I develop a basic theory of content within the framework of truthmaker semantics and, in the second part, consider some of the applications to subject matter, common content, logical subtraction and ground.

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)

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

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

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

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

In this article, we illuminate how a consumption practice in an ephemeral religious organization subverts systems of economic inequality that otherwise prevail in, and structure, society. Drawing on a rich ethnographic study in Pakistan, we show how the practice of food consumption in the Tablighi Jamaat —an Islamic organization originating in South Asia that is practiced intermittently by its followers—represents temporal spaces of egalitarianism. Within these temporal spaces, entrenched economic hierarchies that are salient in organizing Pakistani society are challenged. We (...) found that while the fundamental principles of the Tablighi Jamaat advocate for subversion of the economic hierarchies that propagate myriad inequalities by demarcating local Muslims into spheres of different social and economic classes, it is in the practice of food consumption when ethical transgression from these hierarchies are rendered most intelligible. Finally, we consider the implications of this study to the emerging field of Islamic business ethics. (shrink)

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

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

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

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

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

We investigate the notion of classical negation from a non-classical perspective. In particular, one aim is to determine what classical negation amounts to in a paracomplete and paraconsistent four-valued setting. We first give a general semantic characterization of classical negation and then consider an axiomatic expansion BD+ of four-valued Belnap–Dunn logic by classical negation. We show the expansion complete and maximal. Finally, we compare BD+ to some related systems found in the literature, specifically a four-valued modal (...) logic of Béziau and the logic of classical implication and a paraconsistent de Morgan negation of Zaitsev. (shrink)

Horn (1989) identified a number of irregular or marked negations that are not used in accordance with the standard rule of propositional logic. He concluded that negation was pragmatically ambiguous. Van der Sandt (1991) disputed Horn’s ambiguity claim, and proposed a uniform semantics for all negations. I will provide an informal explanation of van der Sandt’s theory, and develop a number of objections. I show that irregular negations are not anaphoric, as Van der Sandt believes, but compositional. I argue (...) for Geurts’s (1998) view that there are several different rules for irregular negations. I reject the implicature theory in favor of the semantic ambiguity thesis. (shrink)

Within each of these negational ranges two general facets of negation may be distinguished for analysis. I shall call them "free" and "bound." Negation, however, is never completely free nor completely bound, so "free" and "bound" shall refer to negation in a relative sense.

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

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

The problems of the meaning and function of negation are disentangled from ontological issues with which they have been long entangled. The question of the function of negation is the crucial issue separating relevant and paraconsistent logics from classical theories. The function is illuminated by considering the inferential role of contradictions, contradiction being parasitic on negation. Three basic modelings emerge: a cancellation model, which leads towards connexivism, an explosion model, appropriate to classical and intuitionistic theories, and a (...) constraint model, which includes relevant theories. These three modelings have been seriously confused in the modern literature: untangling them helps motivate the main themes advanced concerning traditional negation and natural negation. Firstly, the dominant traditional view, except around scholastic times when the explosion view was in ascendency, has been the cancellation view, so that the mainstream negation of much of traditional logic is distinctively nonclassical. Secondly, the primary negation determinable of natural negation is ·relevant negation. In order to picture relevant negation the traditional idea of negation as otherthanness is progressive) refined, to nonexclusive restricted otherthanness. Several pictures result, a reversal picture, a debate model, a record cabinet (or files of the universe) model which help explain relevant negation. Two appendices are attached, one on negation in Hegel and the Marxist tradition, the other on Wittgenstein's treatment of negation and contradiction. (shrink)

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.

No detailed description available for "Negation".

Seit den Anfängen der Philosophie in der Antike bis heute währt das Fragen um das Nichts. Dabei steht es in unterschiedlichen Kontexten: Theologie, Ontologie, Ethik, Erkenntnistheorie, Logik und Ästhetik. Trotz einer unleugbaren Kontinuität zur antiken und mittelalterlichen Beschäftigung mit dem Nichts bildet die europäische Moderne eine originäre und innovative Erkenntnis und Erfahrung des Nichts aus, die nicht zuletzt das Aufkommen des Nihilismus bescheinigt. Dieser Band versucht einen umfassenden Blick auf das Themenfeld «Nichts - Negation - Nihilismus» zu werfen und (...) insbesondere zu erörtern, worin die Kontinuität und Diskontinuität der europäischen Moderne zu Antike und Mittelalter bestehen. (shrink)

We present substructural negations, a family of negations classified in terms of structural rules of an extended kind of sequent calculus, display calculus. In considering the whole picture, we emphasize the duality of negation. Two types of negative modality, impossibility and unnecessity, are discussed and "self-dual" negations like Classical, De Morgan, or Ockham negation are redefined as the fusions of two negative modalities. We also consider how to identify, using intuitionistic and dual intuitionistic negations, two accessibility relations associated (...) with impossibility and unnecessity. (shrink)

There are two main claims that Bradley makes concerning negative judgment in the Principles of Logic: Negative judgment ‘stands at a different level of reflection’ from affirmative judgment. Negative judgment ‘presupposes a positive ground’.I will consider what Bradley means by these claims, and draw comparisons with Wittgenstein's views on negation as they developed between the Tractatus and the Philosophical Remarks.

