David Nelson’s constructive logics with strong negation may beviewed as alternative paraconsistent logic. These logics have been developedbefore da Costa’s works. We address some philosophical aspects of Nelson’slogics and give technical results concerning Kripke models and tableau calculi. We also suggest possible applications of paraconsistent constructivelogics.

Antonio Rubio, autor de la Logica mexicana, usó la noción de instrumento para elaborar una teoría de la ciencia, un análisis de las relaciones entre la lógica y las ciencias. Las obras lógicas de Aristóteles fueron apodadas el “instrumento” de la filosofía, y para Rubio la lógica entrega los “instrumentos-de-saber” a las demás ciencias. La expresión se remonta al “método de adquirir una ciencia” de Aristóteles, el cual, dijo, tiene que ser adquirido antes de abordar la ciencia misma. Rubio interpretó (...) este instrumental como la técnica de definir y clasificar los entes investigados en la ciencia, formar proposiciones en torno a ella y organizarlas en un sistema. Además, elaboró una explicación de la índole ontológica de los instrumentos. Expondré esta hermenéutica científica de Rubio, parcialmente original, desde la filosofía y lógica actuales. (shrink)

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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This volume is dedicated to Prof. Dag Prawitz and his outstanding contributions to philosophical and mathematical logic. Prawitz's eminent contributions to structural proof theory, or general proof theory, as he calls it, and inference-based meaning theories have been extremely influential in the development of modern proof theory and anti-realistic semantics. In particular, Prawitz is the main author on natural deduction in addition to Gerhard Gentzen, who defined natural deduction in his PhD thesis published in 1934. The book opens with an (...) introductory paper that surveys Prawitz's numerous contributions to proof theory and proof-theoretic semantics and puts his work into a somewhat broader perspective, both historically and systematically. Chapters include either in-depth studies of certain aspects of Dag Prawitz's work or address open research problems that are concerned with core issues in structural proof theory and range from philosophical essays to papers of a mathematical nature. Investigations into the necessity of thought and the theory of grounds and computational justifications as well as an examination of Prawitz's conception of the validity of inferences in the light of three “dogmas of proof-theoretic semantics” are included. More formal papers deal with the constructive behaviour of fragments of classical logic and fragments of the modal logic S4 among other topics. In addition, there are chapters about inversion principles, normalization of proofs, and the notion of proof-theoretic harmony and other areas of a more mathematical persuasion. Dag Prawitz also writes a chapter in which he explains his current views on the epistemic dimension of proofs and addresses the question why some inferences succeed in conferring evidence on their conclusions when applied to premises for which one already possesses evidence. (shrink)

This volume is the first ever collection devoted to the field of proof-theoretic semantics. Contributions address topics including the systematics of introduction and elimination rules and proofs of normalization, the categorial characterization of deductions, the relation between Heyting's and Gentzen's approaches to meaning, knowability paradoxes, proof-theoretic foundations of set theory, Dummett's justification of logical laws, Kreisel's theory of constructions, paradoxical reasoning, and the defence of model theory. The field of proof-theoretic semantics has existed for almost 50 years, but the term (...) itself was proposed by Schroeder-Heister in the 1980s. Proof-theoretic semantics explains the meaning of linguistic expressions in general and of logical constants in particular in terms of the notion of proof. This volume emerges from presentations at the Second International Conference on Proof-Theoretic Semantics in Tübingen in 2013, where contributing authors were asked to provide a self-contained description and analysis of a significant research question in this area. The contributions are representative of the field and should be of interest to logicians, philosophers, and mathematicians alike. (shrink)

Methods available for the axiomatization of arbitrary finite-valued logics can be applied to obtain sound and complete intelim rules for all truth-functional connectives of classical logic including the Sheffer stroke and Peirce’s arrow. The restriction to a single conclusion in standard systems of natural deduction requires the introduction of additional rules to make the resulting systems complete; these rules are nevertheless still simple and correspond straightforwardly to the classical absurdity rule. Omitting these rules results in systems for intuitionistic versions of (...) the connectives in question. (shrink)

