Expressivists, such as Blackburn, analyse sentences such as 'S thinks that it ought to be the case that p' as S hoorays that p'. A problem is that the former sentence can be negated in three different ways, but the latter in only two. The distinction between refusing to accept a moral judgement and accepting its negation therefore cannot be accounted for. This is shown to undermine Blackburn's solution to the Frege-Geach problem.
Spinoza’s letter of June 2, 1674 to his friend Jarig Jelles addresses several distinct and important issues in Spinoza’s philosophy. It explains briefly the core of Spinoza’s disagreement with Hobbes’ political theory, develops his innovative understanding of numbers, and elaborates on Spinoza’s refusal to describe God as one or single. Then, toward the end of the letter, Spinoza writes: With regard to the statement that figure is a negation and not anything positive, it is obvious that matter in its (...) totality, considered without limitation [indefinitè consideratam], can have no figure, and that figure applies only to finite and determinate bodies. For he who says that he apprehends a figure, thereby means to indicate simply this, that he apprehends a determinate thing and the manner of its determination. This determination therefore does not pertain to the thing in regard to its being [esse]; on the contrary, it is its non-being [non-esse]. So since figure is nothing but determination, and determination is negation [Quia ergo figura non aliud, quam determinatio, et determinatio negatio est], figure can be nothing other than negation, as has been said. Arguably, what is most notable about this letter is the fate of a single subordinate clause which appears in the last sentence of this passage: et determinatio negatio est. That clause was to be adopted by Hegel and transformed into the slogan of his own dialectical method: Omnis determinatio est negatio (Every determination is negation). Of further significance is the fact that, while Hegel does credit Spinoza with the discovery of this most fundamental insight, he believes Spinoza failed to appreciate the importance of his discovery. The issue of negation and the possibility of self-negation stand at the very center of the philosophical dialogue between the systems of Spinoza and Hegel, and in this paper I will attempt to provide a preliminary explication of this foundational debate between the two systems. In the first part of the paper I will argue that the “determination is negation” formula has been understood in at least three distinct senses among the German Idealists, and as a result many of the participants in the discussion of this formula were actually talking past each other. The clarification of the three distinct senses of the formula will lead, in the second part of the paper, to a more precise evaluation of the fundamental debate between Spinoza and Hegel (and the German Idealists in general) regarding the possibility (or even necessity) of self-negation. In this part I will evaluate the validity of each interpretation of the determination formula, and motivate the positions of the various participants in the debate. (shrink)
A difficulty is exposed in Allan Gibbard's solution to the embedding/Frege-Geach problem, namely that the difference between refusing to accept a normative judgement and accepting its negation is ignored. This is shown to undermine the whole solution.
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.
This paper has two related goals. Firstly, after briefly clarifying the theoretical core of Solger's thought, it will analyse his metaphysics from Hegel's point of view, emphasizing that sacrifice is, for Solger, the fundamental structure of the relationship between the finite and the Infinite. Secondly, it will investigate the main reasons behind Hegel's criticism of Solger, showing that they have different conceptions of privation and negation and concluding that Solger and Hegel have different aims. Hegel's aim consists in recomposing (...) the unity of finite and infinite, whereas Solger's thought is structured on the rupture between these two. (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.
In a series of articles, Fine (Monist 83:357–361, 2000; Mind 112:195–234, 2003; Mind 115:1059–1082, 2006) 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 Schnieder (Philosophical Quarterly 56:39–54, 2006), 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)
Does (affirmative) judgement have a logical dual, negative judgement? Whether there is such a logical dualism was hotly debated at the beginning of the twentieth century. Frege argued in ?Negation? (1918/9) that logic can dispense with negative judgement. Frege's arguments shaped the views of later generations of analytic philosophers, but they will not have convinced such opponents as Brentano or Windelband. These philosophers believed in negative judgement for psychological, not logical, reasons. Reinach's ?On the Theory of Negative Judgement? (1911) (...) spoke to the concerns of these philosophers. While Frege took the distinction between affirmative and negative judgement to be logically redundant, Reinach argued that it is the result of confusing judgement with a different mental act. In this article, I present Reinach's arguments against the ?old logical dualism? in context, analyse them and discuss Reinach's innovative use of the notion of focus in the theory of judgement. Recently, there has been a revival of the view that sentential negation is grounded in a prior mental act of rejection. In the final section, I argue that Reinach's analysis of rejection poses a challenge for the revivalists. (shrink)
One of Da Costa's motives when he constructed the paraconsistent logic Cw was to dualise the negation of intuitionistic logic. In this paper I explore a different way of going about this task. A logic is defined by taking the Kripke semantics for intuitionistic logic, and dualising the truth conditions for negation. Various properties of the logic are established, including its relation to CWo Tableau and natural deduction systems for the logic are produced, as are appropriate algebraic structures. (...) The paper then investigates dualising the intuitionistic conditional in the same way. This establishes various connections between the logic, and a logic called in the literature 'Brouwerian logic' or 'closed-set logic'. (shrink)
Expressivists have a problem with negation. The problem is that they have not, to date, been able to explain why ‘murdering is wrong’ and ‘murdering is not wrong’ are inconsistent sentences. In this paper, I explain the nature of the problem, and why the best efforts of Gibbard, Dreier, and Horgan and Timmons don’t solve it. Then I show how to diagnose where the problem comes from, and consequently how it is possible for expressivists to solve it. Expressivists should (...) accept this solution, I argue, because it is demonstrably the only way of avoiding the problem, and because it generalizes. Once we see how to solve the negation problem, I show, it becomes easy to state a constructive, compositional expressivist semantics for a purely normative language with the expressive power of propositional logic, in which we can for the first time give explanatory, formally adequate expressivist accounts of logical inconsistency, logical entailment, and logical validity. As a corollary, I give what I take to be the first real expressivist explanation of why Geach’s original moral modus ponens argument is genuinely logically valid. This proves that the problem with expressivism cannot be that it can’t account for the logical properties of complex normative sentences. But it does not show that the same solution can work for a language with both normative and descriptive predicates, let alone that expressivists are able to deal with more complex linguistic constructions like tense, modals, or even quantifiers. In the final section, I show what kind of constraints the solution offered here would place expressivists under, in answering these further questions. (shrink)
This paper advances three necessary conditions on a successful account of sentential negation. First, the ability to explain the constancy of sentential meaning across negated and unnegated contexts (the Fregean Condition). Second, the ability to explain why sentences and their negations are inconsistent, and inconsistent in virtue of the meaning of negation (the Semantic Condition). Third, the ability of the account to generalize regardless of the topic of the negated sentence (the Generality Condition). The paper discusses three accounts (...) of negation available to moral expressivists. The first—the dominant commitment account—fails to meet the Fregean Condition. The two remaining accounts—commitment semantics and the expression account—satisfy all three conditions. A recent argument that the dominant commitment account is the only option available to expressivists is considered and rejected. (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 anti-realist 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)
