This work studies some problems connected to the role of negation in logic, treating the positive fragments of propositional calculus in order to deal with two main questions: the proof of the completeness theorems in systems lacking negation, and the puzzle raised by positive paradoxes like the well-known argument of Haskel Curry. We study the constructive com- pleteness method proposed by Leon Henkin for classical fragments endowed with implication, and advance some reasons explaining what makes difficult to extend this constructive (...) method to non-classical fragments equipped with weaker implications (that avoid Curry's objection). This is the case, for example, of Jan Lukasiewicz's n-valued logics and Wilhelm Ackermann's logic of restricted implication. Besides such problems, both Henkin's method and the triviality phenomenon enable us to propose a new positive tableau proof system which uses only positive meta-linguistic resources, and to mo- tivate a new discussion concerning the role of negation in logic proposing the concept of paratriviality. In this way, some relations between positive reasoning and infinity, the possibilities to obtain a ¯first-order positive logic as well as the philosophical connection between truth and meaning are dis- cussed from a conceptual point of view. (shrink)

http://dx.doi.org/10.5007/1808-1711.2009v13n3p357 Neste trabalho concorda-se com a tese de I. Hacking segundo a qual o significado das constantes lógicas é dado pelas Regras de Introdução e Eliminação do cálculo de sequentes de Gentzen que caracterizam a concepção da noção de consequência lógica abstrata. Perguntamos quais são as regras mínimas que um conectivo deve satisfazer para que seja considerado uma negação genuína. Tomaremos como referência para tratar dessa questão os C-sistemas de Newton da Costa e o sistema LP de Graham Priest. Finalmente, (...) analisaremos esses sistemas na lógica de sequentes a fim de mostrar que a negação paraconsistente ou bem carece das regras puras de eliminação e negação da negação ou ela envolve outros conectivos, o que torna difícil atribuir um significado unívoco à negação paraconsistente. (shrink)

In his paraphasis of Aristotle’s Metaphysics, Avicenna seems to adopt a first principle distintc form the one adopted by the Greek philosopher for this science. In fact, some interpreters consider him as prefering the principle of third excluded instead of the principle of non contradiction. Since I desagree with this thesis, I propose to analyse here Avicenna’s formulation of the first principle. In order to do that, I propose, first, to clarify the meaning of the first principle by looking to (...) one of the most important modern critics: Jan Lukasiewicz. To put it briefly, the question I will try to answer is: how Avicenna and Aristotle justify the nature of the logical relations. The article finishes with make some general remarks on Avicenna’s defense of the first principle. (shrink)

In 1880, when Oliver Wendell Holmes (later to be a Justice of the U.S. Supreme Court) criticized the logical theology of law articulated by Christopher Columbus Langdell (the first Dean of Harvard Law School), neither Holmes nor Langdell was aware of the revolution in logic that had begun, the year before, with Frege's Begriffsschrift. But there is an important element of truth in Holmes's insistence that a legal system cannot be adequately understood as a system of axioms and corollaries; and (...) this element of truth is not obviated by the more powerful logical techniques that are now available. (shrink)

Imperatives cannot be true or false, so they are shunned by logicians. And yet imperatives can be combined by logical connectives: "kiss me and hug me" is the conjunction of "kiss me" with "hug me". This example may suggest that declarative and imperative logic are isomorphic: just as the conjunction of two declaratives is true exactly if both conjuncts are true, the conjunction of two imperatives is satisfied exactly if both conjuncts are satisfied—what more is there to say? Much more, (...) I argue. "If you love me, kiss me", a conditional imperative, mixes a declarative antecedent ("you love me") with an imperative consequent ("kiss me"); it is satisfied if you love and kiss me, violated if you love but don't kiss me, and avoided if you don't love me. So we need a logic of three -valued imperatives which mixes declaratives with imperatives. I develop such a logic. (shrink)

We explore in an experimental spirit the prospects for extending classical propositional logic with a new operator P intended to be interpreted when prefixed to a formula as saying that formula in question is at least partly true. The paradigm case of something which is, in the sense envisaged, false though still "partly" true is a conjunction one of whose conjuncts is false while the other is true. Ideally, we should like such a logic to extend classical logic - or (...) any fragment thereof under consideration - conservatively, to be closed under uniform substitution (of arbitrary formulas for sentence letters or propositional variables), and to allow the substitutivity of provably equivalent formulas salva provabilitate. To varying degrees, we experience some difficulties only with this last ('congruentiality') desideratum in the two four-valued logics we end up giving our most extended consideration to. (shrink)

