that the deductive systems N and NFL ew are definitionally equivalent, and hence that constructive logic with strong negation is a substructural logic over FL ew. (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 (...) class='Hi'>logic by giving explicit derivations for Armstrong's Axioms and the Geiger-Paz-Pearl axioms of dependence and independence atoms. (shrink)

The goal of this two-part series of papers is to show that constructive logic with strong negation N is definitionally equivalent to a certain axiomatic extension NFL ew of the substructural logic FL ew. The main result of Part I of this series [41] shows that the equivalent variety semantics of N and the equivalent variety semantics of NFL ew are term equivalent. In this paper, the term equivalence result of Part I [41] is lifted to the (...) setting of deductive systems to establish the definitional equivalence of the logics N and NFL ew. It follows from the definitional equivalence of these systems that constructive logic with strong negation is a substructural logic. (shrink)

Does there exist any equivalence between the notions of inconsistency and consequence in paraconsistent logics as is present in the classical two valued logic? This is the key issue of this paper. Starting with a language where negation ( ${\neg}$ ) is the only connective, two sets of axioms for consequence and inconsistency of paraconsistent logics are presented. During this study two points have come out. The first one is that the notion of inconsistency of paraconsistent logics turns (...) out to be a formula-dependent notion and the second one is that the characterization (i.e. equivalence) appears to be pertinent to a class of paraconsistent logics which have double negation property. (shrink)

The logic B+ is Routley and Meyer’s basic positive logic. We define the logics BK+ and BK'+ by adding to B+ the K rule and to BK+ the characteristic S4 axiom, respectively. These logics are endowed with a relatively strong non-constructive negation. We prove that all the logics defined lack the K axiom and the standard paradoxes of consistency.

A star-free relational semantics for relevant logic is presented together with a sound and complete sequent proof theory. It is an extension of the dualist approach to negation regarded as modality, according to which de Morgan negation in relevant logic is better understood as the confusion of two negative modalities. The present work shows a way to define them in terms of implication and a new connective, co-implication, which is modeled by respective ternary relations. The defined (...) negations are confused by a special constraint on ternary relation, called the generalized star postulate, which implies definability of the Routley star in the frame. The resultant logic is shown to be equivalent to the well-known relevant logic R. Thus it can be seen as a reconstruction of R in the dualist framework. (shrink)

Here are considered the conditions under which the method of diagrams is liable to include non-classical logics, among which the spatial representation of non-bivalent negation. This will be done with two intended purposes, namely: a review of the main concepts involved in the definition of logical negation; an explanation of the epistemological obstacles against the introduction of non-classical negations within diagrammatic logic.

As part of our small contribution in dialogue toward better peace development and reconciliation studies, and following Toffler & Toffler’s War and Antiwar (1993), the present article delves into a realm of logic beyond the traditional confines of negation and the excluded middle principle, exploring the nuances of "Otherness" that transcend classical and Nagatomo logics. Departing from the foundational premises of classical Aristotelian logic systems, this exploration ventures into alternative realms of reasoning, specifically examining Neutrosophic Logic (...) and Klein bottle logic (cf. Smarandache, 2005). The study challenges conventional boundaries and explores the implications of embracing paradoxes and self-reference in logic systems, aiming to redefine approaches to understanding truth and reasoning. The paper investigates how these alternative logics open avenues for philosophical inquiry, redefining entropy, and potentially influencing innovative perspectives in free energy systems. Through this exploration, it seeks to expand the discourse on logic, welcoming a broader spectrum of thought beyond established frameworks; and we also discuss shortly a number of possible implementations including in risk management and also Klein bottle entropy redefinition (Tang et al, 2018). (shrink)

This paper uses the strengthened liar paradox as a springboard to illuminate two more general topics: i) the negation operator and the speech act of denial among speakers of English and ii) some ways the potential for acceptable language change is constrained by linguistic meaning. The general and special problems interact in reciprocally illuminating ways. The ultimate objective of the paper is, however, less to solve certain problems than to create others, by illustrating how the issues that form the (...) topic of this paper are more intricate than previously realised, and that they are related in delicate and somewhat surprising ways. (shrink)