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)
The focus of this paper are the meaning-theoretical arguments against classical logic that Dummett bases on consideration about the meanings of negation. Using Dummettian principles, I shall outline three such arguments, of increasing strength, and show that they are unsuccessful by giving responses to each argument on behalf of the classical logician. What is crucial is that in responding to these arguments a classicist need not challenge any of the basic assumptions of Dummett's outlook on the theory of meaning. (...) In particular, I shall grant Dummett his general bias towards verificationism or justificationism, encapsulated in the slogan `meaning is use'. The second general assumption I see no need to question is Dummett's particular breed of molecularism. Some of Dummett's assumptions will have to be given up, if classical logic is to be vindicated in his meaning-theoretical framework. A major result of this paper will be that the meaning of negation cannot be defined by rules of inference in the Dummettian framework. (shrink)
A cognitive pragmatic approach is taken to some long-standing problem cases of negation, the so-called presupposition denial cases. It is argued that a full account of the processes and levels of representation involved in their interpretation typically requires the sequential pragmatic derivation of two different propositions expressed. The first is one in which the presupposition is preserved and, following the rejection of this, the second involves the echoic (metalinguistic) use of material falling in the scope of the negation. (...) The semantic base for these processes is the standard anti-presuppositionalist wide-scope negation. A different view, developed by Burton-Roberts (1989a, 1989b), takes presupposition to be a semantic relation encoded in natural language and so argues for a negation operator that does not cancel presuppositions. This view is shown to be flawed, in that it makes the false prediction that presupposition denial cases are semantic contradictions and it is based on too narrow a view of the role of pragmatic inferencing. (shrink)
I introduce a formal language called the language of informational independence (IL-language, for short) that extends an ordinary first-order language in a natural way. This language is interpreted in terms of semantical games of imperfect information. In this language, one can define two negations: (i) strong or dual negation, and (ii) weak or contradictory negation. The latter negation, unlike the former, can occur only sentence-initially. Then I argue that, to a certain extent, the two negations match the (...) distinction existing in natural languages between sentential and constituent negation. As a corollary, I derive the fact that there are no mechanical rules for forming the contradictory negation of an English sentence. (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)
The paper is concerned with negation in artificial and natural languages. "Negation" is an ambiguous word. It can mean three different things: An operation(negating), an operator (a sign of negation), the result of an operation. The threethings, however, are intimately linked. An operation such as negation, is realizedthrough an operator of negation, i.e. consists in adding a symbol of negation to an entity to obtain an entity of the same type; and which operation it (...) is dependson what it applies to and on what results from its application.I argue that negation is not an operation on linguistic acts but rather anoperation on the objects of linguistic acts, namely sentences. And I assume that the negation of a sentence is a sentence that contradicts it. If so, the negation of a sentence may be obtained, in case the sentence is molecular, by applying the operation of negation not to the sentence itself but to a constituent sentence. To put it in a succinct and paradoxically sounding way we could say that in order to negate a sentence it is sufficient but not necessary to negate it.However that negation applies to sentences is true only for artificial languages, in which the sign of negation is a monadic sentential connective. In natural language, negation applies to expressions other than sentences, namely word sand non-sentential phrases. Still words and not sentential phrases are interesting and valuable only as ultimate or immediate constituents of sentences, as a means of saying (something that can be true or false) and the concern with negation is ultimately the concern with the negation of sentences. So the problem is what sub-sentential and non sentential expressions negation should apply to in order to obtain the negation of the containing sentence. The standard answer is that the negation of a natural language sentence is equivalent to the negation of its predicate. Yet, I argue, predicate negation is necessary but not sufficient, due to the existence of molecular sentences.Finally I notice that if to apply negation to an artificial sentence is to put the negation sign in front of it, to negate the predicate of a natural language sentencemay or may not be to put the negation sign in front of it. (shrink)
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)
Although it is not younger than other areas of non-classical logic, paraconsistent logic has received full recognition only in recent years, largely due to the work of, among others, Newton da Costa, Graham Priest, Diderik Batens, and Jerzy Perzanowski. A logical system Λ is paraconsistent if there is a set of Λ-formulas Δ ∪ { A } such that (i) in Λ one may derive from Δ both A and its negation, and (ii) the deductive closure of Δ with (...) respect to Λ is different from the set of all formulas. If from Δ one may derive a formula and its negation, Δ is said to be syntactically inconsistent. But is every syntactically inconsistent set of formulas contradictory? In classical logic and many non-classical logics, every syntactically inconsistent set is unsatisfiable, that is, semantically inconsistent. If contradictoriness means semantical inconsistency, there is, up to logical equivalence, only one contradiction. In paraconsistent logics, there are usually many non-equivalent formulas representing the semantically unique contradiction in a non-paraconsistent logic. So when is a formula the contradiction of another formula, and, moreover, how does the notion of contradiction relate to the notions of contrariety and negation? (shrink)
What I hope to achieve in this paper is some rather deeper understanding of the semantic and pragmatic properties of utterances which are said to involve the phenomenon of metalinguistic negation[FN1]. According to Laurence Horn, who has been primarily responsible for drawing our attention to it, this is a special non-truthfunctional use of the negation operator, which can be glossed as 'I object to U' where U is a linguistic utterance. This is to be distinguished from descriptive truthfunctional (...)negation which operates over a proposition. (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)
We discuss aspects of the logic of negation bearing on an issue raised by Jean-Yves Béziau, recalled in §1. Contrary- and subcontrary-forming operators are introduced in §2, which examines some of their logical behaviour, leading on naturally to a consideration in §3 of dual intuitionistic negation (as well as implication), and some further operators related to intuitionistic negation. In §4, a historical explanation is suggested as to why some of these negation-related connectives have attracted more attention (...) than others. The remaining sections (§§5, 6) briefly address a question about a certain notion of global contrariety and the provision of Kripke semantics for the various operators in play in our discussion. (shrink)
Whether assent (acceptance) and dissent (rejection) are thought of as speech acts or as propositional attitudes, the leading idea of rejectivism is that a grasp of the distinction between them is prior to our understanding of negation as a sentence operator, this operator then being explicable as applying to A to yield something assent to which is tantamount to dissent from A. Widely thought to have been refuted by an argument of Frege"s, rejectivism has undergone something of a revival (...) in recent years, especially in writings by Huw Price and Timothy Smiley. While agreeing that Frege"s argument does not refute the position, we shall air some philosophical qualms about it in Section 5, after a thorough examination of the formal issues in Sections 1–4. This discussion draws on – and seeks to draw attention to – some pertinent work of Kent Bendall in the 1970s. (shrink)
I present an argument that negation is a problem for proof-theoretic semantics: it's meaning cannot be defined by rules of inference, and that's particularly problematic for Dummett's and Prawitz' Justification of Deduction. I won the Jacobsen Essay Price of the University of London for this essay a few years ago.
