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)

Our question is: can we embed minimal negation in implicative logics weaker than I→? Previous results show how to define minimal negation in the positive fragment of the logic of relevance R and in contractionless intuitionistic logic. Is it possible to endow weaker positive logics with minimal negation? This paper prooves that minimal negation can be embedded in even such a weak system as Anderson and Belnap’s minimal positive logic.

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)

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

Two algebraic structures, the contrapositionally complemented Heyting algebra (ccHa) and the contrapositionally |$\vee $| complemented Heyting algebra (c|$\vee $|cHa), are studied. The salient feature of these algebras is that there are two negations, one intuitionistic and another minimal in nature, along with a condition connecting the two operators. Properties of these algebras are discussed, examples are given and comparisons are made with relevant algebras. Intuitionistic Logic with Minimal Negation (ILM) corresponding to ccHas and its extension |${\textrm {ILM}}$|-|${\vee (...) }$| for c|$\vee $|cHas are then investigated. Besides its relations with intuitionistic and minimal logics, ILM is observed to be related to Peirce’s logic and Vakarelov’s logic MIN. With a focus on properties of the two negations, relational semantics for ILM and |${\textrm {ILM}}$|-|${\vee }$| are obtained with respect to four classes of frames, and inter-translations between the classes preserving truth and validity are provided. ILM and |${\textrm {ILM}}$|-|${\vee }$| are shown to have the finite model property with respect to these classes of frames and proved to be decidable. Extracting features of the two negations in the algebras, a further investigation is made, following logical studies of negations that define the operators independently of the binary operator of implication. Using Dunn’s logical framework for the purpose, two logics |$K_{im}$| and |$K_{im-{\vee }}$| are discussed, where the language does not include implication. The |$K_{im}$|-algebras are reducts of ccHas and are different from relevant algebraic structures having two negations. The negations in the |$K_{im}$|-algebras and |$K_{im-{\vee }}$|-algebras are shown to occupy distinct positions in an enhanced form of Dunn’s kite of negations. Relational semantics for |$K_{im}$| and |$K_{im-{\vee }}$| is provided by a class of frames that are based on Dunn’s compatibility frames. It is observed that this class coincides with one of the four classes giving the relational semantics for ILM and |${\textrm {ILM}}$|-|${\vee }$|⁠. (shrink)

The concept of constructive negation we refer to in this paper is (minimally) intuitionistic in character (see [1]). The idea is to understand the negation of a proposition A as equivalent to A implying a falsity constant of some sort. Then, negation is introduced either by means of this falsity constant or, as in this paper, by means of a propositional connective defined with the constant. But, unlike intuitionisitc logic, the type of negation we develop here (...) is, of course, devoid of paradoxes of relevance. (shrink)

The present contribution might be regarded as a kind of defense of the common sense in logic. It is demonstrated that if the classical negation is interpreted as the minimal negation with n = 2 truth values, then deviant logics can be conceived as extension of the classical bivalent frame. Such classical apprehension of negation is possible in non- classical logics as well, if truth value is internalized and bivalence is replaced by bipartition.

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 the first part of this paper (Méndez and Robles 2008) a minimal and an intuitionistic negation is introduced in a wide spectrum of relevance logics extending Routley and Meyer's basic positive logic B+. It is proved that although all these logics have the characteristic paradoxes of consistency, they lack the K rule (and so, the K axiom). Negation is introduced with a propositional falsity constant. The aim of this paper is to build up logics definitionally equivalent (...) to those in the aforementioned paper, negation being now introduced with the unary connective. Relational ternary semantics are provided for the new logics and soundness and completeness results are proved. (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)

There is a relatively recent trend in treating negation as a modal operator. One such reason is that doing so provides a uniform semantics for the negations of a wide variety of logics and arguably speaks to a longstanding challenge of Quine put to non-classical logics. One might be tempted to draw the conclusion that negation is a modal operator, a claim Francesco Berto, 761–793, 2015) defends at length in a recent paper. According to one such modal account, (...) the negation of a sentence is true at a world x just in case all the worlds at which the sentence is true are incompatible with x. Incompatibility is taken to be the key notion in the account, and what minimal properties a negation has comes down to which minimal conditions incompatibility satisfies. Our aims in this paper are twofold. First, we wish to point out problems for the modal account that make us question its tenability on a fundamental level. Second, in its place we propose an alternative, non-modal, account of negation as a contradictory-forming operator that we argue is superior to, and more natural than, the modal account. (shrink)

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

The minimal weakening \ of Belnap-Dunn logic under the polarity semantics for negation as a modal operator is formulated as a sequent system which is characterized by the class of all birelational frames. Some extensions of \ with additional sequents as axioms are introduced. In particular, all three modal negation logics characterized by a frame with a single state are formalized as extensions of \. These logics have the finite model property and they are decidable.