This essay demonstrates proof-theoretically the consistency of a type-free theoryC with an unrestricted principle of comprehension and based on a predicate logic in which contraction (A (A B)) (A B), although it cannot holds in general, is provable for a wide range ofA's.C is presented as an axiomatic theoryCH (with a natural-deduction equivalentCS) as a finitary system, without formulas of infinite length. ThenCH is proved simply consistent by passing to a Gentzen-style natural-deduction systemCG that allows countably infinite conjunctions and in (...) which all theorems ofCH are provable.CG is seen to be a consistent by a normalization argument. It also shown that in a senseC is highly non-extensional. (shrink)

The knowability paradox is an instance of a remarkable reasoning pattern (actually, a pair of such patterns), in the course of which an occurrence of the possibility operator, the diamond, disappears. In the present paper, it is pointed out how the unwanted disappearance of the diamond may be escaped. The emphasis is not laid on a discussion of the contentious premise of the knowability paradox, namely that all truths are possibly known, but on how from this assumption the conclusion is (...) derived that all truths are, in fact, known. Nevertheless, the solution offered is in the spirit of the constructivist attitude usually maintained by defenders of the anti-realist premise. In order to avoid the paradoxical reasoning, a paraconsistent constructive relevant modal epistemic logic with strong negation is defined semantically. The system is axiomatized and shown to be complete. (shrink)

We conducted a computer-based psychological experiment in which a random mix of 40 tautologies and 40 non-tautologies were presented to the participants, who were asked to determine which ones of the formulas were tautologies. The participants were eight university students in computer science who had received tuition in propositional logic. The formulas appeared one by one, a time-limit of 45 s applied to each formula and no aids were allowed. For each formula we recorded the proportion of the participants who (...) classified the formula correctly before timeout and the mean response time among those participants. We propose a new proof formalism for modeling propositional reasoning with bounded cognitive resources. It models declarative memory, visual memory, working memory, and procedural memory according to the memory model of Atkinson and Shiffrin and reasoning processes according to the model of Newell and Simon. We also define two particular proof systems, T and NT, for showing propositional formulas to be tautologies and non-tautologies, respectively. The accuracy was found to be higher for non-tautologies than for tautologies ( p T was.89 and for non-tautologies the correlation between latency and minimum proof length in NT was.87. (shrink)

First, we describe a psychological experiment in which the participants were asked to determine whether sentences of first-order logic were true or false in finite graphs. Second, we define two proof systems for reasoning about truth and falsity in first-order logic. These proof systems feature explicit models of cognitive resources such as declarative memory, procedural memory, working memory, and sensory memory. Third, we describe a computer program that is used to find the smallest proofs in the aforementioned proof systems when (...) capacity limits are put on the cognitive resources. Finally, we investigate the correlation between a number of mathematical complexity measures defined on graphs and sentences and some psychological complexity measures that were recorded in the experiment. (shrink)

Two common forms of natural deduction proof systems are found in the Gentzen–Prawitz and Jaśkowski–Fitch systems. In this paper, I provide translations between proofs in these systems, pointing out the ways in which the translations highlight the structural rules implicit in the systems. These translations work for classical, intuitionistic, and minimal logic. I then provide translations for classical S4 proofs.

Computationalism, the notion that cognition is computation, is a working hypothesis of many AI researchers and Cognitive Scientists. Although it has not been proved, neither has it been disproved. In this paper, I give some refutations to some well-known alleged refutations of computationalism. My arguments have two themes: people are more limited than is often recognized in these debates; computer systems are more complicated than is often recognized in these debates. To underline the latter point, I sketch the design and (...) abilities of a possible embodied computer system. (shrink)

What all contemporary so-called aristotelian realists have in common has been identified by David Armstrong as the principle of instantiation. This principle has been put forward in different versions, but all of them have the following simple consequence in common: uninstantiated universals do not exist. Such entities are for the lotus-eating Platonist to countenance, but not for any sort of moderate realist. I shall argue that this principle, in any guise, is not the best way to differentiate aristotelianism from Platonism. (...) In its place, I shall suggest that the best way to differentiate the two versions of realism from each other is by means of a far more powerful idea: naturalism. And the surprising conclusion given this means of differentiation will be that contrary to the usual proclamations, Platonism will be the more naturalistic theory, whereas aristotelianism will come to be seen for what it really is, namely, non-naturalistic. (shrink)