This paper examines the relationships between the many-valued logics G~ and Gn~ of Esteva, Godo, Hajek, and Navara, i.e., Godel logic G enriched with Łukasiewicz negation, and neighbors of intuitionistic logic. The popular fragments of Rauszer's Heyting-Brouwer logic HB admit many-valued extensions similar to G which may likewise be enriched with Łukasiewicz negation; the fuzzy extensions of these logics, including HB, are equivalent to G ~, as are their n-valued extensions equivalent to Gn~ for any (...) n ≥ 2. These enriched systems extend Wansing's logic I4C4, showing that Łukasiewicz negation is a species of Nelson's negation of constructible falsity and yielding a Kripke-style semantics for G~ and Gn~ to complement the many-valued semantics. (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)

We present a meta-analytic review on the processing of negations in conditional reasoning about affirmation problems (Modus Ponens: "MP", Affirmation of the Consequent "AC") and denial problems (Denial of the Antecedent "DA", and Modus Tollens "MT"). Findings correct previous generalisations about the phenomena. First, the effects of negation in the part of the conditional about which an inference is made, are not constrained to denial problems. These inferential-negation effects are also observed on AC. Second, there generally are reliable (...) effects of a negation in the clause referred to by the categorical premise, and these referred-negation effects are constrained to the logically fallacious AC and DA inferences. All findings are presented and discussed in relation to contemporary mental model (MM) and mental logic (ML) theories. It is argued that a double-negation elimination hypothesis provides a sufficient explanation of inferential-negation effects within both MM theory and ML theory, if the latter is extended by a validating search for counter examples. Both MM and ML theories adhere to a processing scheme that allows them to incorporate an account of referred-negation effects based on the thesis that counter-example frequency is modulated by the scope of a contrast class delineated by a false affirmative. We conclude that MM and ML theories provide adequate processing schemes to accommodate for the explanatory hypotheses, at least in principle. In practice, both approaches remain equivocal as regards the connectivity and interactivity with long-term memory knowledge invoked in generating, manipulating, and testing the mental representations of negative state of affairs. (shrink)

In [1] J.-Y. Bèziau formulated a logic called Z. Bèziau’s idea was generalized independently in [6] and [7]. A family of logics to which Z belongs is denoted in [7] by K. In particular; it has been shown in [6] and [7] that there is a correspondence between normal modal logics and logics from the class K. Similar; but only partial results has been obtained also for regular logics. In a logic N has been investigated in the language (...) with negation; implication; conjunction and disjunction by axioms of positive intuitionistic logic; the right-to-left part of the second de Morgan law; and the rules of modus ponens and contraposition. From the semantical point of view the negation used by Došen is the modal operator of impossibility. It is known this operator is a characteristic of the modal interpretation of intuitionistic negation. In the present paper we consider an extension of N denoted by N+. We will prove that every extension of N+ that is closed under the same rules as N+; corresponds to a regular logic being an extension of the regular deontic logic D21. The proved correspondence allows to obtain from soundnesscompleteness result for any given regular logic containing D2, similar adequacy theorem for the respective extension of the logic N+. (shrink)

This chapter contains sections titled: Negation and Nonexistence Designation and Existence Logical Truth, Modality, and Ontology.

I try to explain the difference between three kinds of negation: external negation, negation of the predicate and privation. Further I use polygons of opposition as heuristic devices to show that a logic which contains all three mentioned kinds of negation must be a fragment of a Łukasiewicz-four-valued predicate logic. I show, further, that, this analysis can be elaborated so as to comprise additional kinds of privation. This would increase the truth-values in question and (...) bring fragments of (more generally speaking) Łukasiewicz-n-valued predicate logics into the scene. (shrink)

In general, there is only one fuzzy logic in which the standard interpretation of the strong conjunction is a strict triangular norm, namely, the product logic. We study several equations which are satisfied by some strict t-norms and their dual t-conorms. Adding an involutive negation, these equations allow us to generate countably many logics based on strict t-norms which are different from the product logic.

. A paraconsistent logic is a logical system that attempts to deal with contradictions in a discriminating way. In an earlier paper [Notre Dame J. Form. Log. 49, 401–424], we developed the systems of weakening of intuitionistic negation logic, called and, in the spirit of da Costa's approach by preserving, differently from da Costa, the fundamental properties of negation: antitonicity, inversion and additivity for distributive lattices. Taking into account these results, we make some observations on the (...) modified systems of and, and their paraconsistent properties. (shrink)

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)

We investigate properties of monadic purely negational fragment of Intuitionistic Control Logic ). This logic arises from Intuitionistic Propositional Logic ) by extending language of \ by additional new constant for falsum. Having two different falsum constants enables to define two forms of negation. We analyse implicational relations between negational monadic formulae and present a poset of non equivalent formulae of this fragment of \.