The problem of negative truth is the problem of how, if everything in the world is positive, we can speak truly about the world using negative propositions. A prominent solution is to explain negation in terms of a primitive notion of metaphysical incompatibility. I argue that if this account is correct, then minimal logic is the correct logic. The negation of a proposition A is characterised as the minimal incompatible of A composed of it and the (...) logical constant ¬. A rule based account of the meanings of logical constants that appeals to the notion of incompatibility in the introduction rule for negation ensures the existence and uniqueness of the negation of every proposition. But it endows the negation operator with no more formal properties than those it has in minimal logic. (shrink)

In models for paraconsistent logics, the semantic values of sentences and their negations are less tightly connected than in classical logic. In ‘American Plan’ logics for negation, truth and falsity are, to some degree, independent. The truth of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\sim }p}$$\end{document} is given by the falsity of p, and the falsity of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\sim }p}$$\end{document} is given by the truth of p. (...) Since truth and falsity are only loosely connected, p and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\sim }p}$$\end{document} can both hold, or both fail to hold. In ‘Australian Plan’ logics for negation, negation is treated rather like a modal operator, where the truth of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\sim }p}$$\end{document} in a situation amounts to p failing in certain other situations. Since those situations can be different from this one, p and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\sim }p}$$\end{document} might both hold here, or might both fail here. So much is well known in the semantics for paraconsistent logics, and for first-degree entailment and logics like it, it is relatively easy to translate between the American Plan and the Australian Plan. It seems that the choice between them seems to be a matter of taste, or of preference for one kind of semantic treatment or another. This paper explores some of the differences between the American Plan and the Australian Plan by exploring the tools they have for modelling a language in which we have two negations. (shrink)

Karl Popper developed a theory of deductive logic in the late 1940s. In his approach, logic is a metalinguistic theory of deducibility relations that are based on certain purely structural rules. Logical constants are then characterized in terms of deducibility relations. Characterizations of this kind are also called inferential definitions by Popper. In this paper, we expound his theory and elaborate some of his ideas and results that in some cases were only sketched by him. Our focus is on Popper's (...) notion of duality, his theory of modalities, and his treatment of different kinds of negation. This allows us to show how his works on logic anticipate some later developments and discussions in philosophical logic, pertaining to trivializing connectives, the duality of logical constants, dual-intuitionistic logic, the conservativeness of language extensions, the existence of a bi-intuitionistic logic, the non-logicality of minimal negation, and to the problem of logicality in general. (shrink)

Constructive logic with Nelson 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 Nelson 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 Nelson negation which is an extension of the Johansson's minimal logic with Nelson negation and its in a sense dual version — the co-minimal logic with Nelson 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)

ABSTRACT This work presents efficient algorithms to update knowledge bases in the presence of integrity constraints. The algorithms ensure that the changes to the knowledge bases are minimal. We use the deductive database paradigm to represent knowledge. Minimality is defined as a natural partial order over possible models of the database and expresses a preference for data explicity stored in the database over the data deduced by default. This requirement seems rational for many applications and yet it is hard (...) to be expressed with a set of integrity constraints. We handle safe negation and local variables in the bodies of the rules. We tackle the issue of modifying the active domain (i.e. adding new constants or removing some of the constants which are present in the knowledge base), and we demonstrate how interaction with the user can be used to handle recursive definitions. (shrink)

Our purpose is to formulate a complete logic of propositions that takes into account the fact that propositions are both senses provided with truth values and contents of conceptual thoughts. In our formalization, propositions are more complex entities than simple functions from possible worlds into truth values. They have a structure of constituents (a content) in addition to truth conditions. The formalization is adequate for the purposes of the logic of speech acts. It imposes a stronger criterion of propositional identity (...) than strict equivalence. Two propositions P and Q are identical if and only if, for any illocutionary force F, it is not possible to perform with success a speech act of the form F(P) without also performing with success a speech act of the form F(Q). Unlike hyperintensional logic, our logic of propositions is compatible with the classical Boolean laws of propositional identity such as the symmetry and the associativity of conjunction and the reduction of double negation. (shrink)