We put together several observations on constructive negation. First, Russell anticipated intuitionistic logic by clearly distinguishing propositional principles implying the law of the excluded middle from remaining valid principles. He stated what was later called Peirce’s law. This is important in connection with the method used later by Heyting for developing his axiomatization of intuitionistic logic. Second, a work by Dragalin and his students provides easy embeddings of classical arithmetic and analysis into intuitionistic negationless systems. In the last section, (...) we present in some detail a stepwise construction of negation which essentially concluded the formation of the logical base of the Russian constructivist school. Markov’s own proof of Markov’s principle (different from later proofs by Friedman and Dragalin) is described. (shrink)
Russell's criticisms force Meinong to adopt a distinction between two types of negation. Logical expositions of Meinong's theory show the distinction is easily drawn in formal terms, but that alone does not justify the distinction intuitively.I criticise Routley'streatment of the distinction and argue that only Terence Parsons'theory retains and preserves the tight network of conceptual connections between the notions of negation, contradiction and impossibility. Hence, Parsons' approach best expresses the Meinongian perspective.
Poverty -of-stimulus arguments have taken new ground recently, augmented by experimental findings from th e study of child language. In this paper, we briefly review two variants of the poverty-of-stimulus argument that have received empirical support from studies of child language; then we examine a third argument of this kind in more detail. The case under discussion involves the structural notion of c-command as it pertains to children’s interpretation of disjunction in the scope of negation.
(2) Peter wollte Potsdam nicht verlassen bevor das Projekt in ruhigem Fahrwasser war. There are other well-known examples of non-interpreted negation, viz. cases of so-called negative concord in Slavic and Romance languages, but also in dialects of German and English. But arguably, in those cases the “superfluous” negation has to be present for grammatical reasons, which is not the case here. I will show that the negation is in fact interpreted, and that, due to a complex interplay (...) of semantic and pragmatic factors, we do get truth conditions for the two sentences that are not quite identical, but very similar. (shrink)
In positive logic the negation of a propositionA is defined byA X whereX is some fixed proposition. A number of standard properties of negation, includingreductio ad absurdum, can then be proved, but not the law of noncontradiction so that this forms a paraconsistent logic. Various stronger paraconsistent logics are then generated by putting in particular propositions forX. These propositions range from true through contingent to false.
The utterance of a negative statement invites the pragmatic inference that some reason exists for the proposition it negates to be true; this pragmatic inference paves the way for the logically unexpected Modus Shmollens inference: “If p then q ; not- q ; therefore, p .” Experiment 1 shows that a majority of reasoners endorse Modus Shmollens from an explicit major conditional premise and a negative utterance as a minor premise: e.g., reasoners conclude that “the soup tastes like garlic” from (...) the premises “If a soup tastes like garlic, then there is garlic in the soup; Carole tells Didier that there is no garlic in the soup they are eating.” Experiment 2 shows that this effect is mediated by the derivation of a pragmatic inference from negation. We discuss how theories of conditional reasoning can integrate such a pragmatic effect. (shrink)
The present paper is an attempt at the investigation of the nature of polarity contrast in natural languages. Truth conditions for natural language sentences are incomplete unless they include a proper definition of the conditions under which they are false. It is argued that the tertium non datur principle of classical bivalent logical systems is empirically invalid for natural languages: falsity cannot be equated with non-truth. Lacking a direct intuition about the conditions under which a sentence is false, we need (...) an independent foundation of the concept of falsity. The solution I offer is a definition of falsity in terms of the truth of a syntactic negation of the sentence. A definition of syntactic negation is proposed for English (Section 1). The considerations are applied to the analysis of definites in non-generic sentences and the analysis of generic indefinites. These two domains are investigated in breadth and some depth and the analyses compared and connected. During the discussion of non-generic predications with definite arguments and their respective negations (Section 2), a theory of predication is developed, basic to which is the distinction between integrative and summative predication. Summative predication, e.g., distributive plural, leads to contrary, all-or-no-thing, polarity contrasts due to the fundamental Presupposition of Indivisibility. Further-more, levels of predication are distinguished that are built up by various processes of constructing macropredications from lexical predicates. Given this analysis, particular (i.e., non-generic) quantification (Section 3) can be reanalyzed as an integrative, first-order form of predication that fills the truth-value gaps created by summative predication. The account comprises both nominal and adverbial quantification and relates quantification to the simpler types of predication discussed in Section 2. (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.1..
There are two natural ways of thinking about negation: (i) as a form of complementation and (ii) as an operation of reversal, or inversion (to deny that p is to say that things are "the other way around"). A variety of techniques exist to model conception (i), from Euler and Venn diagrams to Boolean algebras. Conception (ii), by contrast, has not been given comparable attention. In this note we outline a twofold geometric proposal, where the inversion metaphor is understoood (...) as involving a rotation o a reflection, respectively. These two options are equivalent in classical two-valued logic but they differ significantly in many-valued logics. Here we show that they correspond to two basic sorts of negation operators— Posts and Kleenes—and we provide a simple group-theoretic argument demonstrating their generative power. (shrink)
This paper will articulate an underappreciated side of the psychoanalytical Deleuze: his relation to Melanie Klein, particularly as it appears in The Logic of Sense. Deleuze's engagement with Klein largely follows his familiar strategy of re-reading a thinker off of a twist in one or two of that thinker's key concepts. With Klein, this twist involves re-reading her story of psychic development on the basis of disjunction rather than negation, so that the psychic surface that emerges generates a persistent (...) non-correspondence between self and other and between concept and thing. Deleuze thereby makes Klein a central figure in his ontology of sense and his analysis of how the physical surface of bodies generates a metaphysical surface of thought. However, Deleuze's ultimate turn is a Nietzschean one towards overcoming, the thought of eternal return, and the demolition of the Oedipal Law. As this final turn makes clear, even in his early writings that engaged more directly and affirmatively with psychoanalytical thought, Deleuze was already on an anti-Oedipal path. (shrink)
Negation is a fundamental component of communication (no-answers), cognition (logical negation), perception (different color), attitude (dislike), emotion (hatred), and volition (disagreement). Its many uses make it difficult to provide an integrated definition of the concept. The aim of this paper is to show that an integrated definition of the concept can be arrived at by means of a phenomenological method structuring it into three general essences labelled lack, otherness and obstruction.