Given a three-valued definition of validity, which choice of three-valued truth tables for the connectives can ensure that the resulting logic coincides exactly with classical logic? We give an answer to this question for the five monotonic consequence relations $st$, $ss$, $tt$, $ss\cap tt$, and $ts$, when the connectives are negation, conjunction, and disjunction. For $ts$ and $ss\cap tt$ the answer is trivial (no scheme works), and for $ss$ and $tt$ it is straightforward (they are the collapsible schemes, in which (...) the middle value acts like one of the classical values). For $st$, the schemes in question are the Boolean normal schemes that are either monotonic or collapsible. (shrink)

This discussion explores the possibility of distinguishing a tighter notion of contrariety evident in the Square of Opposition, especially in its modal incarnations, than as that binary relation holding statements that cannot both be true, with or without the added rider ‘though can both be false’. More than one theorist has voiced the intuition that the paradigmatic contraries of the traditional Square are related in some such tighter way—involving the specific role played by negation in contrasting them—that distinguishes them from (...) other pairs of incompatible statements constructed from the same conceptual materials. Prominent among examples, these other nonstandard pairs are the ‘new contraries’ presented by Robert Blanché’s hexagon(s) of opposition. With special, though not exclusive, attention to these cases, we investigate whether contrariety in the distinguished sense can be captured by adding to the incompatibility condition the further demand that the pair of statements concerned can be represented as the results of applying some sentence operator to the content in its scope, for one of the pair, and, for the other, the application of that same operator to the negation of that content. For one of the two cases, a Blanché case, of nonstandard contrariety singled out for attention, the question of whether such a representation is available is settled at the end of Section 4, and then in a more satisfying way in Section 5, though for the other case, noticed by Peter Simons, the question remains open, after some tentative discussion in one subsection, 6.2, of an Appendix (Section 6). (shrink)

Meta-ethical realism faces the serious epistemological problem of how to explain our epistemic access to moral reality. In the face of this challenge many are sceptical about non-naturalist realism. Nonetheless, there is good reason to acknowledge moral objectivity: morality shows all the signs of a truth-apt discourse but doesn’t exhibit the typical relativity inducing features. This suggests a middle-ground position, a theory that embraces the virtues of realism but does avoid its vices: objectivist antirealism. In this paper, I’ll discuss, mainly (...) following Crispin Wright’s account of moral truth as superassertibility, a promising version of objectivist antirealism and show how to cope with notorious problems, notably those arising from the thought that moral disagreement might be possible in which nobody is guilty of a cognitive shortcoming, which contradicts the antirealist claim that moral truth is not beyond our epistemic reach. The solution is to deny the possibility of cognitively faultless moral disagreement by arguing that cognitively blameless thinkers either agree or stay agnostic and, therefore, never disagree about any moral proposition. Since assuming an agnostic stance on the part of such thinkers contradicts the antirealist’s conception of truth—even within the limits of intuitionistic logic—I’ll propose an alternative logical revision for the moral discourse: a three-valued logic with epistemically construed truth-values. (shrink)

We provide a logical matrix semantics and a Gentzen-style sequent calculus for the first-degree entailments valid in W. T. Parry’s logic of Analytic Implication. We achieve the former by introducing a logical matrix closely related to that inducing paracomplete weak Kleene logic, and the latter by presenting a calculus where the initial sequents and the left and right rules for negation are subject to linguistic constraints.

