A classical paraconsistent logic, which is regarded as a modified extension of first-degree entailment logic, is introduced as a Gentzen-type sequent calculus. This logic can simulate the classical negation in classical logic by paraconsistent double negation in CP. Theorems for syntactically and semantically embedding CP into a Gentzen-type sequent calculus LK for classical logic and vice versa are proved. The cut-elimination and completeness theorems for CP are also shown using these embedding theorems. Similar results are also obtained (...) for an intuitionistic paraconsistent logic, and several versions of Glivenko and Gödel-Gentzen translation theorems are proved for CP and IP. (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)

In this paper, a way of constructing many-valued paraconsistent logics with weak double negation axioms is proposed. A hierarchy of weak double negation axioms is addressed in this way. The many-valued paraconsistent logics constructed are defined as Gentzen-type sequent calculi. The completeness and cut-elimination theorems for these logics are proved in a uniform way. The logics constructed are also shown to be decidable.

N. Kamide introduced a pair of classical and constructive logics, each with a peculiar type of negation: its double negation behaves as classical and intuitionistic negation, respectively. A consequence of this is that the systems prove contradictions but are non-trivial. The present paper aims at giving insights into this phenomenon by investigating subsystems of Kamide’s logics, with a focus on a system in which the double negation behaves as the negation of minimal logic. (...) We establish the negation inconsistency of the system and embeddability of contradictions from other systems. In addition, we attempt at an informational interpretation of the negation using the dimathematical framework of H. Wansing. (shrink)

In his paper 'Generalised Ortho Negation' [2] J.Michael Dunn mentions a claim of mine to the effect that there is no condition on 'perp frames' equivalent to the holding of double negation elimination ∼∼A ⊩ A. That claim is wrong. In this paper I correct my error and analyse the behaviour of conditions on frames for negations which verify a number of different theses.

In the context of intuitionistic analysis, we consider the set consisting of all continuous functions from [0,1] to such that =0 and =1, and the set consisting of ’s in where there exists x[0,1] such that . It is well-known that there are weak counterexamples to the intermediate value theorem, and with Brouwer’s continuity principle we have . However, there exists no satisfying answer to . We try to answer to this question by reducing it to a schema about intuitionistic (...) decidability that asserts “there exists an intuitionistically enumerable set that is not intuitionistically decidable”. We also introduce the notion of strong Specker double sequence, and prove that the existence of such a double sequence is equivalent to the existence of a function where. (shrink)

We investigate intuitionistic propositional modal logics in which a modal operator is equivalent to intuitionistic double negation. Whereas ¬¬ is divisible into two negations, is a single indivisible operator. We shall first consider an axiomatization of the Heyting propositional calculus H, with the connectives →,∧,∨ and ¬, extended with . This system will be called Hdn . Next, we shall consider an axiomatization of the fragment of H without ¬ extended with . This system will be called Hdn (...) + . We shall show that these systems are sound and complete with respect to specific classes of Kripke-style models with two accessibility relations, one intuitionistic and the other modal. This type of models is investigated in [2] and [3], and here we try to apply the techniques of these papers to an intuitionistic modal operator with a natural interpretation. The full results of our investigation will be published in [4] and [1]. (shrink)

This paper presents an algebraic framework for investigating proposed translations of classical logic into intuitionistic logic, such as the four negative translations introduced by Kolmogorov, Gödel, Gentzen and Glivenko. We view these asvariant semanticsand present a semantic formulation of Troelstra’s syntactic criteria for a satisfactory negative translation. We consider how each of the above-mentioned translation schemes behaves on two generalisations of Heyting algebras: bounded pocrims and bounded hoops. When a translation fails for a particular class of algebras, we demonstrate that (...) failure via specific finite examples. Using these, we prove that the syntactic version of these translations will fail to satisfy Troelstra’s criteria in the corresponding substructural logical setting. (shrink)

An efficient variant of the double-negation translation explains the relationship between Shoenfield’s and G¨odel’s versions of the Dialectica interpretation.

Moerdijk has introduced a topos of sheaves on a category of filters. Following his suggestion, we prove that its double negation subtopos is the topos of sheaves on the subcategory of ultrafilters - the ultrasheaves. We then use this result to establish a double negation translation of results between the topos of ultrasheaves and the topos on filters.

We propose an extension of minimal intuitionistic predicate logic, based on delimited control operators, that can derive the predicate-logic version of the double-negation shift schema, while preserving the disjunction and existence properties.

We prove that the (non-intuitionistic) law of the double negation shift has a bounded functional interpretation with bar recursive functionals of finite type. As an application. we show that full numerical comprehension is compatible with the uniformities introduced by the characteristic principles of the bounded functional interpretation for the classical case.