In this paper, I argue that temporality, as described in Being and Nothingness , is a central theme in Nausea . In the first section I make the point that one of Sartre's guiding concerns at the time of publishing Nausea is temporality and the temporal nature of freedom. In the second section, the theme of melancholy and its relationship to temporality is explored. The third section explores Sartre's use of this image of being taken 'from behind'. I use this (...) temporal imagery as a guide for interpreting Roquentin's reaction to the rape and murder of Lucienne. By interpreting this scene by way of the temporality of Being and Nothingness , we can duly recognize the early Sartre's concern with temporality, understand the melancholia that arises because of the 'internal' negation of the past, and give a more satisfying account of a scene which is often ignored in the secondary literature. (shrink)
Since antiquity two different negations in natural languages have been noted: predicate negation (not honest) and predicate term negation (dishonest). The extensive literature offers no models. We propose category-theoretic models with two distinct negation operators, neither of them in general Boolean. We study combinations of the two (not dishonest) and sentential counterparts of each. We emphasize the relevance of our work for the theory of cognition.
One of the mainstays of the theory of definite descriptions since Russell (1905) has been their interaction with negation. In particular, Russellians, who advocate the view that definite descriptions are a kind of quantifier, point to these interactions as evidence in favor of the their view. The argument runs roughly as follows.
Studies 6, 1–15. Korean has three forms that express negation: short-form negation, long-form negation and inherently lexical verbs. The goal of this paper is to argue that there are three separate notions related to the expression and interpretation of negation in Korean, which must be kept separate. They are the notions of a negative clause, of the surface c-command domain of a negative element, and of the semantic scope of a negative element. The main arguments derive (...) from the interactions of the negative-sensitive adverb yekan with different forms of negation, and of the interaction of examples with both yekan and a negative-sensitive item like awmu-to (‘anyone’). (Stanford University). (shrink)
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 ultraiilters — the ultrasheaves. We then use this result to estab-.
The mediaeval logic of Aristotelian privation, represented by Ockham's expositionof All A is non-P as All S is of a type T that is naturally P and no S is P, iscritically evaluated as an account of privative negation. It is argued that there aretwo senses of privative negation: (1) an intensifier (as in subhuman), the dualof Neoplatonic hypernegation (superhuman), which is studied in linguistics asan operator on scalar adjectives, and (2) a (often lexicalized) Boolean complementrelative to the (...) extension of a privative negation in sense (1) (e.g., Brute). Thissecond sense, which is the privative negation discussed in modern linguistics, isshown to be Aristotle's. It is argued that Ockham's exposition fails to capture muchof the logic of Aristotelian privation due to limitations in the expressive power of thesyllogistic. (shrink)
Constructive logic with <span class='Hi'>Nelson</span> negation is an extension of the intuitionistic logic with a special type of negation expressing some features of constructive falsity and refutation by counterexample. In this paper we generalize this logic weakening maximally the underlying intuitionistic negation. The resulting system, called subminimal logic with <span class='Hi'>Nelson</span> negation, is studied by means of a kind of algebras called generalized N-lattices. We show that generalized N-lattices admit representation formalizing the intuitive idea of refutation (...) by means of counterexamples giving in this way a counterexample semantics of the logic in question and some of its natural extensions. Among the extensions which are near to the intuitionistic logic are the minimal logic with <span class='Hi'>Nelson</span> negation which is an extension of the Johansson's minimal logic with <span class='Hi'>Nelson</span> negation and its in a sense dual version — the co-minimal logic with <span class='Hi'>Nelson</span> negation. Among the extensions near to the classical logic are the well known 3-valued logic of Lukasiewicz, two 12-valued logics and one 48-valued logic. Standard questions for all these logics — decidability, Kripke-style semantics, complete axiomatizability, conservativeness are studied. At the end of the paper extensions based on a new connective of self-dual conjunction and an analog of the Lukasiewicz middle value ½ have also been considered. (shrink)
This paper deals with, prepositional calculi with strong negation (N-logics) in which the Craig interpolation theorem holds. N-logics are defined to be axiomatic strengthenings of the intuitionistic calculus enriched with a unary connective called strong negation. There exists continuum of N-logics, but the Craig interpolation theorem holds only in 14 of them.
The paper is devoted to the contributions of Helena Rasiowa to the theory of non-classical negation. The main results of Rasiowa in this area concerns–constructive logic with strong (Nelson) negation.