Minimizers are widely acknowledged cross-linguistically to denote a minimal quantity, extent or degree. With respect to minimizers in Mandarin Chinese, Shyu claims that their so-called negative polarity is purely syntactically determined and is facilitated by the lian…dou EVEN construction. Within the framework of Dynamic Syntax which allows for interaction between syntactic, semantic and pragmatic information, we demonstrate that the total negation is actually derived from the interaction between syntax, semantics and pragmatics, rather than being determined by purely syntactic (...) means. (shrink)

The notion of negation is basic to any formal or informal logical system. When any such system is presented to us, it is presented either as a system without negation or as a system with some form of negation. In both cases, we are supposed to know intuitively whether there is no negation in the system or whether the form of negation presented in the system is indeed as claimed. To be more specific, suppose Robinson (...) Crusoe writes a logical system with Hilbert-type axioms and rules, which includes a unary connective ∗A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$*A$$\end{document}. He puts the document in a bottle and lets it lose at sea. We find it and take a look. We ask: is the connective “∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$*$$\end{document}” a negation in the system? Yet the notion of what is negation in a formal system is not clear. When we see a unary connective ∗A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$*A$$\end{document}, together with some other axioms for some additional connectives, how can we tell whether ∗A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$*A$$\end{document} is indeed a form of negation of A? Are there some axioms which the connective “∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$*$$\end{document}” must satisfy in order to qualify ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$*$$\end{document} as a negation? (shrink)

Proof-theoretic methods are developed for subsystems of Johansson’s logic obtained by extending the positive fragment of intuitionistic logic with weak negations. These methods are exploited to establish properties of the logical systems. In particular, cut-free complete sequent calculi are introduced and used to provide a proof of the fact that the systems satisfy the Craig interpolation property. Alternative versions of the calculi are later obtained by means of an appropriate loop-checking history mechanism. Termination of the new calculi is proved, and (...) used to conclude that the considered logical systems are PSPACE-complete. (shrink)

Intuitionistic logic formalises the foundational ideas of L.E.J. Brouwer’s mathematical programme of intuitionism. It is one of the earliest non-classical logics, and the difference between classical and intuitionistic logic may be interpreted to lie in the law of the excluded middle, which asserts that either a proposition is true or its negation is true. This principle is deemed unacceptable from the constructive point of view, in whose understanding the law means that there is an effective procedure to determine the (...) truth of all propositions. This understanding of the distinction between the two logics supports the view that negation plays a vital role in the formulation of intuitionistic logic.Nonetheless, the formalisation of negation in intuitionistic logic has not been universally accepted, and many alternative accounts of negation have been proposed. Some seek to weaken or strengthen the negation, and others actively supporting negative inferences that are impossible with it.This thesis follows this tradition and investigates various aspects of negation in intuitionistic logic. Firstly, we look at a problem proposed by H. Ishihara, which asks how effectively one can conserve the deducibility of classical theorems into intuitionistic logic, by assuming atomic classes of non-constructive principles. The classes given in this section improve a previous class given by K. Ishii in two respects: (a) instead of a single class for the law of the excluded middle, two classes are given in terms of weaker principles, allowing a finer analysis and (b) the conservation now extends to a subsystem of intuitionistic logic called Glivenko’s logic. This section also discusses the extension of Ishihara’s problem to minimal logic.Secondly, we study the relationship between two frameworks for weak constructive negation, the approach of D. Vakarelov on one hand and the framework of subminimal negation by A. Colacito, D. de Jongh, and A. L. Vargas on the other hand. We capture a version of Vakarelov’s logic with the semantics of the latter framework, and clarify the relationship between the two semantics. This also provides proof-theoretic insights, which results in the formulation of a cut-free sequent calculus for the aforementioned system.Thirdly, we investigate the ways to unify the formalisations of some logics with contra-intuitionistic inferences. The enquiry concerns paraconsistent logics by R. Sylvan and A. B. Gordienko, as well as the logic of co-negation by G. Priest and of empirical negation by M. De and H. Omori. We take Sylvan’s system as basic, and formulate the frame conditions of the defining axioms of the other systems. The conditions are then used to obtain cut-free labelled sequent calculi for the systems.Finally, we consider L. Humberstone’s actuality operator for intuitionistic logic, which can be seen as the dualisation of a contra-intuitionistic negation. A compete axiomatisation of intuitionistic logic with actuality operator is given, and comparisons are made for some related operators.Abstract prepared by Satoru Niki.E-mail:[email protected]. (shrink)

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

The logic TW+ is positive Ticket Entailment without the contraction axiom. Constructive negation is understood in the (minimal) intuitionistic sense but without paradoxes of relevance. It is shown how to introduce a constructive negation of this kind in positive logics at least as strong as TW+. Special attention is paid to the reductio axioms. Concluding remarks about relevance, modal and entailment logics are stated. Complete relational ternary semantics are provided for the logics introduced in this paper.