The double negation has always been considered by the logical systems from ancient times to the present. In fact, that is an issue that the current syntactic theories studying human reasoning, for example, the mental logic theory, address today. However, in this paper, I claim that, in the case of some languages such as Spanish, the double negation causes problems for the cognitive theories mainly based on formal schemata and supporting the idea of a universal syntax (...) of thought in the human mind. Thus, I propose that, given those problems, semantic frameworks such as that of the mental models theory seem to be more appropriate for explaining the human inferential activity. (shrink)

I shall present in this article a double negation account of the distinction between causes and background conditions. Such an account will be based on the idea that, unlike causes, background conditions allow for certain effects by way of double prevention. In Section 1 I shall introduce objective and non-objective theories of the causes-background conditions distinction and I shall discuss and reject some non-objective theories. In Section 2 I shall examine some existing objective theories and argue that (...) they need to be supplemented in two relevant respects. In Section 3 I shall present my double negation account. Background conditions will be taken to allow for their effects by acting as early double preventers, late double preventers or effect double preventers. In Section 4 I shall deal with two main problems, i.e., the possibility of interpreting background conditions as purely indirect causes and the introduction of negative entities in my account, and with further issues. (shrink)

The aim of the paper is to clarify the theoretical core of Solger's thought, the foundation for his aesthetics. I first analyze Solger's dialectic of double negation. Secondly I focus on Solger's gnoseology, which is orientated toward grasping the equilibrium between the Infinite (God) and the finite (world) consisting in this double negation. Lastly I investigate the notion of sacrifice, connecting it with Solger's ironic dialectic and showing its relevance to a complete understanding of his thought.

I prove that the three basic propositions of Vasiliev's paraconsistent logic have a semantic interpretation by means of the intuitionist logic. The interpèretation is confirmed by amens of the da Costa's model of Vasiliev's paraconsistent logic.

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)

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)

Strong negation is a well-known alternative to the standard negation in intuitionistic logic. It is defined virtually by giving falsity conditions to each of the connectives. Among these, the falsity condition for implication appears to unnecessarily deviate from the standard negation. In this paper, we introduce a slight modification to strong negation, and observe its comparative advantages over the original notion. In addition, we consider the paraconsistent variants of our modification, and study their relationship with non-constructive (...) principles and connexivity. (shrink)

Litotes, “a figure of speech in which an affirmative is expressed by the negative of the contrary” has had some tough reviews. For Pope and Swift, litotes—stock examples include “no mean feat”, “no small problem”, and “not bad at all”—is “the peculiar talent of Ladies, Whisperers, and Backbiters”; for Orwell, it is a means to affect “an appearance of profundity” that we can deport from English “by memorizing this sentence: A not unblack dog was chasing a not unsmall rabbit across (...) a not ungreen field.” But such ridicule is not without equivocation, given that litotes, or “logical” double negation, may or may not be semantically redundant. When the negation of a logical contrary yields an unexcluded middle, it contributes to expressive power: someone who is not unhappy may not be happy either, and an occurrence may not be infrequent without being frequent. But if something is not possible, what can it be but possible? Why does Crashaw’s “not impossible she” survive rhetorically while Orwell’s “not unsmall rabbit” is doomed? How is Robbie being “not not friends” with Mary on 7th Heaven distinct from being friends with her, if not not-p reduces to p? The key is recognizing in litotes a corollary of MaxContrary, the tendency for contradictory sentential negation ¬p to strengthen pragmatically to a contrary ©p, as when the formal contradictory Fr. “Il ne faut pas partir” is reinterpreted as expressing a contrary. Just as the Law of Excluded Middle can apply where it “shouldn’t”, resulting in pragmatically presupposed disjunctions between semantic contraries, so that “p v ©p” amounts to an instance of “p v ¬p”, the Law of Double Negation can fail to apply where it “should”. When not not-p conveys ¬©p, the negation of a virtual contrary, the middle between p and not-p is no longer excluded, rendering the Fregean dictum that “Wrapping up a thought in double negation does not alter its truth value” not unproblematic. (shrink)

I propose a comprehensive account of negation as a modal operator, vindicating a moderate logical pluralism. Negation is taken as a quantifier on worlds, restricted by an accessibility relation encoding the basic concept of compatibility. This latter captures the core meaning of the operator. While some candidate negations are then ruled out as violating plausible constraints on compatibility, different specifications of the notion of world support different logical conducts for negations. The approach unifies in a philosophically motivated picture (...) the following results: nothing can be called a negation properly if it does not satisfy Contraposition and Double Negation Introduction; the pair consisting of two split or Galois negations encodes a distinction without a difference; some paraconsistent negations also fail to count as real negations, but others may; intuitionistic negation qualifies as real negation, and classical Boolean negation does as well, to the extent that constructivist and paraconsistent doubts on it do not turn on the basic concept of compatibility but rather on the interpretation of worlds. (shrink)