Deaf children whose hearing losses are so severe that they cannot acquire spoken language, and whose hearing parents have not exposed them to sign language, use gestures called homesigns to communicate. Homesigns have been shown to contain many of the properties of natural languages. Here we ask whether homesign has structure building devices for negation and questions. We identify two meanings (negation, question) that correspond semantically to propositional functions, that is, to functions that apply to a sentence (whose (...) semantic value is a proposition, φ) and yield another proposition that is more complex (¬φ for negation; ?φ for question). Combining φ with.. (shrink)
In this paper being a sequel to our [1] the logic with semi-negation is chosen as an example to elucidate some basic notions of the semantics for sentential calculi. E.g., there are shown some links between the Post number and the degree of complexity of a sentential logic, and it is proved that the degree of complexity of the sentential logic with semi-negation is 20. This is the first known example of a logic with such a degree of (...) complexity. The results of the final part of the paper cast a new light on the scope of the Kripke-style semantics in comparison to the matrix semantics. (shrink)
The paper is a study of the logic of existence, negation, and order in the Neoplatonic tradition. The central idea is that Neoplatonists assume a logic in which the existence predicate is a comparative adjective and in which monadic predicates function as scalar adjectives that nest the background order. Various scalar predicate negations are then identifiable with various Neoplatonic negations, including a privative negation appropriate for the lower orders of reality and a hyper-negation appropriate for the higher. (...) Reversion to the One can then be explained as the logical inference of hyper-negations from mundane knowledge. Part I develops the relevant linguistic and logical theory, and Part II defends Wolfson and the scalar interpretation against the more traditional Aristotelian understanding of Whittaker and others of reversion as intensional abstraction. (shrink)
In this paper the arguments for optimal data selection and the contrast class account of negations in the selection task and the conditional inference task are summarised, and contrasted with the matching bias approach. It is argued that the probabilistic contrast class account provides a unified, rational explanation for effects across these tasks. Moreover, there are results that are only explained by the contrast class account that are also discussed. The only major anomaly is the explicit negations effect in the (...) selection task (Evans, Clibbens, & Rood, 1996), which it is argued may not be the result of normal interpretative processes. It is concluded that the effects of negation on human reasoning provide good evidence for the view that human reasoning processes may be rational according to a probabilistic standard. (shrink)
This work is part of a wider investigation into lattice-structured algebras and associated dual representations obtained via the methodology of canonical extensions. To this end, here we study lattices, not necessarily distributive, with negation operations. We consider equational classes of lattices equipped with a negation operation ¬ which is dually self-adjoint (the pair (¬,¬) is a Galois connection) and other axioms are added so as to give classes of lattices in which the negation is De Morgan, orthonegation, (...) antilogism, pseudocomplementation or weak pseudocomplementation. These classes are shown to be canonical and dual relational structures are given in a generalized Kripke-style. The fact that the negation is dually self-adjoint plays an important role here, as it implies that it sends arbitrary joins to meets and that will allow us to define the dual structures in a uniform way. (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)
A refutation mechanism is introduced into logic programming, dual to the usual proof mechanism; then negation is treated via refutation. A four-valued logic is appropriate for the semantics: true, false, neither, both. Inconsistent programs are allowed, but inconsistencies remain localized. The four-valued logic is a well-known one, due to Belnap, and is the simplest example of Ginsberg’s bilattice notion. An efficient implementation based on semantic tableaux is sketched; it reduces to SLD resolution when negations are not involved. The resulting (...) system can give reasonable answers to queries that involve both negation and free variables. Also it gives the same results as Prolog when there are no negations. Finally, an implementation in Prolog is given. (shrink)
The puzzle of English until is well-known. Karttunen 1974 argues that until is ambiguous between a durative and a punctual negative polarity (NPI) meaning. Mittwoch 1977 claims that there is no ambiguity and that the two meanings are due to scope differences: NPI-until is in fact until above negation. Mittwoch’s account relies crucially on the assumption that negation is an aspectual operator that ‘stativizes’ verb meanings (a position recently argued for in de Swart 1996, and de Swart and (...) Molendijk 1999; see also Klima 1964, Seuren 1974, Verkuyl 1993). Thus far, the correct analysis of until remains an open issue. (shrink)
A general Gentzen-style framework for handling both bilattice (or strong) negation and usual negation is introduced based on the characterization of negation by a modal-like operator. This framework is regarded as an extension, generalization or re- finement of not only bilattice logics and logics with strong negation, but also traditional logics including classical logic LK, classical modal logic S4 and classical linear logic CL. Cut-elimination theorems are proved for a variety of proposed sequent calculi including CLS (...) (a conservative extension of CL) and CLScw (a conservative extension of some bilattice logics, LK and S4). Completeness theorems are given for these calculi with respect to phase semantics, for SLK (a conservative extension and fragment of LK and CLScw, respectively) with respect to a classical-like semantics, and for SS4 (a conservative extension and fragment of S4 and CLScw, respectively) with respect to a Kripke-type semantics. The proposed framework allows for an embedding of the proposed calculi into LK, S4 and CL. (shrink)
We introduce modal propositional substructural logics with strong negation, and prove the completeness theorems (with respect to Kripke models) for these logics.
Wansing’s extended intuitionistic linear logic with strong negation, called WILL, is regarded as a resource-conscious refinment of Nelson’s constructive logics with strong negation. In this paper, (1) the completeness theorem with respect to phase semantics is proved for WILL using a method that simultaneously derives the cut-elimination theorem, (2) a simple correspondence between the class of Petri nets with inhibitor arcs and a fragment of WILL is obtained using a Kripke semantics, (3) a cut-free sequent calculus for WILL, (...) called twist calculus, is presented, (4) a strongly normalizable typed λ-calculus is obtained for a fragment of WILL, and (5) new applications of WILL in medical diagnosis and electric circuit theory are proposed. Strong negation in WILL is found to be expressible as a resource-conscious refutability, and is shown to correspond to inhibitor arcs in Petri net theory. (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 (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. 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)
A new logic, quantized intuitionistic linear logic (QILL), is introduced, and is closely related to the logic which corresponds to Mulvey and Pelletier's (commutative) involutive quantales. Some cut-free sequent calculi with a new property quantization principle and some complete semantics such as an involutive quantale model and a quantale model are obtained for QILL. The relationship between QILL and Wansing's extended intuitionistic linear logic with strong negation is also observed using such syntactical and semantical frameworks.
This paper addresses the two interpretations that a combination of negative indefinites can get in concord languages like French: a concord reading, which amounts to a single negation, and a double negation reading. We develop an analysis within a polyadic framework, where a sequence of negative indefinites can be interpreted as an iteration of quantifiers or via resumption. The first option leads to a scopal relation, interpreted as double negation. The second option leads to the construction of (...) a polyadic negative quantifier corresponding to the concord reading. Given that sentential negation participates in negative concord, we develop an extension of the polyadic approach which can deal with non-variable binding operators, treating the contribution of negation in a concord context as semantically empty. Our semantic analysis, incorporated into a grammatical analysis formulated in HPSG, crucially relies on the assumption that quantifiers can be combined in more than one way upon retrieval from the quantifier store. We also consider cross-linguistic variation regarding the participation of sentential negation in negative concord. (shrink)
We present a semantics for strong negation systems on the basis of the subformula property of the sequent calculus. The new models, called subformula models, are constructed as a special class of canonical Kripke models for providing the way from the cut-elimination theorem to model-theoretic results. This semantics is more intuitive than the standard Kripke semantics for strong negation systems.