This essay is a logico?philosophical critique of the Marxian system of sociology with special reference to the theory of social change. To every change in the natural order (taken in conjunction with the technological order) corresponds an appropriate change in the human order, that is, in the system of social relations. This, it is shown, is the fundamental Marxian thesis regarding social equilibrium. And accordingly the key idea regarding social change is that a gradually maturing inherent disproportion between the two (...) orders is the basic condition of social change, eventually leading to the establishment of a new social equilibrium at a higher level. A close examination of the concept of mode of production and changes therein is attempted in the course of elucidating the above thesis. It is argued that the technological interpretation of the concept of mode of production conflicts with the Marxian anthropology and that we should have a structural interpretation according to which it will refer to the economy as a whole and will include among other elements production relations. This interpretation leads the discussion into one of the central problems of Marxism, namely, the relation between the mode of production and the social system. It is argued that the social system is basic to Marx and after a close analysis of the concept of system it is shown that the Marxian theory of base and superstructure rests mainly on confusion of conceptual orientations. This, it is shown, is connected with a fundamental idea of Marxism, viz., that of a system without a transcendent centre. This essay endeavours to prove that such an idea is logically untenable. Hence the Marxian Laws of Transformation of Quantity into Quality and Negation of Negation are examined and shown logically untenable. The last section discusses the question about the determinant of technological change and eventually of social change. Three possible Marxian solutions are examined: (a) in terms of needs ? it is argued that this is inconsistent with Marx's theory of man and labour; (b) in terms of science and new ideas ? it is shown that this ultimately goes contrary to Marxian materialism and positivism; and finally, (c) the systemic solution in terms of the dialectic of nature and man is subjected to logical scrutiny and it is shown that this cannot be consistent and valid without involving a trans?historical standpoint; for, it is argued, the idea of the self?directing Humanity on which it rests is a self?contradictory notion. (shrink)

Curry's paradox, sometimes described as a general version of the better known Russell's paradox, has intrigued logicians for some time. This paper examines the paradox in a natural deduction setting and critically examines some proposed restrictions to the logic by Fitch and Prawitz. We then offer a tentative counterexample to a conjecture by Tennant proposing a criterion for what is to count as a genuine paradox.

Logical inferentialism claims that the meaning of the logical constants should be given, not model-theoretically, but by the rules of inference of a suitable calculus. It has been claimed that certain proof-theoretical systems, most particularly, labelled deductive systems for modal logic, are unsuitable, on the grounds that they are semantically polluted and suffer from an untoward intrusion of semantics into syntax. The charge is shown to be mistaken. It is argued on inferentialist grounds that labelled deductive systems are as syntactically (...) pure as any formal system in which the rules define the meanings of the logical constants. (shrink)

Proofs from assumptions are amongst the most fundamental reasoning techniques. Yet the precise nature of assumptions is still an open topic. One of the most prominent conceptions is the placeholder view of assumptions generally associated with natural deduction for intuitionistic propositional logic. It views assumptions essentially as holes in proofs, either to be filled with closed proofs of the corresponding propositions via substitution or withdrawn as a side effect of some rule, thus in effect making them an auxiliary notion subservient (...) to proper propositions. The Curry–Howard correspondence is typically viewed as a formal counterpart of this conception. I will argue against this position and show that even though the Curry–Howard correspondence typically accommodates the placeholder view of assumptions, it is rather a matter of choice, not a necessity, and that another more assumption-friendly view can be adopted. (shrink)