PLEASE NOTE: This is the corrected 2nd eBook edition, 2021. ●●●●● _Critique of Impure Reason_ has now also been published in a printed edition. To reduce the otherwise high price of this scholarly, technical book of nearly 900 pages and make it more widely available beyond university libraries to individual readers, the non-profit publisher and the author have agreed to issue the printed edition at cost. ●●●●● The printed edition was released on September 1, 2021 and is now available through (...) all booksellers, including Barnes & Noble, Amazon, and brick-and-mortar bookstores under ISBN 978-0-578-88646-6. ●●●●● In light of the length of this book, readers who would like to have a more detailed description of the book's objectives and method may find it helpful to read the detailed and clearly written Wikipedia entry about this work: From the Wikipedia search page, use the search phrase "Critique of Impure Reason". At least at the time of this writing (11/29/2021), the Wikipedia entry is well-researched and accurate. ●●●●● In addition, a "Primer on Bartlett's CRITIQUE OF IMPURE REASON" has been made available by the author. It is available under its title through PhilPapers and other philosophy online archives. ●●●●● COMMENDATIONS OF THIS WORK, from the back cover of the published edition: ●●●●● “I admire its range of philosophical vision.” – Nicholas Rescher, Distinguished University Professor of Philosophy, University of Pittsburgh, author of more than 100 books. ●●●●● “Bartlett’s _Critique of Impure Reason_ is an impressive, bold, and ambitious work. Careful scholarship is balanced by original analyses that lead the reader to recognize the limits of meaning, knowledge, and conceptual possibility. The work addresses a host of traditional philosophical problems, among them the nature of space, time, causality, consciousness, the self, other minds, ontology, free will and determinism, and others. The book culminates in a fascinating and profound new understanding of relativity physics and quantum theory.” – Gerhard Preyer, Professor of Philosophy, Goethe-University, Frankfurt am Main, Germany, author of many books including _Concepts of Meaning_, _Beyond Semantics and Pragmatics_, _Intention and Practical Thought_, and _Contextualism in Philosophy_. ●●●●● “[This work’s] goal is of a unique and difficult species: Dr. Bartlett seeks to develop a formal logical calculus on the basis of transcendental philosophical arguments; in fact, he hopes that this calculus will be the formal expression of the transcendental foundation of knowledge.... I consider Dr. Bartlett’s work soundly conceived and executed with great skill.” – C. F. von Weizsäcker, philosopher and physicist, former Director, Max-Planck-Institute, Starnberg, Germany. ●●●●● “Bartlett has written an American “Prolegomena to All Future Metaphysics.” He aims rigorously to eliminate meaningless assertions, reach bedrock, and place philosophy on a firm foundation that will enable it, like science and mathematics, to produce lasting results that generations to come can build on. This is a great book, the fruit of a lifetime of research and reflection, and it deserves serious attention.” — Martin X. Moleski, former Professor, Canisius College, Buffalo, NY, studies of scientific method, the presuppositions of thought, and the self-referential nature of epistemology. ●●●●● “Bartlett has written a book on what might be called the underpinnings of philosophy. It has fascinating depth and breadth, and is all the more striking due to its unifying perspective based on the concepts of reference and self-reference.” – Don Perlis, Professor of Computer Science, University of Maryland, author of numerous publications on self-adjusting autonomous systems and philosophical issues concerning self-reference, mind, and consciousness. ●●●●● ●●●●● The _Critique of Impure Reason: Horizons of Possibility and Meaning_ comprises a major and important contribution to philosophy. Thanks to the generosity of its publisher, this massive 885-page volume has been published as a free open access eBook (3.75MB) as well as an open access printed edition. It inaugurates a revolutionary paradigm shift in philosophical thought by providing compelling and long-sought-for solutions to a wide range of philosophical problems. In the process, the work fundamentally transforms the way in which the concepts of reference, meaning, and possibility are understood. The book includes a Foreword by the celebrated German philosopher and physicist Carl Friedrich von Weizsäcker. ●●●●● In Kant’s _Critique of Pure Reason_ we find an analysis of the preconditions of experience and of knowledge. In contrast, but yet in parallel, the new _Critique_ focuses upon the ways—unfortunately very widespread and often unselfconsciously habitual—in which many of the concepts that we employ _conflict_ with the very preconditions of meaning and of knowledge. ●●●●● This is a book about the boundaries of frameworks and about the unrecognized conceptual confusions in which we become entangled when we attempt to transgress beyond the limits of the possible and meaningful. We tend either not to recognize