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)
We study an application of gaggle theory to unary negative modal operators. First we treat negation as impossibility and get a minimal logic system Ki that has a perp semantics. Dunn's kite of different negations can be dealt with in the extensions of this basic logic Ki. Next we treat negation as “unnecessity” and use a characteristic semantics for different negations in a kite which is dual to Dunn's original one. Ku is the minimal logic that (...) has a characteristic semantics. We also show that Shramko's falsification logic FL can be incorporated into some extension of this basic logic Ku. Finally, we unite the two basic logics Ki and Ku together to get a negative modal logic K-, which is dual to the positive modal logic K+ in [7]. Shramko has suggested an extension of Dunn's kite and also a dual version in [12]. He also suggested combining them into a “united” kite. We give a united semantics for this united kite of negations. (shrink)
In this paper, we identify a paradigm of metalinguistic comparatives in Greek headed by the preposition para. Para clauses are lexically distinct from other comparatives clauses in Greek (headed by apo, apoti). Building on earlier intuitions, we propose a semantics of metalinguistic MORE as a contrast between two propositions in terms of how appropriate of preferred they are by some individual. Syntactically, metalinguistic comparison appears to behave like a co-ordinate structure with ellipsis in the para-clause. Our account is extended to (...) metalinguistic negation, lexicalized by oxi in Greek, which, on a par with metalinguistic comparison, is defined as a binary operator, also contrasting two propositions. (shrink)
We consider algebras on binary relations with two main operators: relational composition and dynamic negation. Relational composition has its standard interpretation, while dynamic negation is an operator familiar to students of Dynamic Predicate Logic (DPL) (Groenendijk and Stokhof, 1991): given a relation R its dynamic negation R is a test that contains precisely those pairs (s,s) for which s is not in the domain of R. These two operators comprise precisely the propositional part of DPL.This paper contains (...) a finite equational axiomatization for these dynamic relation algebras. The completenessresult uses techniques from modal logic. We also lookat the variety generated by the class of dynamic relation algebras and note that there exist nonrepresentable algebras in this variety, ones which cannot be construedas spaces of relations. These results are also proved for an extension to a signature containing atomic tests and union. (shrink)
Jespersen (1860-1934:73-75) described what he called resumptive negation: “A second class [of emphatic negation] comprises what may be termed resumptive negation, the characteristic of which is that after a negative sentence has been completed, something is added in a negative form with the obvious result that the negative result is heightened. . . . In its pure form, the supplementary negative is added outside the frame of the first sentence, generally as a afterthought, as in ‘I shall (...) never do it, not under any circumstances, not on any condition, neither at home nor abroad’, etc.” Such examples have, to the best my knowledge, received no attention in the modern linguistic literature with the exception of the discussion by van der Wouden (1997) and mention (without discussion) in (Lawler 1977), plus brief comments by Klima and by McCawley (see van der Wouden) . A somewhat similar type of example (1) is known from Ross’s (1967) “Slifting” paper, where it is presented with along with extensive p.c. comments from Larry Horn; later in passim mention appears in (Horn 1989) and occasionally elsewhere. (shrink)
In this paper we will study the properties of the least extension n() of a given intermediate logic by a strong negation. It is shown that the mapping from to n() is a homomorphism of complete lattices, preserving and reflecting finite model property, frame-completeness, interpolation and decidability. A general characterization of those constructive logics is given which are of the form n (). This summarizes results that can be found already in [13,14] and [4]. Furthermore, we determine the structure (...) of the lattice of extensions of n(LC). (shrink)
The Russian Genitive of Negation construction (Gen Neg) involves case alternation between Genitive and the two structural cases, Nominative and Accusative.1 The factors governing the alternation have been a matter of debate for many decades, and there is a huge literature. Here we focus on one central issue and its theoretical ramifications. The theoretical issue is the following. The same truth-conditional content can often be structured in more than one way; we believe that there is a distinction between choices (...) in how to structure a situation to be described, and choices in how to structure a sentence describing the (already structured) situation. The distinction may not always be sharp, and the term Information Structure may perhaps cover both, but we believe that the distinction is important and needs closer attention. Babby (1980), in a masterful work on the Russian Genitive of Negation, argued that the choice depended principally on Theme-Rheme structure; after initially following Babby (Borschev & Partee 1998), we later argued (Borschev & Partee 2002a,b) that the choice reflects not Theme-Rheme structure but a structuring of the described situation which we call Perspectival Structure. Here we briefly review the phenomenon, Babby’s Theme-Rheme-based analysis, and our arguments for a different analysis. We then consider Hanging Topics, partitive Genitives, and broader licensing conditions of Genitive case, raising the possibility that our counterexamples to Babby’s use of Theme-Rheme structure might be explained away as examples involving Hanging Topics rather than (Praguian) Themes. We argue against that idea as well, but leave open the possibility that our Perspectival Structure may eventually be construable as a kind of information structure itself, if that notion can include some kinds of structuring of the situation as well as of the discourse. (shrink)
We study axiomatic extensions of the propositional constructive logic with strong negation having the disjunction property in terms of corresponding to them varieties of Nelson algebras. Any such varietyV is characterized by the property: (PQWC) ifA,B V, thenA×B is a homomorphic image of some well-connected algebra ofV.We prove:• each varietyV of Nelson algebras with PQWC lies in the fibre –1(W) for some varietyW of Heyting algebras having PQWC, • for any varietyW of Heyting algebras with PQWC the least and (...) the greatest varieties in –1(W) have PQWC, • there exist varietiesW of Heyting algebras having PQWC such that –1(W) contains infinitely many varieties (of Nelson algebras) with PQWC. (shrink)
The central topic of this inquiry is a cross-linguistic contrast in the interaction of conjunction and negation. In Hungarian (Russian, Serbian, Italian, Japanese), in contrast to English (German), negated definite conjunctions are naturally and exclusively interpreted as `neither’. It is proposed that Hungarian-type languages conjunctions simply replicate the behavior of plurals, their closest semantic relatives. More puzzling is why English-type languages present a different range of interpretations. By teasing out finer distinctions in focus on connectives, syntactic structure, and context, (...) the paper tracks down missing readings and argues that it is eventually not necessary to postulate a radical cross-linguistic semantic difference. In the course of making that argument it is observed that negated conjunctions on the `neither’ reading carry the expectation that the predicate hold of both conjuncts. The paper investigates several hypotheses concerning the source of this expectation. (shrink)
An efficient variant of the double-negation translation explains the relationship between Shoenfield’s and G¨odel’s versions of the Dialectica interpretation.
We consider the variety of Dynamic Relation Algebras V(DRA). We show that the monoid of an algebra in this variety determines dynamic negation uniquely.