States of affairs involving a non-symmetric relation such as loving are said to have a relational order, something that distinguishes, for instance, Romeo’s loving Juliet from Juliet’s loving Romeo. Relational order can be properly understood by appealing to o-roles, i.e., ontological counterparts of what linguists call thematic roles, e.g., agent, patient, instrument, and the like. This move allows us to meet the appropriate desiderata for a theory of relational order. In contrast, the main theories that try to do without o-roles, (...) proposed by philosophers such as Russell, Hochberg, and Fine, are in trouble with one or another of these desiderata. After discussing some alternatives, it is proposed that o-roles are best viewed as very generic properties characterizable as ways in which objects jointly exemplify a relation. This makes for exemplification relations understood as complex entities having o-roles as building blocks. (shrink)

In classical quantificational logic, every individual constant is assigned a value from the domain of discourse, thus ensuring that every instance of \\) is valid and so a theorem of a complete logic. Standard tense and modal logics validate a rule of necessitation, according to which, crudely, every theorem is always and necessarily true. Combining these two generates the result that everything always and necessarily exists. In a number of works from the late 1950s through to his death in 1969, (...) Prior worked to develop tense and modal logics that avoided these results. Prior’s key idea was to reject the rule of necessitation. In this essay, I present and criticize Prior’s account of the contingency of existence and offer a more satisfying account that shares many of the virtues of Prior’s account. (shrink)

Discontinuity refers to the character of many natural language constructions wherein signs differ markedly in their prosodic and semantic forms. As such it presents interesting demands on monostratal computational formalisms which aspire to descriptive adequacy. Pied piping, in particular, is argued by Pollard (1988) to motivate phrase structure-style feature percolation. In the context of categorial grammar, Bach (1981, 1984), Moortgat (1988, 1990, 1991) and others have sought to provide categorial operators suited to discontinuity. These attempts encounter certain difficulties with respect (...) to model theory and/or proof theory, difficulties which the current proposals are intended to resolve.Lambek calculus is complete for interpretation byresiduation with respect to the adjunction operation of groupoid algebras (Buszkowski 1986). In Moortgat and Morrill (1991) it is shown how to give calculi for families of categorial operators, each defined by residuation with respect to an operation of prosodic adjunction (associative, non-associative, or with interactive axioms). The present paper treats discontinuity in this way, by residuation with respect to three adjunctions: + (associative), (.,.) (split-point marking), andW (wrapping) related by the equations 1+s 2+s 3=(s 1,s 3)Ws 2. We show how the resulting methods apply to discontinuous functors, quantifier scope and quantifier scope ambiguity, pied piping, and subject and object antecedent reflexivisation. (shrink)

In this paper we present a new hierarchy of analytical tableaux systems TNDC n, 1≤n<ω, for da Costa's hierarchy of propositional paraconsistent logics Cn, 1≤n<ω. In our tableaux formulation, we introduce da Costa's “ball” operator “o”, the generalized operators “k” and “(k)”, for 1≤k, and the negations “~k”, for k≥1, as primitive operators, differently to what has been done in the literature, where these operators are usually defined operators. We prove a version of Cut Rule for the TNDC n, 1≤n<ω, (...) and also prove that these systems are logically equivalent to the corresponding systems Cn, 1≤n<ω. The systems TNDC n constitute completely automated theorem proving systems for the systems of da Costa's hierarchy Cn, 1≤n<ω. (shrink)

We critically investigate and refine Dunn's relevant predication, his formalisation of the notion of a real property. We argue that Dunn's original dialectical moves presuppose some interpretation of relevant identity, though none is given. We then re-motivate the proposal in a broader context, considering the prospects for a classical formalisation of real properties, particularly of Geach's implicit distinction between real and ''Cambridge'' properties. After arguing against these prospects, we turn to relevance logic, re-motivating relevant predication with Geach's distinction in mind. (...) Finally we draw out some consequences of Dunn's proposal for the theory of identity in relevance logic. (shrink)