or not to accept that we all-too-often attempt to trespass beyond the boundaries of the frameworks that make knowledge possible and the world meaningful. ●●●●● The _Critique of Impure Reason_ proposes a bold, ground-breaking, and startling thesis: that a great many of the major philosophical problems of the past can be solved through the recognition of a viciously deceptive form of thinking to which philosophers as well as non-philosophers commonly fall victim. For the first time, the book advances and justifies the criticism that a substantial number of the questions that have occupied philosophers fall into the category of “impure reason,” violating the very conditions of their possible meaningfulness. ●●●●● The purpose of the study is twofold: first, to enable us to recognize the boundaries of what is referentially forbidden—the limits beyond which reference becomes meaningless—and second, to avoid falling victims to a certain broad class of conceptual confusions that lie at the heart of many major philosophical problems. As a consequence, the boundaries of _possible meaning_ are determined. ●●●●● Bartlett, the author or editor of more than 20 books, is responsible for identifying this widespread and delusion-inducing variety of error, _metalogical projection_. It is a previously unrecognized and insidious form of erroneous thinking that undermines its own possibility of meaning. It comes about as a result of the pervasive human compulsion to seek to transcend the limits of possible reference and meaning. ●●●●● Based on original research and rigorous analysis combined with extensive scholarship, the _Critique of Impure Reason_ develops a self-validating method that makes it possible to recognize, correct, and eliminate this major and pervasive form of fallacious thinking. In so doing, the book provides at last provable and constructive solutions to a wide range of major philosophical problems. ●●●●● CONTENTS AT A GLANCE ▪▪▪▪▪ Preface ▪▪▪▪▪ Foreword by Carl Friedrich von Weizsäcker ▪▪▪▪▪ Acknowledgments ▪▪▪▪▪ Avant-propos: A philosopher’s rallying call ▪▪▪▪▪ Introduction ▪▪▪▪▪ A note to the reader ▪▪▪▪▪ A note on conventions ▪▪▪▪▪ ▪▪▪▪▪ ▪▪▪▪▪ PART I ▪▪▪▪▪ ▪▪▪▪▪ WHY PHILOSOPHY HAS MADE NO PROGRESS AND HOW IT CAN ▪▪▪▪▪ 1 Philosophical-psychological prelude ▪▪▪▪▪ 2 Putting belief in its place: Its psychology and a needed polemic ▪▪▪▪▪ 3 Turning away from the linguistic turn: From theory of reference to metalogic of reference ▪▪▪▪▪ 4 The stepladder to maximum theoretical generality ▪▪▪▪▪ ▪▪▪▪▪ ▪▪▪▪▪ PART II ▪▪▪▪▪ THE METALOGIC OF REFERENCE ▪▪▪▪▪ A New Approach to Deductive, Transcendental Philosophy ▪▪▪▪▪ 5 Reference, identity, and identification ▪▪▪▪▪ 6 Self-referential argument and the metalogic of reference ▪▪▪▪▪ 7 Possibility theory ▪▪▪▪▪ 8 Presupposition logic, reference, and identification ▪▪▪▪▪ 9 Transcendental argumentation and the metalogic of reference ▪▪▪▪▪ 10 Framework relativity ▪▪▪▪▪ 11 The metalogic of meaning ▪▪▪▪▪ 12 The problem of putative meaning and the logic of meaninglessness ▪▪▪▪▪ 13 Projection ▪▪▪▪▪ 14 Horizons ▪▪▪▪▪ 15 De-projection ▪▪▪▪▪ 16 Self-validation ▪▪▪▪▪ 17 Rationality: Rules of admissibility ▪▪▪▪▪ ▪▪▪▪▪ ▪▪▪▪▪ PART III ▪▪▪▪▪ PHILOSOPHICAL APPLICATIONS OF THE METALOGIC OF REFERENCE ▪▪▪▪▪ Major Problems and Questions of Philosophy and the Philosophy of Science ▪▪▪▪▪ 18 Ontology and the metalogic of reference ▪▪▪▪▪ 19 Discovery or invention in general problem-solving, mathematics, and physics ▪▪▪▪▪ 20 The conceptually unreachable: “The far side” ▪▪▪▪▪ 21 The projections of the external world, things-in-themselves, other minds, realism, and idealism ▪▪▪▪▪ 22 The projections of time, space, and space-time ▪▪▪▪▪ 23 The projections of causality, determinism, and free will ▪▪▪▪▪ 24 Projections of the self and of solipsism ▪▪▪▪▪ 25 Non-relational, agentless reference and referential fields ▪▪▪▪▪ 26 Relativity physics as seen through the lens of the metalogic of reference ▪▪▪▪▪ 27 Quantum theory as seen through the lens of the metalogic of reference ▪▪▪▪▪ 28 Epistemological lessons learned from and applicable to relativity physics and quantum theory ▪▪▪▪▪ ▪▪▪▪▪ PART IV ▪▪▪▪▪ HORIZONS ▪▪▪▪▪ 29 Beyond belief ▪▪▪▪▪ 30 _Critique of Impure Reason_: Its results in retrospect ▪▪▪▪▪ ▪▪▪▪▪ SUPPLEMENT ▪▪▪▪▪ The Formal Structure of the Metalogic of Reference ▪▪▪▪▪ APPENDIX I ▪▪▪▪▪ The Concept of Horizon in the Work of Other Philosophers ▪▪▪▪▪ APPENDIX II ▪▪▪▪▪ Epistemological Intelligence ▪▪▪▪▪ References ▪▪▪▪▪ Index ▪▪▪▪▪ About the author. 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Neste artigo discutimos alguns aspectos da lógica nos dias atuais. O propósito central é mostrar a evolução dessa disciplina. Começamos com uma breve introdução onde especificamos o que queremos dizer com o termo “lógica”. A seguir, exporemos e discutiremos o que consideramos ser algumas das principais áreas de investigação da lógica atual. Concluímos o artigo com algumas observações sobre lógicas não clássicas e seus impactos sobre a filosofia. Ao final do texto se encontram mais detalhes que apontam para um aprofundamento (...) e uma ampliação das questões aqui tratadas. (shrink)