Abstract This essay discusses the paradox of the N?g?rjunian negation as presented in his Vigrahavy?vartani. In Part One it is argued that as the Naiy?yika remarks, N?g?rjuna's speech act ?No proposition has its own intrinsic thesis? seemingly contradicts his famous claim that he has no negation whatsoever. In Parts Two and Three I consider the traditional as well as modem responses to this paradox and offer my own. I argue that N?g?rjuna's speech act does not generate a paradox (...) for two reasons: (a) the equivalence thesis of the kind??P = ?P is obviously false; and (b) since N?g?rjuna's speech act is situated in the dialogical/conversational universe of discourse as opposed to the argumentative/systematic universe of discourse, the teaching of the non?intrinsic thesis of all statements that it purports, holds for all statements in its class, including itself. Lastly, it is argued that even though the N?g?rjunian speech act is not a negation situated in the argumentative universe of discourse, it serves both philosophical and soteriological purposes. (shrink)
The present paper provides novel results on the model theory of Independence friendly modal logic. We concentrate on its particularly well-behaved fragment that was introduced in Tulenheimo and Sevenster (Advances in Modal Logic, 2006). Here we refer to this fragment as ‘Simple IF modal logic’ (IFML s ). A model-theoretic criterion is presented which serves to tell when a formula of IFML s is not equivalent to any formula of basic modal logic (ML). We generalize the notion of bisimulation familiar (...) from ML; the resulting asymmetric simulation concept is used to prove that IFML s is not closed under complementation. In fact we obtain a much stronger result: the only IFML s formulas admitting their classical negation to be expressed in IFML s itself are those whose truth-condition is in fact expressible in ML. (shrink)
L'auteur met en évidence l'ambiguïté de la théorie phénoménologique de la négation telle qu'elle est soutenue par Husserl. Husserl hésite entre une conception de la négation comme acte et l'incorporation de la négation au sens lui-même : entre une conception illocutionnaire et une conception propositionnelle de la négation. En définitive, il choisit la seconde conception, mais en l'étendant au niveau infrapropositionnel (à la perception). L'auteur traite ce problème comme révélateur de l'ambiguïté de la philosophie phénoménologique, suspendue entre acte et sens, (...) langage et perception. The author shows the ambiguity of the phenomenological theory of negation, such as it is held by Husserl. Husserl hesitates between a conception of negation as an act and of negation belonging to the meaning itself : that is, between an illocutionary and a propositional view of negation. Eventually, he chooses the later, but by extending it to the subpropositional level (i.e. to perception). The author takes this problem to be revealing of the ambiguity of phenomenological philosophy, standing between act and meaning, and between language and perception. (shrink)
This paper deals with some structural properties of the sequent calculus and describes strong symmetries between cut-free derivations and derivations, which do not make use of identity axioms. Both of them are discussed from a semantic and syntactic point of view. Identity axioms and cuts are closely related to the treatment of negation in the sequent calculus, so the results of this article explain some nice symmetries of negation.
Wittgenstein hérite de Frege l'idée d'une égalité de statut entre affirmation et négation, mais au lieu d'en tirer la thèse d'une absence de force de la négation, il en restaure au contraire la force alors même qu'il ne lui correspond aucune objectivité. D'où vient cette force ? Cette force serait d'expression. Dans cet article, je montre que Wittgenstein n'est finalement pas intéressé par la question sémantique de la négation, mas plutôt par cette attitude propre au philosophe consistant à ne pas (...) faire cas de son symbolisme opératoire, ce qui l'entraîne indûment à s'interroger sur son essence cachée.S'agissant du cas de la négation, Wittgenstein montre comment s'attaquer à la source de notre errement plutôt qu ' à la source de la signification justifiant son usage. Il « traite » ainsi ce qu'il appelle dans une de ses Dictées le « problème de Socrate » . L'impasse sur le symbolisme de la négation est un symptôme d'aveuglement au symbolisme. Reste donc à saisir l'articulation du signe avec le symptôme, soit entre deux espèces de traits que Wittgenstein tient pour hétérogènes. Pourtant, dans son combat contre les préjugés grammaticaux, Wittgenstein entend bien redonnera l'expression du signe une importance qui permet de comprendre en même temps son action sur le symptôme (sa disparition).Dans la « résolution » , une certaine concomitance freudienne entre les deux est donc présupposée. Nous examinons ce point de rencontre avec Freud tout en distinguant leurs conceptions respectives du grammatical. Wittgenstein inherits Frege's conception of equality of status of negative and positive assertions. Yet, far from concluding with Frege that the negation is weak, he restores its strength while at the same time he thinks it is devoid of objectivity since negative facts do not exist. Where does this strength come from ? The answer is : from its sole expressivity. My contention is therefore to show that Wittgenstein is less interested in the semantics of negation than in the philosopher's attitude which consists in neglecting the symbolic operation of negating, hence his questioning the essence of negation.Against this « symptom » Wittgenstein calls « Socrates ' problem » , the right thing to do is to cure it by tracing back the wot of our being misled by grammatical prejudices rather than to look for the source of meaning. What is to be treated is our blindness to symbolism. However we are left with a question concerning the connection between the sign of negation and the « symptom » resulting from omitting the latter. The difficulty arises from the fact that for Wittgenstein they belong to two different realms, while at the same time he seems to assume that the sole strength of expressing negation as a symbol could effect the desappearing of the symptom e.g. the « solution » of the « problem » . It is impossible to appraise this meeting-point between Wittgenstein and Freud without taking into account their respective conceptions of « the grammatical ». (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)
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)
This paper presents the observation that negative non-wh-questions with inverted negation do not have an alternative (alt-)question reading. In English, a simple question like (1) has two possible readings: a yes-no (yn-)question reading, paraphrased in (1a), and an alt-question reading, disambiguated in (1b). Under the yn-question reading, the question can be answered as in (2); under the alt-question reading, acceptable answers are (3).
We prove that any positive elementary (least fixed point) induction expressing the negation of transitive closure on finite nondirected graphs requires at least two recursion variables.
A prepositional logic S has the Converse Ackermann Property (CAP) if (AB)C is unprovable in S when C does not contain . In A Routley-Meyer semantics for Converse Ackermann Property (Journal of Philosophical Logic, 16 (1987), pp. 65–76) I showed how to derive positive logical systems with the CAP. There I conjectured that each of these positive systems were compatible with a so-called semiclassical negation. In the present paper I prove that this conjecture was right. Relational Routley-Meyer type semantics (...) are provided for each one of the resulting systems (the positive systems plus the semiclassical negation). (shrink)
Feature logics are the logical basis for so-called unification grammars studied in computational linguistics. We investigate the expressivity of feature terms with negation and the functional uncertainty construct needed for the description of long-distance dependencies and obtain the following results: satisfiability of feature terms is undecidable, sort equations can be internalized, consistency of sort equations is decidable if there is at least one atom, and consistency of sort equations is undecidable if there is no atom.