The logic B$_{+}$ is Routley and Meyer's basic positive logic. The logic B$_{K+}$ is B$_{+}$ plus the $K$ rule. We add to B$_{K+}$ four intuitionistic-type negations. We show how to extend the resulting logics within the modal and relevance spectra. We prove that all the logics defined lack the K axiom.

The so-called paradoxes of material implication have motivated the development of many non-classical logics over the years, such as relevance logics, paraconsistent logics, fuzzy logics and so on. In this note, we investigate some of these paradoxes and classify them, over minimal logic. We provide proofs of equivalence and semantic models separating the paradoxes where appropriate. A number of equivalent groups arise, all of which collapse with unrestricted use of double negation elimination. Interestingly, the principle ex falso quodlibet, (...) and several weaker principles, turn out to be distinguishable, giving perhaps supporting motivation for adopting minimal logic as the ambient logic for reasoning in the possible presence of inconsistency. (shrink)

In this paper, we investigate the representation of negated sentences in Minimal Recursion Semantics (Copestake, Flickinger, Pollard, & Sag, 2005). We begin with its treatment in the English Resource Grammar (Flickinger, 2000, 2011), a broad-coverage implemented HPSG (Pollard & Sag, 1994), and argue that it is largely a suitable representation for English, despite possible objections. We then explore whether it is suitable for typologically different languages: namely, those that express sentential negation via inflection on the verb, particularly Turkish (...) and Inuktitut. We find that the interaction between negation and intersective modifiers requires a change to the way in which (at least) one of them contributes to semantic composition, and we argue for adapting the analysis of intersective modifiers. (shrink)

In this paper, consistency is understood in the standard way, i.e. as the absence of a contradiction. The basic constructive logic BKc4, which is adequate to this sense of consistency in the ternary relational semantics without a set of designated points, is defined. Then, it is shown how to define a series of logics by extending BKc4 up to minimal intuitionistic logic. All logics defined in this paper are paraconsistent logics.

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

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

Two logics L1 and L2 are negatively equivalent if for any set of formulas X and any negated formula ¬, ¬ can be deduced from the set of hypotheses X in L1 if and only if it can be done in L2. This article is devoted to the investigation of negative equivalence relation in the class of extensions of minimal logic.

Constructive dualities have recently been proposed for some lattice-based algebras and a related project has been outlined by Holliday and Bezhanishvili, aiming at obtaining “choice-free spatial dualities for other classes of algebras [ $$\ldots $$ ], giving rise to choice-free completeness proofs for non-classical logics”. We present in this article a way to complete the Holliday–Bezhanishvili project (uniformly, for any normal lattice expansion). This is done by recasting in a choice-free manner recent relational representation and duality results by the author. (...) These results addressed the general representation and duality problem for lattices with quasi-operators, extending the Jónsson–Tarski approach for BAOs, and Dunn’s follow-up approach for distributive generalized Galois logics, to contexts where distributivity may not be assumed. To illustrate, we apply the framework to lattices (and their logics) with some form or other of a (quasi)complementation operator, obtaining correspondence results and canonical extensions in relational frames and choice-free dualities for lattices with a minimal, or a Galois quasi-complement, or involutive lattices, including De Morgan algebras, as well as Ortholattices and Boolean algebras, as special cases. (shrink)