Logical pluralism is the view that there is more than one correct logic. This very general characterization gives rise to a whole family of positions. I argue that not all of them are stable. The main argument in the paper is inspired by considerations known as the “collapse problem”, and it aims at the most popular form of logical pluralism advocated by JC Beall and Greg Restall. I argue that there is a more general argument available that challenges all variants (...) of logical pluralism that meet the following three conditions: that there are at least two correct logical systems characterized in terms of different consequence relations, that there is some sort of rivalry among the correct logics, and that logical consequence is normative. The hypothesis I argue for amounts to conditional claim: If a position satisfies all these conditions, then that position is unstable in the sense that it collapses into competing positions. (shrink)

Correspondence analysis is Kooi and Tamminga’s universal approach which generates in one go sound and complete natural deduction systems with independent inference rules for tabular extensions of many-valued functionally incomplete logics. Originally, this method was applied to Asenjo–Priest’s paraconsistent logic of paradox LP. As a result, one has natural deduction systems for all the logics obtainable from the basic three-valued connectives of LP -language) by the addition of unary and binary connectives. Tamminga has also applied this technique to the paracomplete (...) analogue of LP, strong Kleene logic \. In this paper, we generalize these results for the negative fragments of LP and \, respectively. Thus, the method of correspondence analysis works for the logics which have the same negations as LP or \, but either have different conjunctions or disjunctions or even don’t have them as well at all. Besides, we show that correspondence analyses for the negative fragments of \ and LP, respectively, are also suitable without any changes for the negative fragments of Heyting’s logic \ and its dual \ and LP). (shrink)

Esta enciclopédia abrange, de uma forma introdutória mas desejavelmente rigorosa, uma diversidade de conceitos, temas, problemas, argumentos e teorias localizados numa área relativamente recente de estudos, os quais tem sido habitual qualificar como «estudos lógico-filosóficos». De uma forma apropriadamente genérica, e apesar de o território teórico abrangido ser extenso e de contornos por vezes difusos, podemos dizer que na área se investiga um conjunto de questões fundamentais acerca da natureza da linguagem, da mente, da cognição e do raciocínio humanos, bem (...) como questões acerca das conexões destes com a realidade não mental e extralinguística. A razão daquela qualificação é a seguinte: por um lado, a investigação em questão é qualificada como filosófica em virtude do elevado grau de generalidade e abstracção das questões examinadas (entre outras coisas); por outro, a investigação é qualificada como lógica em virtude de ser uma investigação logicamente disciplinada, no sentido de nela se fazer um uso intenso de conceitos, técnicas e métodos provenientes da disciplina de lógica. O agregado de tópicos que constitui a área de estudos lógico-filosóficos é já visível, pelo menos em parte, no Tractatus Logico-Philosophicus de Ludwig Wittgenstein, uma obra publicada em 1921. E uma boa maneira de ter uma ideia sinóptica do território disciplinar abrangido por esta enciclopédia, ou pelo menos de uma porção substancial dele, é extrair do Tractatus uma lista dos tópicos mais salientes aí discutidos; a lista incluirá certamente tópicos do seguinte género, muitos dos quais se podem encontrar ao longo desta enciclopédia: factos e estados de coisas; objectos; representação; crenças e estados mentais; pensamentos; a proposição; nomes próprios; valores de verdade e bivalência; quantificação; funções de verdade; verdade lógica; identidade; tautologia; o raciocínio matemático; a natureza da inferência; o cepticismo e o solipsismo; a indução; as constantes lógicas; a negação; a forma lógica; as leis da ciência; o número. (shrink)

The proof theory of many-valued systems has not been investigated to an extent comparable to the work done on axiomatizatbility of many-valued logics. Proof theory requires appropriate formalisms, such as sequent calculus, natural deduction, and tableaux for classical (and intuitionistic) logic. One particular method for systematically obtaining calculi for all finite-valued logics was invented independently by several researchers, with slight variations in design and presentation. The main aim of this report is to develop the proof theory of finite-valued first order (...) logics in a general way, and to present some of the more important results in this area. In Systems covered are the resolution calculus, sequent calculus, tableaux, and natural deduction. This report is actually a template, from which all results can be specialized to particular logics. (shrink)

O objetivo geral da pesquisa da qual esse artigo faz parte é investigar o sistema metafísico que emerge dos trabalhos de David Lewis. Esse sistema pode ser decomposto em pelo menos duas teorias. A primeira nomeada como realismo modal genuíno (RMG) e a segunda como mosaico neo-humeano. O RMG é, sem dúvida, mais popular e defende a hipótese metafísica da existência de uma pluralidade de mundos possíveis. A principal razão em favor dessa hipótese é a sua aplicabilidade na discussão de (...) problemas losó cos, esse motivo não será diretamente abordado neste artigo. Pois, o foco está em compreender o mosaico neo-humea- no e como ele relaciona-se com a metafísica de mundos possíveis. Em outras palavras, o meu objetivo é entender as bases losó cas que sus- tentam o realismo modal genuíno. (shrink)