Entendemos el concepto de “negación mínima” en el sentido clásico definido por Johansson. El propósito de este artículo es definir la lógica positiva mínima Bp+, y probar que la negación mínima puede introducirse en ella. Además, comentaremos algunas de las múltiples extensiones negativas de Bp+.“Minimal negation” is classically understood in a Johansson sense. The aim of this paper is to define the minimal positive logic Bp+ and prove that a minimal negation can be inroduced in it. In addition, (...) some of the many possible negation extensions of Bp+ are commented. (shrink)
This essay represents part of an effort to rewrite the history metaphysics in terms of what philosophy never said, nor could say. It works from the Neoplatonic commentary tradition on Plato's Parmenides as the matrix for a distinctively apophatic thinking that takes the truth of metaphysical doctrines as something other than anything that can be logically articulated. It focuses on Damascius in the 5—6th century AD as the culmination of this tradition in the ancient world and emphasizes that Neoplatonism represents (...) the crisis of Greek metaphysics on account of the inability to give a rational account of foundations for knowing and of the ultimate principle of beings. Neoplatonism discovered how all such ultimate principles were necessarily beyond the reach of reason and speech. This apophatic insight is drawn out with the help of contemporary criticism of Neoplatonic philosophy, defining also some points of divergence. The essay then discusses the motives for thinking the unsayable in postmodern times on the basis of this parallel with Neoplatonic thought. Discourse's becoming critical of itself to the point of self-subversion animates them both. However, the tendency in postmodern thought to totally reject theology, including negative theology, is a betrayal of its own deepest motivations. This tendency is debated through an examination of the thought of Jean-Luc Nancy. While any traditional discourse can be negated, the negating and self-negating capacity of discourse itself is infinite, and this is where a perennial negative theological philosophy of the unsayable is to be located. Language, eminently the language of philosophy, as infinitely open, points in a direction which becomes equally and ineluctably theological. (shrink)
G. H. von Wright proposed that a temporal interval exemplifies a real contradiction if at least one part of any division of this interval involves the presence of contradictorily related (though non-simultaneous) states. In connection with intervals, two negations must be discerned: 'does not hold at an interval' and 'fails throughout an interval'. Von Wright did not distinguish the two. As a consequence, he made a mistake in indicating how to use his logical symbolism to express the notion of real (...) contradiction. The present paper aims to reconstruct and philosophically motivate von Wright's argument for the possibility of real contradictions. (shrink)
Negative polarity is one of the more elusive aspects of linguistics and a subject which has been gaining in importance in recent years. Written from within the well-defined theoretical framework of Generalized Quantifiers, the three main areas considered in this study are collocations, polarity items and multiple negations. In this mature piece of research, van der Wouden takes into account, not only semantic and syntactic considerations, but also to a large extent, pragmatic ones illustrating a wide array of linguistic approaches.
This paper examines a recent attempt to provide a negative answer to the question of the existence of illocutionary negations. It argues that the attempt is unsuccessful both because it presupposes a misinterpretation of the question's theoretical import and because, even granting that misinterpretation, it bases its proposed answer on certain assumptions that can independently be shown to be untenable.
The plank of the dependency theory is that unless there is a transition to socialism and a complete break with the metropolitan countries, the peripheral status of the dependent countries would continue. After the Second World War with the emergence of many new nations, as a consequence of decolonization, the question of development assumed paramount importance for these countries. Raul Prebisch (1950) understood the nineteenth century paradigm of free trade as inoperative and disadvantageous to the raw materials exporting countries. The (...) spectacular success of the Newly Industrialized countries‐ Hong Kong,Singapore, South Korea and Taiwan by integrating with the developed nations, have achieved a higher standard of living and negated the basic assumptions of the dependency theory. (shrink)
ABSTRACT: The 3rd BCE Stoic logician "Chrysippus says that the number of conjunctions constructible from ten propositions exceeds one million. Hipparchus refuted this, demonstrating that the affirmative encompasses 103,049 conjunctions and the negative 310,952." After laying dormant for over 2000 years, the numbers in this Plutarch passage were recently identified as the 10th (and a derivative of the 11th) Schröder number, and F. Acerbi showed how the 2nd BCE astronomer Hipparchus could have calculated them. What remained unexplained is why Hipparchus’ (...) logic differed from Stoic logic, and consequently, whether Hipparchus actually refuted Chrysippus. This paper closes these explanatory gaps. (1) I reconstruct Hipparchus’ notions of conjunction and negation, and show how they differ from Stoic logic. (2) Based on evidence from Stoic logic, I reconstruct Chrysippus’ calculations, thereby (a) showing that Chrysippus’ claim of over a million conjunctions was correct; and (b) shedding new light on Stoic logic and – possibly – on 3rd century BCE combinatorics. (3) Using evidence about the developments in logic from the 3rd to the 2nd centuries, including the amalgamation of Peripatetic and Stoic theories, I explain why Hipparchus, in his calculations, used the logical notions he did, and why he may have thought they were Stoic. (shrink)
It is sometimes held that rules of inference determine the meaning of the logical constants: the meaning of, say, conjunction is fully determined by either its introduction or its elimination rules, or both; similarly for the other connectives. In a recent paper, Panu Raatikainen (2008) argues that this view - call it logical inferentialism - is undermined by some "very little known" considerations by Carnap (1943) to the effect that "in a definite sense, it is not true that the standard (...) rules of inference" themselves suffice to "determine the meanings of [the] logical constants" (p. 2). In a nutshell, Carnap showed that the rules allow for non-normal interpretations of negation and disjunction. Raatikainen concludes that "no ordinary formalization of logic ... is sufficient to `fully formalize' all the essential properties of the logical constants" (ibid.). We suggest that this is a mistake. Pace Raatikainen, intuitionists like Dummett and Prawitz need not worry about Carnap's problem. And although bilateral solutions for classical inferentialists - as proposed by Timothy Smiley and Ian Rumfitt - seem inadequate, it is not excluded that classical inferentialists may be in a position to address the problem too. (shrink)