Let R+ be the positive fragment of Anderson and Belnap’s Logic of Relevance, R. And let RMO+ be the result of adding the Mingle principle ) to R+. We have shown in [2] that either a minimal negation or else a semiclassical one can be added to RMO+ preserving the variable-sharing property. Moreover, each of there systems is given a semantics in the Routley-Meyer style. In describing in [2] the models for RMO+ plus minimal negation, we (...) noted that a similar strategy would give us a semantics for R+ with minimal negation; that is, constructive R. The aim of this paper is to prove this claim. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:God, Evil and the Limits of Theology by Karen KilbyVincent BirchGod, Evil and the Limits of Theology by Karen Kilby (London: T&T Clark, 2020), 176 pp.Karen Kilby's God, Evil and the Limits of Theology is a collection of essays reminiscent in multiple respects of Herbert McCabe's God Matters. Kilby cites McCabe on only a handful of occasions, but, more so than the references, the form and the content (...) of Kilby's volume bring McCabe to mind. Like McCabe's book, God, Evil and the Limits of Theology consists of mostly short and readable but highly provocative essays, notwithstanding that Kilby's essays are more academic in form and hang together thematically better than McCabe's text. Regarding content, it is clear that Kilby is aligning herself in certain respects with the tradition of grammatical Thomism that McCabe helped initiate, though it might be better to consider Kilby's work an exercise in "grammatical apophaticism," since Thomas figures prominently in only three of the eleven essays. Even the "grammatical" attribution should be applied with caution, though Kilby does use the term herself, because Ludwig Wittgenstein, whose influence puts the "grammatical" in grammatical Thomism, is not at all explicitly drawn upon for theological purposes. Nevertheless, as will become clear through our discussion of the essays in particular, the arguments and themes of the collection build upon and assist in providing a valuable expansion of the line of thinking carried on by grammatical Thomists.The first four essays treat various concerns in Trinitarian theology, and the first, entitled "Perichoresis and Projection: Problems with Social Doctrines of the Trinity," does some important preliminary critical work before Kilby makes her own constructive contribution to Trinitarian theology. For Kilby, social Trinitarianism (Jurgen Moltmann, Colin Gunton, and Patricia Wilson-Kastner are some proponents with whom she engages) is characterized by the idea that the Trinity, conceived as the interpenetration or perichoresis of three persons ("person" applied with fairly minimal negation), should be understood more as a rich source of [End Page 733] insight for apprehending the proper nature of human sociality than as a fundamentally mysterious doctrine. As the title of the chapter suggests, Kilby's primary critique of social theories of the Trinity is that they entail projection of human experience and reality into the Godhead, and she takes this critique a step further in arguing that it is of the very nature of social theories of the Trinity that they do just that. Her defense consists of a key claim that recurs throughout the rest of the Trinitarian essays—that human beings cannot know what it means for God to be three persons but one God. Consequently, social Trinitarians' bold descriptions of insights pertaining to how the three persons are united in perichoresis could be only a product of projection. Kilby concludes this essay by gesturing toward her constructive alternative to social theories of the Trinity: the doctrine of the Trinity does not give insight into God; rather, it determines the grammar for the structure of Christianity.Kilby follows up her first chapter with a study of Thomas Aquinas's Trinitarian theology, and as might be suggested by the conclusion of her first essay, she offers a thoroughly apophatic interpretation of Thomas, very much in the vein of David Burrell's Aquinas: God and Action. That is, Kilby claims that Thomas is not giving us insight into God but only manifesting the limits of our God-talk. Kilby is critical of various contemporary retrievals of Thomas that attempt to show some sort of "making sense" in Thomas's Trinitarian theology, and she focuses her argument regarding what might be called its "nonsense" on Thomas's uses of the terms "procession" and "relation." Kilby claims these terms appear to lose all intelligible connection to human experience when applied to the Trinity, especially noting the tension between, on the one hand, the subsisting relations being identical to the divine essence and, on the other, their being really distinct from each other. It is precisely the apparent incoherencies that arise within Thomas's Trinitarian theology that corroborate for Kilby that Thomas's theology offers... (shrink)

In this paper, consistency is understood as the absence of the negation of a theorem, and not, in general, as the absence of any contradiction. We define the basic constructive logic BKc1 adequate to this sense of consistency in the ternary relational semantics without a set of designated points. Then we show how to define a series of logics extending BKc1 within the spectrum delimited by contractionless minimal intuitionistic logic. All logics defined in the paper are paraconsistent logics.

What is a minimal proof-theoretical foundation of logic? Two different ways to answer this question may appear to offer themselves: reduce the whole of logic either to the relation of inference, or else to the property of incompatibility. The first way would involve defining logical operators in terms of the algebraic properties of the relation of inference—with conjunction $$\hbox {A}\wedge \hbox {B}$$ A ∧ B as the infimum of A and B, negation $$\lnot \hbox {A}$$ ¬ A as (...) the minimal incompatible of A, etc. The second way involves introducing logical operators in terms of the relation of incompatibility, such that X is incompatible with $$\{\lnot \hbox {A}\}$$ { ¬ A } iff every Y incompatible with X is incompatible with {A}; and X is incompatible with $$\{\hbox {A}\!\wedge \!\hbox {B}\}$$ { A ∧ B } iff X is incompatible with {A,B}; etc. Whereas the first route leads us naturally to intuitionistic logic, the second leads us to classical logic. The aim of this paper is threefold: to investigate the relationship of the two approaches within a very general framework, to discuss the viability of erecting logic on such austere foundations, and to find out whether choosing one of the ways we are inevitably led to a specific logical system. (shrink)