This paper outlines an approach to the principle of bivalence based on very general, but still elementary, semantic considerations. The principle of bivalence states that “every sentence is either true or false”. Clearly, some logics are bivalent while others are not. A more general formulation of uses the concept of designated and non-designated logical values and is captured by “every sentence is either designated or non-designated”. Yet this formulation seems trivial, because the concept of non-designated value is negative. In order (...) to refine the analysis, the class of anti-designated values has been distinguished. The non-trivial version of the principle of bivalence is expressed by “every sentence is either designated or anti-designated”. The last part of the paper mentions some extralogical reasons for considering the principle of bivalence with truth being a designated value as intimately connected to human thinking and behavior. (shrink)

The new paradigm in the psychology of reasoning adopts a Bayesian, or prob- abilistic, model for studying human reasoning. Contrary to the traditional binary approach based on truth functional logic, with its binary values of truth and falsity, a third value that represents uncertainty can be introduced in the new paradigm. A variety of three-valued truth table systems are available in the formal literature, including one proposed by de Finetti. We examine the descriptive adequacy of these systems for natural language (...) indicative condi- tionals and bets on conditionals. Within our framework the so-called “defective” truth table, in which participants choose a third value when the antecedent of the indicative conditional is false, becomes a coherent response. We show that only de Finetti’s system has a good descriptive fit when uncer- tainty is the third value. (shrink)

Though my ultimate concern is with issues in epistemology and metaphysics, let me phrase the central question I will pursue in terms evocative of philosophy of religion: What are the implications of our logic-in particular, of Cantor and G6del-for the possibility of omniscience?

The volume includes twenty-five research papers presented as gifts to John L. Bell to celebrate his 60th birthday by colleagues, former students, friends and admirers. Like Bell’s own work, the contributions cross boundaries into several inter-related fields. The contributions are new work by highly respected figures, several of whom are among the key figures in their fields. Some examples: in foundations of maths and logic ; analytical philosophy, philosophy of science, philosophy of mathematics and decision theory and foundations of economics. (...) Most articles are contributions to current philosophical debates, but contributions also include some new mathematical results, important historical surveys, and a translation by Wilfrid Hodges of a key work of arabic logic. (shrink)

The article demonstrates that to describe the property of atomicity of transactions in database systems, we need a three-valued logic with propositional connective characterized in the same way as Blamey’s interjunction. However, the article explains that since Blamey’s partial logic with interjunction is a logic without tautologies, it does not satisfy some salient conditions of being a logic of atomic transactions. The article introduces a logic of the considered kind, and provides an example of the formal exposition of the case (...) of an atomic transaction. Finally, the article explains the philosophical significance of the introduced logic. (shrink)

A many-valued (aka multiple- or multi-valued) semantics, in the strict sense, is one which employs more than two truth values; in the loose sense it is one which countenances more than two truth statuses. So if, for example, we say that there are only two truth values—True and False—but allow that as well as possessing the value True and possessing the value False, propositions may also have a third truth status—possessing neither truth value—then we have a many-valued semantics in the (...) loose but not the strict sense. A many-valued logic is one which arises from a many-valued semantics and does not also arise from any two-valued semantics [Malinowski, 1993, 30]. By a ‘logic’ here we mean either a set of tautologies, or a consequence relation. We can best explain these ideas by considering the case of classical propositional logic. The language contains the usual basic symbols (propositional constants p, q, r, . . .; connectives ¬, ∧, ∨, →, ↔; and parentheses) and well-formed formulas are defined in the standard way. With the language thus specified—as a set of well-formed formulas—its semantics is then given in three parts. (i) A model of a logical language consists in a free assignment of semantic values to basic items of the non-logical vocabulary. Here the basic items of the non-logical vocabulary are the propositional constants. The appropriate kind of semantic value for a proposition is a truth value, and so a model of the language consists in a free assignment of truth values to basic propositions. Two truth values are countenanced: 1 (representing truth) and 0 (representing falsity). (ii) Rules are presented which determine a truth value for every proposition of the language, given a model. The most common way of presenting these rules is via truth tables (Figure 1). Another way of stating such rules—which will be useful below—is first to introduce functions on the truth values themselves: a unary function ¬ and four binary functions ∧, ∨, → and ↔ (Figure 2).. (shrink)