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)

We investigate an information based generalization of the incompatibility-frame treatment of logics with non-classical negation connectives. Our framework can be viewed as an alternative to the neighbourhood semantics for extensions of lattice logic by various negation connectives, investigated by Hartonas. We set out the basic semantic framework, along with some correspondence results for extensions. We describe three kinds of constructions of canonical models and show that double negation law is not canonical with respect to any of (...) these constructions. We also compare our semantics to Hartonas’. (shrink)

Let denote a first‐order logic in a language that contains infinitely many constant symbols and also containing intuitionistic logic. By, we mean the associated logic axiomatized by the double negation of the universal closure of the axioms of plus. We shall show that if is strongly complete for a class of Kripke models, then is strongly complete for the class of Kripke models that are ultimately in.

This volume offers insights on experimental and empirical research in theoretical linguistic issues of negation and polarity, focusing on how negation is marked and how negative polarity is emphatic and how it interacts with double negation. Metalinguistic negation and neg-raising are also explored in the volume. Leading specialists in the field present novel ideas by employing various experimental methods in felicity judgments, eye tracking, self-paced readings, prosody and ERP. Particular attention is given to extensive crosslinguistc (...) data from French, Catalan and Korean along with analyses using semantic and pragmatic methods, corpus linguistics, diachronic perspectives, and longitudinal acquisitional studies as well as signed and gestural negation. Each contribution is situated with regards to major previous studies, thereby offering readers insights on the current state of the art in research on negation and negative polarity, highlighting how theory and data together contributes to the understanding of cognition and mind.. (shrink)

Using methods of abstract logic and the theory of valuation, we prove that there is no paraconsistent negation obeying the law of double negation and such that $\neg(a\wedge\neg a)$ is a theorem which can be algebraized by a technique similar to the Tarski-Lindenbaum technique.

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)

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)

This investigation draws from research on negative polarity item (NPI) illusions in order to explore a new and interesting instance of misalignment observed for grammatical sentences containing two negative markers. Previous research has shown that unlicensed NPIs can be perceived as acceptable when occurring soon after a structurally inaccessible negation (e.g. ever in *The bills that no senators voted for have ever become law). Here we examine the opposite configuration: grammatical sentences created by substituting the NPI ever with the (...) negative adverb never (e.g. The bills that no senators voted for have never become law). The processing and acceptability of these sentences was studied using three tasks: a speeded acceptability judgment (experiment 1), a self-paced reading task (experiment 2), and an offline acceptability rating (experiment 3). The results are consistent across measures in showing that the integration of the adverb never is disrupted by the linearly preceding but structurally inaccessible negative quantifier no in the relative clause. In our view, this pattern of results is in line with Parker and Phillips’ (2016) proposal that NPI illusions arise when the context containing the inaccessible negation has not been fully encoded by the time the NPI ever is encountered, making the embedded negative quantifier transparently available as a licensor. In a similar vein, the disruption effects observed for grammatical sentences containing two negative elements could arise if the negative quantifier is still being integrated when never is encountered, forcing the parser to deal with two negative elements simultaneously. This interpretation suggests that the same incomplete encodings that could be ameliorating the online perception of unlicensed NPIs could also be responsible for deteriorating the perception of the sentences under investigation here. This would represent an illusion of ungrammaticality. Furthermore, these results provide evidence against the speculation that NPI illusions are the consequence of misrepresenting ever as its near neighbor never, given that continuations with never are judged as unacceptable in spite of their grammaticality. Together, these findings inform the landscape of hypotheses on NPI illusions and offer valuable insights on the complexity of multiple negation and the relation between processing difficulty and acceptability. (shrink)

Since Fodor 1970, negation has worn a Homogeneity Condition to the effect that homogeneous predicates, ) denote homogeneously—all or nothing —to characterize the meaning of – when uttered out-of-the blue, in contrast to –:The mirrors are smooth. The mirrors are not smooth. The mirrors circle the telescope’s reflector. The mirrors do not circle the telescope’s reflector. It has been a problem for philosophical logic and for the semantics of natural language that – appear to defy the Principle of Excluded (...) Middle while – do not—Smooth ¬Smooth Circle ¬Circle. An impoverished logical form – has been the occasion to embellish all else—Boolean algebra, lexical presuppositions, Strongest Meaning Hypothesis, trivalence, supervaluation, double strengthening, etc., enriching the semantics and pragmatics with what remains a special theory of negation, which may be dismissed when the logical syntax and semantics of negation reflects that negated sentences are also tensed sentences. (shrink)