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

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

In order to understand negation as such, at least since Aristotle’s time, there have been many ways of conceptually modelling it. In particular, negation has been studied as inconsistency, contradictoriness, falsity, cancellation, an inversion of arrangements of truth values, etc. In this paper, making substantial use of category theory, we present three more conceptual and abstract models of negation. All of them capture negation as turning upside down the entire structure under consideration. The first proposal turns (...) upside down the structure almost literally; it is the well known construction of opposite category. The second one treats negation as a contravariant functor and the third one captures negation as adjointness. Traditionally, negation was investigated in the context of language as negation of sentences or parts of sentences, e.g. names. On the contrary we propose to negate structures globally. As a consequence of our approach we provide a solution to the ontological problem of the existence of negative states of affairs. (shrink)

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

Intuitionism can be seen as a verificationism restricted to mathematical discourse. An attempt to generalize intuitionism to empirical discourse presents various challenges. One of those concerns the logical and semantical behavior of what has been called ' empirical negation'. An extension of intuitionistic logic with empirical negation was given by Michael De and a labelled tableaux system was there shown sound and complete. However, a Hilbert-style axiom system that is sound and complete was missing. In this paper we (...) provide the missing axiom system which is shown sound and complete with respect to its intended semantics. Along the way we consider some further applications of empirical negation. (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)

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

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

Those philosophers who gave a negative answer to all four questions were called "eel-wrigglers" by the Buddhists. It was impossible to fix their position either for approval or for rejection. They would criticize any view, positive or negative, but would not themselves hold any. And it was difficult for a serious person to enter into any controversy with them.

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

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.

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

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)

As we’ve seen in the last chapter, there is good linguistic reason to categorize negations (and negative operators in general) by which De Morgan laws they support. The weakest negative operators (merely downward monotonic) support only two De Morgan laws;1 medium-strength negative operators support a third;2 and strong negative operators support all four. As we’ve also seen, techniques familiar from modal logic are of great use in giving unifying theories of negative operators. In particular, Dunn’s (1990) distributoid theory allows us (...) to generate relational semantics for many negations. However, the requirements of distributoid theory are a bit too strict for use in modeling the weakest negations. For a relational semantics to work, an operator must either distribute or antidistribute over either conjunction or disjunction; but the merely downward monotonic operators do not. Thus, a unifying semantics cannot be had in distributoid theory. In the (more familiar) study of positive modalities, there is a parallel result. Normal necessities distribute over conjunction, and normal possibilities over disjunction. When these distributions break down, a relational semantics is no longer appropriate. Here, there is a somewhat familiar solution: neighborhood semantics. In this chapter, I’ll adapt neighborhood semantics to the less familiar case of negative modalities, showing how it can be used to give a single semantic framework appropriate to all the pertinent sorts of negative operators. (shrink)

One of the oldest inquiries of philosophical and theological thought is addressed to the question who or what "God" is. Negative theology provides a way of answering that question by identifying what God is not. In recent years certain critical trends, such as deconstruction, have resembled features of negative theology in that they cancel the apparent meanings of the written word. Similarly, the Buddhist concept of nothingness has raised questions about the meaning of negation and theism. These trends have (...) caused philosophers and theologians to reconsider negation and theology in new ways, which are explored in this collection. In Negation and Theology a group of distinguished philosophers and theologians take up issues that illuminate the character of negative thinking and seek to make connections both with traditional negative theologies and with current theological and philosophical discussions. One essay, for example, is concerned with the problem of reconciling Paul Tillich's statement that God is being-itself with the Buddhist conception of an ultimate nothingness or with the tradition of negative theology. The format of the book distinguishes it from mere collections of essays. Scharlemann assumes the role of editor, introducer, and interlocutor, questioning the contributors and allowing them to respond. (shrink)

The paper consists of two parts. In the first one I present some general remarks regarding the history of negation and attempt to answer the philosophical question concerning the essence of negation. In the second part I resume the theological teaching on the degrees of certainty and point to five forms of negation – known from other areas of research -- as applied in the framework of theological investigations.