Popper emphasised both the problem-solving nature of human knowledge, and the need to criticise a scientific theory as strongly as possible. These aims seem to contradict each other, in that the former stresses the problems that motivate scientific theories while the one ignores the character of the problems that led to the formation of the theories against which the criticism is directed. A resolution is proposed in which problems as such are taken as prime in the search for knowledge, and (...) subject to discussion. This approach is then applied to the problem of induction. Popper set great stake to his solution of it, but others doubted its legitimacy, in ways that are clarified by changing the form of the induction problem itself. That change draws upon logic, which is the subject of another application: namely, in contrast to Popper’s adhesion to classical logic as the only welcome form (because of the maximal strength of criticism that it dispenses), can other logics be used without abandoning his philosophy of criticism? (shrink)

The book presents a thoroughly elaborated logical theory of generalized truth-values understood as subsets of some established set of truth values. After elucidating the importance of the very notion of a truth value in logic and philosophy, we examine some possible ways of generalizing this notion. The useful four-valued logic of first-degree entailment by Nuel Belnap and the notion of a bilattice constitute the basis for further generalizations. By doing so we elaborate the idea of a multilattice, and most notably, (...) a trilattice of truth values – a specific algebraic structure with information ordering and two distinct logical orderings, one for truth and another for falsity. Each logical order not only induces its own logical vocabulary, but determines also its own entailment relation. We consider both semantic and syntactic ways of formalizing these relations and construct various logical calculi. (shrink)

Psychological research on people’s understanding of natural language connectives has traditionally used truth table tasks, in which participants evaluate the truth or falsity of a compound sentence given the truth or falsity of its components in the framework of propositional logic. One perplexing result concerned the indicative conditional if A then C which was often evaluated as true when A and C are true, false when A is true and C is false but irrelevant“ (devoid of value) when A is (...) false (whatever the value of C). This was called the “psychological defective table of the conditional.” Here we show that far from being anomalous the “defective” table pattern reveals a coherent semantics for the basic connectives of natural language in a trivalent framework. This was done by establishing participants’ truth tables for negation, conjunction, disjunction, conditional, and biconditional, when they were presented with statements that could be certainly true, certainly false, or neither. We review systems of three-valued tables from logic, linguistics, foundations of quantum mechanics, philosophical logic, and artificial intelligence, to see whether one of these systems adequately describes people’s interpretations of natural language connectives. We find that de Finetti’s (1936/1995) three-valued system is the best approximation to participants’ truth tables. (shrink)

In this paper, we introduce the notion of dual Post’s negation and an infinite class of Dual Post’s finitely-valued logics which differ from Post’s ones with respect to the definitions of negation and the sets of designated truth values. We present adequate natural deduction systems for all Post’s k-valued ) logics as well as for all Dual Post’s k-valued logics.

B. Kooi and A. Tamminga present a correspondence analysis for extensions of G. Priest’s logic of paradox. Each unary or binary extension is characterizable by a special operator and analyzable via a sound and complete natural deduction system. The present paper develops a sound and complete proof searching technique for the binary extensions of the logic of paradox.

I extend theories of nonmonotonic reasoning to account for reasons allowing free choice. My approach works with a wide variety of approaches to nonmonotonic reasoning and explains the connection between reasons for kinds of action and reasons for actions or subkinds falling under them. I use an Anderson–Kanger reduction of reason statements, identifying key principles in the logic of reasons.

This paper shows how complex argumentation, analyzed as dialectical structures, can be evaluated within a Bayesian framework by interpreting them as coherence constraints on subjective degrees of belief. A dialectical structure is a set of arguments (premiss-conclusion structure) among which support- and attack-relations hold. This approach addresses the observation that some theses in a debate can be better justified than others and thus fixes a shortcoming of a theory of defeasible reasoning which applies the bivalence principle to argument evaluations by (...) assigning them the status of being either defeated or undefeated. Evaluation procedures which are based on the principle of bivalence can, however, be embedded as a special case within the Bayesian framework. The approach developed in this paper rests on the assumptions that arguments can be reconstructed as deductively valid and that complex argumentation can be reconstructed such that premisses of arguments with equivalent conclusions are pairwise independent. (shrink)