A conviction had by many Christians over many centuries is that natural language is inadequate for describing God. This is the doctrine of divine ineffability. Apophaticism understands divine ineffability as it being justified or proper to negate statements that describe God. This paper develops and defends a version of apophaticism in which the negation involved is metalinguistic. The interest of this metalinguistic apophaticism is two-fold. First, it provides a philosophical model of historical apophaticisms that shows their rational coherence. Second, (...) metalinguistic apophaticism enables a minimal understanding of ineffability that is independently plausible given its minor commitments. (shrink)

Semantic paradoxes like the liar are notorious challenges to truth theories. A paradox can be phrased with minimal resources and minimal assumptions. It is not surprising, then, that the liar is also a challenge to minimalism about truth. Horwich (1990) deals swiftly with the paradox, after discriminating between other strategies for avoiding it without compromising minimalism. He dismisses the denial of classical logic, the denial that the concept of truth can coherently be applied to propositions, and the denial (...) that the liar sentence expresses a proposition, but he endorses the denial that the liar is an acceptable instance of the equivalence schema (E). This paper has two main parts. It first shows that Horwich’s preferred denial is also problematic. As Simmons (1999), Beall and Armour-Garb (2003), and Asay (2015) argued, the solution is ad hoc, faces a possible loss of expressibility, and is ultimately unstable. Finally, the paper explores a different combination of possibilities for minimalism: treating the truth-predicate as context-dependent, rejecting the notion that the liar expresses a proposition, and reinterpreting negation in some contexts as metalinguistic denial. The paper argues that these are preferable options, but signposts possible dangers ahead. (shrink)

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

THERE IS NO DOUBT THAT THE HEIDEGGERIAN CRITIQUE of subjectivity has left a profound mark on the philosophy of the twentieth century. Anyone who has read Sartre, Lacan, Lévinas, Foucault, or Derrida can attest to this. Paradoxically, this critique resulted less in a complete disappearance of the subject from the philosophical scene than in its preservation under the minimal form of what one could call “a subject without qualities.” Like the Heideggerian Dasein before them, “consciousness” for Sartre, “the subject (...) of the unconscious” for Lacan, or “the subject of ethical responsibility” as understood by Lévinas can all be seen as representative figures of such a minimal subjectivity. In these various forms, the subject’s unstable existence is dominated, respectively, by the questioning of an impromptu givenness, by a blind drive for negation, by the arbitrary facticity of a chain of signifiers, or by the appeal of the face of the other. (shrink)

Sen's classic social choice result supposedly demonstrates a conflict between Pareto and even minimal forms of liberalism. By providing the first direct mathematical proof of this seminal result, we underscore a significantly different interpretation: rather than conflicts among rights, Sen's result occurs because the liberalism assumption negates the assumption that voters have transitive preferences. This explanation enriches interpretations of Sen's conclusion by including radically new kinds of societal conflicts, it suggests ways to sidestep these difficulties, and it explains earlier (...) approaches to avoid the difficulties. (shrink)

This article proposes a minimalist concept of autonomy that is consistent with determinism, but negates fatalism. Drawing on Nicolai Hartmann’s stratified ontology, it argues that autonomy is achieved not by suspending physical laws, but by introducing new, higher-level determinations unique to individual entities. The tension between general laws and individual autonomy is resolved by emphasizing the unique properties and individual laws that apply to each entity. The article also explains how this minimal autonomy makes sense of setting goals and (...) attempting to achieve them, demonstrating that even within a deterministic framework, individuals can have meaningful influence over their actions and outcomes. (shrink)

It was shown in the previous work of the author that one can avoid the paradox of minimal logic { ϕ , ¬ ϕ } ¬ ψ defining the negation operator via reduction not a constant of absurdity, but to a unary operator of absurdity. In the present article we study in details what does it mean that negation in a logical system can be represented via an absurdity or contradiction operator. We distinguish different sorts of such (...) presentations. Finally, we consider the possibility to represent the negation via absurdity and contradiction operators in such well known systems of paraconsistent logic as D.Batens's logic CLuN and Sette's logic P 1. (shrink)