ABSTRACTIn this paper, we present a generalisation of proof simulation procedures for Frege systems by Bonet and Buss to some logics for which the deduction theorem does not hold. In particular, we study the case of finite-valued Łukasiewicz logics. To this end, we provide proof systems and which augment Avron's Frege system HŁuk with nested and general versions of the disjunction elimination rule, respectively. For these systems, we provide upper bounds on speed-ups w.r.t. both the number of steps in proofs (...) and the length of proofs. We also consider Tamminga's natural deduction and Avron's hypersequent calculus GŁuk for 3-valued Łukasiewicz logic and generalise our results considering the disjunction elimination rule to all finite-valued Łukasiewicz logics. (shrink)

The standard natural deduction rules for the identity predicate have seemed to some not to be harmonious. Stephen Read has suggested an alternative introduction rule that restores harmony but presupposes second-order logic. Here it will be shown that the standard rules are in fact harmonious. To this end, natural deduction will be enriched with a theory of definitional identity. This leads to a novel conception of canonical derivation, on the basis of which the identity elimination rule can be justified in (...) a proof-theoretical manner. (shrink)

By maintaining that a conditional sentence can be taken to express the validity of a rule of inference, we offer a solution to the Miners Paradox that leaves both modus ponens and disjunction elimination intact. The solution draws on Sundholm's recently proposed account of Fitch's Paradox.

This article discusses deontic modality in the context of legal documents and its divergence from the natural, conventional, interpretation. This work demonstrates that the meaning of the performative verb is not purely linguistic. A number of non-linguistic factors cause the variation of meanings of performatives, in this case, when expressing prohibition, permission, recommendation, advice, proposal or request. These factors include: status of the addressee, type of the relationship between the author and the addressee, type of the document, possibility of control (...) and subsequent punishment for the breach of a norm, methodology of legal drafting, traditions of the Russian document processes. There is no direct association between the type of the legal norm and the meaning of the performative. To express its will, a regional law-maker, to a varying degree, uses orders, prohibition and permissions. These are dictated by the type of the document. (shrink)

In moral dilemmas, where circumstances prevent two or more equally justified prima facie ethical requirements from being fulfilled, it is often maintained that, since the agent cannot do both, conjoint obligation is overridden by Kant's principle that ought implies can, but that the agent nevertheless has a disjunctive obligation to perform one of the otherwise obligatory actions or the other. Against this commonly received view, it is demonstrated that although Kant's ought-can principle may avoid logical inconsistency, the principle is incompatible (...) with disjunctive obligation in standard deontic logic, and that it entails paradoxically that none of the conflicting dilemma actions will in fact occur. The principle appears to provide the only plausible safeguard against deontic antinomy, but cannot be admitted because of its collision with considered moral judgments. (shrink)

The paper is devoted to the concise description of some Natural Deduction System (ND for short) for Linear Temporal Logic. The system's distinctive feature is that it is labelled and analytical. Labels convey necessary semantic information connected with the rules for temporal functors while the analytical character of the rules lets the system work as a decision procedure. It makes it more similar to Labelled Tableau Systems than to standard Natural Deduction. In fact, our solution of linearity representation is rather (...) independent of the underlying proof method, provided that some form of (analytic) cut is admissible. We will also discuss some generalisations of the system and compare it with other formalizations of linearity. (shrink)

A sentence mentioning an object can be regarded as saying any one of several things about that object, without thereby being ambiguous. Some of the (logical) repercussions of this commonplace observation are recorded, and some critical discussion is provided of views which would appear to go against it.

The state of affairs of some things falling under a predicate is supposedly a single entity that collects these things as its constituents. But whether we think of a state of affairs as a fact, a proposition or a possibility, problems will arise if we adopt a plural logic. For plural logic says that any plurality include themselves, so whenever there are some things, the state of affairs of their plural self-inclusion should be a single thing that collects them all. (...) This leads to paradoxes analogous to those that afflict naïve set theory. Here I suggest that they are the very same paradoxes, because sets can be reduced to states of affairs. However, to obtain a consistent theoretical reduction we must restrict the usual axiom scheme of Comprehension for plural logic to ‘stratified’ formulas, to avoid viciously circular definitions. I prove that with this modification to the background plural logic, the theory of states of affairs is consistent; moreover, it yields the axioms of the familiar set theory NFU. (shrink)