In this transdisciplinary article which stems from philosophical considerations (that depart from phenomenology—after Merleau-Ponty, Heidegger and Rosen—and Hegelian dialectics), we develop a conception based on topological (the Moebius surface and the Klein bottle) and geometrical considerations (based on torsion and non-orientability of manifolds), and multivalued logics which we develop into a unified world conception that surmounts the Cartesian cut and Aristotelian logic. The role of torsion appears in a self-referential construction of space and time, which will be further related to (...) the commutator of the True and False operators of matrix logic, still with a quantum superposed state related to a Moebius surface, and as the physical field at the basis of Spencer-Brown’s primitive distinction in the protologic of the calculus of distinction. In this setting, paradox, self-reference, depth, time and space, higher-order non-dual logic, perception, spin and a time operator, the Klein bottle, hypernumbers due to Musès which include non-trivial square roots of ±1 and in particular non-trivial nilpotents, quantum field operators, the transformation of cognition to spin for two-state quantum systems, are found to be keenly interwoven in a world conception compatible with the philosophical approach taken for basis of this article. The Klein bottle is found not only to be the topological in-formation for self-reference and paradox whose logical counterpart in the calculus of indications are the paradoxical imaginary time waves, but also a classical-quantum transformer (Hadamard’s gate in quantum computation) which is indispensable to be able to obtain a complete multivalued logical system, and still to generate the matrix extension of classical connective Boolean logic. We further find that the multivalued logic that stems from considering the paradoxical equation in the calculus of distinctions, and in particular, the imaginary solutions to this equation, generates the matrix logic which supersedes the classical logic of connectives and which has for particular subtheories fuzzy and quantum logics. Thus, from a primitive distinction in the vacuum plane and the axioms of the calculus of distinction, we can derive by incorporating paradox, the world conception succinctly described above. (shrink)

Meta-ethical realism faces the serious epistemological problem of how to explain our epistemic access to moral reality. In the face of this challenge many are sceptical about non-naturalist realism. Nonetheless, there is good reason to acknowledge moral objectivity: morality shows all the signs of a truth-apt discourse but doesn’t exhibit the typical relativity inducing features. This suggests a middle-ground position, a theory that embraces the virtues of realism but does avoid its vices: objectivist antirealism. In this paper, I’ll discuss, mainly (...) following Crispin Wright’s account of moral truth as superassertibility, a promising version of objectivist antirealism and show how to cope with notorious problems, notably those arising from the thought that moral disagreement might be possible in which nobody is guilty of a cognitive shortcoming, which contradicts the antirealist claim that moral truth is not beyond our epistemic reach. The solution is to deny the possibility of cognitively faultless moral disagreement by arguing that cognitively blameless thinkers either agree or stay agnostic and, therefore, never disagree about any moral proposition. Since assuming an agnostic stance on the part of such thinkers contradicts the antirealist’s conception of truth—even within the limits of intuitionistic logic—I’ll propose an alternative logical revision for the moral discourse: a three-valued logic with epistemically construed truth-values. (shrink)

The book provides a historical and systematic exposition of the semantic theory of truth formulated by Alfred Tarski in the 1930s. This theory became famous very soon and inspired logicians and philosophers. It has two different, but interconnected aspects: formal-logical and philosophical. The book deals with both, but it is intended mostly as a philosophical monograph. It explains Tarski’s motivation and presents discussions about his ideas as well as points out various applications of the semantic theory of truth to philosophical (...) problems. (shrink)

Anievas and Nişancıoğlu’s attempt to shift the terms of the debate about early modern capitalism by a major widening of its perspectives is a welcome move. Accepting this, the paper suggests that their argument can be more forcefully made if the theoretical residues of earlier traditions of Marxist historical explanation are purged from the way they expound that argument. The most ambivalent of these relates to their continued use of the idea of a ‘coexistence of modes of production’. This permeates (...) the confused way they present Atlantic slavery. A second, comparable source of confusion concerns their description of the relationship between merchant capital and the absolutist state. The alliance between the modern state and mercantile capital is radically misrecognised thanks to an uncritical espousal of Anderson’s view of absolutism. The paper suggests that Anievas and Nişancıoğlu might have written a stronger book had they reconceptualised the economic history of capitalism by allowing for a whole epoch dominated by powerful groups of merchant capitalists. In conclusion, I argue that the commercial capital of the later middle ages/early modern period was the first form in which production began to be subordinated to capital. (shrink)

According to Suszko’s Thesis, there are but two logical values, true and false. In this paper, R. Suszko’s, G. Malinowski’s, and M. Tsuji’s analyses of logical twovaluedness are critically discussed. Another analysis is presented, which favors a notion of a logical system as encompassing possibly more than one consequence relation. [A] fundamental problem concerning many-valuedness is to know what it really is. [13, p. 281].

In this paper, we develop a duality for the varieties of a Łukasiewicz n + 1-valued modal System. This duality is an extension of Stone duality for modal algebras. Some logical consequences (such as completeness results, correspondence theory...) are then derived and we propose some ideas for future research.

The method of approximate reasoning using a fuzzy logic introduced by Baldwin (1978 a,b,c), is used to model human reasoning in the resolution of two well known paradoxes. It is shown how classical propositional logic fails to resolve the paradoxes, how multiple valued logic partially succeeds and that a satisfactory resolution is obtained with fuzzy logic. The problem of precise representation of vague concepts is considered in the light of the results obtained.