Beginning from the texts of Colectivo Situaciones on the insurrectional movements of December 2001 in Argentina, this article raises the question of institutional critique on the basis of a reflection on destituent practices. Far from being reducible to the aim of a reinstitution – that is, the aim of accomplishing the classical functions of political power – these practices refer instead to a « positive no », to the self-transforming actualization of the potentials of social action before and beyond the (...) figures of institutional representation. Thus, what appears amidst destitution is not a « labor of negation », but rather the contours of an instituting activity that engenders new forms of coexistence and coaction, even if this activity is partially haunted by the staging of past conflicts. To better understand this instituent activity, we must consider a second meaning of the word « destitution », which is desubjectivation as the destruction of the faculty of subjectivation, as Agamben describes it in Remnants of Auschwitz. It is only on the basis of the double meaning of destitution that the complex co-involvement of institution and destitution can be brought to light and that the privilege long accorded to « constitution » can be questioned, on the level of the theory of subjectity as well as that of politics. (shrink)

Vygotsky was a brilliant literary scholar whose role in psychology borrows substantially from his interests in and fascination with literature and theatre. The central question for Vygotsky’s theory was aesthetic synthesis – the emergence of generalized feelings in human life-experiences. The critical empirical example for the emergence of affective synthesis for Vygotsky was the short story by Ivan Bunin, ‘Legkoe dykhanie’. My task in this article is to analyse Vygotsky’s way of conceptualizing dialectical synthesis as a general psychological process. I (...) demonstrate that Vygotsky succeeded in locating the empirical phenomena where such syntheses can be observed, and proceeded halfway towards creating a general model of affective synthesis. Yet he failed to complete the task due to the limitations of the analogical transfer of the notion of short circuit into the psychological realm. The problem of synthesis remains unsolved up to the present day. Possible ways of solving it require formalization of the notion of double negation that has been used in dialectical philosophies, but has not been encoded into psychology’s research methodologies. (shrink)

It is often said that in a purely formal perspective, intuitionistic logic has no obvious advantage to deal with the liar-type paradoxes. In this paper, we will argue that the standard intuitionistic natural deduction systems are vulnerable to the liar-type paradoxes in the sense that the acceptance of the liar-type sentences results in inference to absurdity (⊥). The result shows that the restriction of the Double Negation Elimination (DNE) fails to block the inference to ⊥. It is, however, (...) not the problem of the intuitionistic approaches to the liar-type paradoxes but the lack of expressive power of the standard intuitionistic natural deduction system. We introduce a meta-level negation for a given system and a meta-level absurdity, ⋏, to the intuitionistic system. We shall show that in the system, the inference to ⊥ is not given without the assumption that the system is complete. Moreover, we consider the Double Meta-Level Negation Elimination rules (DMNE) which implicitly assume the completeness of the system. Then, the restriction of DMNE can rule out the inference to ⊥. (shrink)

Le terme d'athéisme produit un sentiment trompeur de familiarité. Car l'athéisme ne s'arrête pas à la négation de Dieu et les Dieux ont cette propension à ressusciter sous d'autres formes, à quitter le royaume de la religion et de la théologie pour se réfugier dans le quotidien de nos vies. C'est ce que Marx dénomme la " religion quotidienne " qui déifie des réalités sociales et économiques comme la valeur, le capital, la marchandise, transformant celles-ci en puissances tutélaires et transcendantes (...) face auxquelles les hommes seraient à la fois impuissants et irrémédiablement soumis. Trois textes de Marx permettent ici de saisir le passage de l'opium du peuple au fétichisme de la marchandise, et le type de subversion politique que produit l'athéisme. Véritable " sport de combat ", l'athéisme consiste alors en une pensée de l'émancipation et de la démocratie par et pour un peuple qui se constitue par la critique toujours renouvelée des pouvoirs. Cette critique est le legs précieux de Marx : toutes les religions scellent la croyance que l'Histoire serait commandée par des forces sur lesquelles les hommes n'auraient pas de prise, et c'est pourquoi il faut les combattre. L'athéisme conséquent n'appelle pas à la réforme, mais à la révolution. (shrink)