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)

The logic B+ is Routley and Meyer's basic positive logic. We show how to introduce a minimal intuitionistic negation and an intuitionistic negation in B+. The two types of negation are introduced in a wide spectrum of relevance logics built up from B+. It is proved that although all these logics have the characteristic paradoxes of consistency, they lack the K rule (and so, the K axioms).

The law of negation of the negation as it appears in the economic writings of Marx in general, and in Capital in particular, has repeatedly been treated in our philosophical literature in one way or another. To this day, however, as judged from the literature, attention has not been directed to the principle of negation of the negation as it is manifested not only in Marx's historical but also in his logical analysis. Negation of the (...) negation has been examined primarily as a form of expression of historical analysis . In the present article the emphasis is upon negation of the negation in logical analysis, upon the function of this principle considered as form of motion dealt with by historical analysis. All the examples of negation of the negation in historical analysis that have already been treated in the literature are omitted quite consciously, to avoid repetition. Nor is anything said about the structure of Capital as a "dialectically subdivided whole" in connection with negation of the negation, as that question has been splendidly handled in the book by L. A. Man'kovskii referred to in the footnote. (shrink)

This article answers two questions (posed in the literature), each concerning the guaranteed existence of proofs free of double negation. A proof is free of double negation if none of its deduced steps contains a term of the formn(n(t)) for some term t, where n denotes negation. The first question asks for conditions on the hypotheses that, if satisfied, guarantee the existence of a double-negation-free proof when the conclusion is free of double negation. The second (...) question asks about the existence of an axiom system for classical propositional calculus whose use, for theorems with a conclusion free of double negation, guarantees the existence of a double-negation-free proof. After giving conditions that answer the first question, we answer the second question by focusing on the Lukasiewicz three-axiom system. We then extend our studies to infinite-valued sentential calculus and to intuitionistic logic and generalize the notion of being double-negation free. The double-negation proofs of interest rely exclusively on the inference rule condensed detachment, a rule that combines modus ponens with an appropriately general rule of substitution. The automated reasoning program Otter played an indispensable role in this study. (shrink)

The book Paraconsistent Logic: Consistency, Contradiction and Negation by Walter Carnielli and Marcelo Coniglio is the most thorough study of Logics of Formal Inconsistency...

The general aim of this article is to study the negation fragment of classical logic within the framework of contemporary (Abstract) Algebraic Logic. More precisely, we shall find the three classes of algebras that are canonically associated with a logic in Algebraic Logic, i.e. we find the classes |$\textrm{Alg}^*$|⁠, |$\textrm{Alg}$| and the intrinsic variety of the negation fragment of classical logic. In order to achieve this, firstly, we propose a Hilbert-style axiomatization for this (...) fragment. Then, we characterize the reduced matrix models and the full generalized matrix models of this logic. Also, we classify the negation fragment in the Leibniz and Frege hierarchies. (shrink)

Intuitionistic propositional logic with Galois negations ( \(\mathsf {IGN}\) ) is introduced. Heyting algebras with Galois negations are obtained from Heyting algebras by adding the Galois pair \((\lnot,{\sim })\) and dual Galois pair \((\dot{\lnot },\dot{\sim })\) of negations. Discrete duality between GN-frames and algebras as well as the relational semantics for \(\mathsf {IGN}\) are developed. A Hilbert-style axiomatic system \(\mathsf {HN}\) is given for \(\mathsf {IGN}\), and Galois negation logics are defined as extensions of \(\mathsf {IGN}\). We give (...) the bi-tense logic \(\mathsf {S4N}_t\) which is obtained from the minimal tense extension of the modal logic \(\mathsf {S4}\) by adding tense operators. We give a new extended Gödel translation \(\tau \) and prove that \(\mathsf {IGN}\) is embedded into \(\mathsf {S4N}_t\) by \(\tau \). Moreover, every Kripke-complete Galois negation logic _L_ is embedded into its tense companion \(\tau (L)\). (shrink)