Glivenko’s theorem says that classical provability of a propositional formula entails intuitionistic provability of the double negation of that formula. This stood right at the beginning of the success story of negative translations, indeed mainly designed for converting classically derivable formulae into intuitionistically derivable ones. We now generalise this approach: simultaneously from double negation to an arbitrary nucleus; from provability in a calculus to an inductively generated abstract consequence relation; and from propositional logic to any set (...) of objects whatsoever. In particular, we give sharp criteria for the generalisation of classical logic to be a conservative extension of the one of intuitionistic logic with double negation. (shrink)

Building on the notational principles of C. S. Peirce’s graphical logic, Pietarinen has tried to develop a propositional logic unfolding in the medium of sound. Apart from its intrinsic interest, this project serves as a concrete test of logic’s range. However, I argue that Pietarinen’s inaugural proposal, while promising, has an important shortcoming, since it cannot portray double-negation without thereby portraying a contradiction.

We give a simple proof-theoretic argument showing that Glivenko’s theorem for propositional logic and its version for predicate logic follow as an easy consequence of the deduction theorem, which also proves some Glivenko type theorems relating intermediate predicate logics between intuitionistic and classical logic. We consider two schemata, the double negation shift (DNS) and the one consisting of instances of the principle of excluded middle for sentences (REM). We prove that both schemata combined derive classical logic, while each (...) one of them provides a strictly weaker intermediate logic, and neither of them is derivable from the other. We show that over every intermediate logic there exists a maximal intermediate logic for which Glivenko’s theorem holds. We deduce as well a characterization of DNS, as the weakest (with respect to derivability) scheme that added to REM derives classical logic. (shrink)

We show that in elementary analysis (EL) the contrapositive of countable choice is equivalent to double negation elimination for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Sigma_{2}^{0}}$$\end{document}-formulas. By also proving a recursive adaptation of this equivalence in Heyting arithmetic (HA), we give an instance of the conservativity of EL over HA with respect to recursive functions and predicates. As a complement, we prove in HA enriched with the (extended) Church thesis that every decidable predicate is recursive.

This review admires Michael Marder’s inquiry as a parallel for which biosemiotics can find points of conceptual resonance, even as methodological differences remain. By looking at the dump of ungrounded semiosis – the semiotics of dislocating referents from objects, and its effects – we can better do the work of applying biosemiotics not just towards the wonders of living relations, but also to the manifold ways in which industrial civilization is haphazardly yet systematically destroying the possibility for spontaneous yet contextualized (...) semiogenesis. Biosemiotics has much to gain by understanding the ways, gross and subtle, in which Anthropocenic hubris undercuts our own ability to make sense of the world, doubling down on overconfidence at the expense of meaning-making. (shrink)

Stephen Mulhall presents a series of multiply interrelated essays which explore the idea of selfhood as a matter of non-self-identity: for example, as becoming or self-overcoming, or as being doubled or divided. He draws on Nietzsche, Sartre, and Wittgenstein, but also on works of opera, cinema, and fiction.

I shall show that Levinas’s idea of infinite responsibility draws on Blanchot’s mechanism of “worklessness” which in turn explicitly draws on Gide’s mechanism of retroaction and the mise en abyme—a story that doubles itself within itself—which the latter accounts for. However, a false picture of mise en abyme and worklessness brought Levinas to two interrelated misconceptions. First, of the act of responsibility as inherently futile. Second, of repetition as “mechanical,” comprising instances which are allocated to pre-established loci upon a plenitude: (...) A totalitarianism which fits badly with Levinas’s pursuit of otherness. Deleuze, on the other hand, drawing the correct lesson from mise en abyme, established that an instance of repetition—reconstituting all previous ones—generates the very ground upon which it emerges. Accordingly, in a Deleuzian ethics, the absolute Other has no existence prior to the act of attending his needs. The act is therefore “fated to succeed.”. (shrink)

As a basic work of a translation I suggest to be attentive to the doubly negated propositions because they do not belong to classical logic, rather to intuitionist logic. I offer several instances of these propositions and classify the typical ways of their occurrences in a text.

Litotes, often confused with meiosis and understatement, has long suffered neglect. By comparing synonymous key words in previous definitions, this essay defines litotes as “a trope in which an affirmative is expressed by the negation of its opposite,” and for the first time classifies litotes into three subtypes based upon Aristotle’s study of opposition: contradictory, contrary and relative. Focusing particularly on the strong contradictory type of litotes and its realization in The Analects of Confucius, with its nearly one hundred (...) double negation litotes, we find they prominently serve the logical functions of achieving high probability, providing a sophisticated major premise, and elaborating strict definitions. Further, the essay argues that this strong argumentative function in logic leads to litotes’ epistemic power, a promising dimension for future research. (shrink)