Chapter 1 constitutes an introduction to Gentzen calculi from two perspectives, logical and philosophical. It introduces the notion of generalisations of Gentzen sequent calculus and the discussion on properties that characterize good inferential systems. Among the variety of Gentzen-style sequent calculi, I divide them in two groups: syntactic and semantic generalisations. In the context of such a discussion, the inferentialist philosophy of the meaning of logical constants is introduced, and some potential objections – mainly concerning the choice (...) of working with semantic generalizations – are addressed. Finally, I’ll introduce the case studies that I’ll be dealing with in part II. Chapter 2 is concerned with the origins and development of Jaśkowski’s discussive logic. The main idea of this chapter is to systematize the various stages of the development of discussive logic related researches from two different angles, i.e., its connections to modal logics and its proof theory, by highlighting virtues and vices. Chapter 3 focuses on the Gentzen-style proof theory of discussive logic, by providing a characterization of it in terms of labelled sequent calculi. Chapter 4 deals with the Gentzen-style proof theory of relevant logics, again by employing the methodology of labelled sequent calculi. This time, instead of working with a single logic, I’ll deal with a whole family of them. More precisely, I’ll study in terms of proof systems those relevant logics that can be characterised, at the semantic level, by reduced Routley-Meyer models, i.e., relational structures with a ternary relation between states and a unique base element. Chapter 5 investigates the proof theory of a modal expansion of intuitionistic propositional logic obtained by adding an ‘actuality’ operator to the connectives. This logic was introduced also using Gentzen sequents. Unfortunately, the original proof system is not cut-free. This chapter shows how to solve this problem by moving to hypersequents. Chapter 6 concludes the investigations and discusses the future of the research presented throughout the dissertation. (shrink)

This revised and considerably expanded 2nd edition brings together a wide range of topics, including modal, tense, conditional, intuitionist, many-valued, paraconsistent, relevant, and fuzzy logics. Part 1, on propositional logic, is the old Introduction, but contains much new material. Part 2 is entirely new, and covers quantification and identity for all the logics in Part 1. The material is unified by the underlying theme of world semantics. All of the topics are explained clearly using devices such as (...) tableau proofs, and their relation to current philosophical issues and debates are discussed. Students with a basic understanding of classical logic will find this book an invaluable introduction to an area that has become of central importance in both logic and philosophy. It will also interest people working in mathematics and computer science who wish to know about the area. (shrink)

Relevant logics are a family of non-classical logics characterized by the behavior of their implication connectives. Unlike some other non-classical logics, such as intuitionistic logic, there are multiple philosophical views motivating relevant logics. Further, different views seem to motivate different logics. In this article, we survey five major views motivating the adoption of relevant logics: Use Criterion, sufficiency, meaning containment, theory construction, and truthmaking. We highlight the philosophical differences as well as the (...) different logics they support. We end with some questions for future research. (shrink)

We present a framework for machine implementation of families of non-classical logics with Kripke-style semantics. We decompose a logic into two interacting parts, each a natural deduction system: a base logic of labelled formulae, and a theory of labels characterizing the properties of the Kripke models. By appropriate combinations we capture both partial and complete fragments of large families of non-classical logics such as modal, relevance, and intuitionistic logics. Our approach is modular and supports uniform (...) proofs of soundness, completeness and proof normalization. We have implemented our work in the Isabelle Logical Framework. (shrink)

Until not too many years ago, all logics except classical logic (and, perhaps, intuitionistic logic too) were considered to be things esoteric. Today this state of a airs seems to have completely been changed. There is a growing interest in many types of nonclassical logics: modal and temporal logics, substructural logics, paraconsistent logics, non-monotonic logics { the list is long. The diversity of systems that have been proposed and studied is so great that a need (...) is felt by many researchers to try to put some order in the present logical jungle. Thus Cl91], Ep90] and Wo88] are three recent books in which an attempt is made to develop a general theoretical framework for the study of logics. On the more pragmatic side, several systems have been developed with the goal of providing a computerized logical framework in which many di erent logical systems can be implemented in a uniform way. An example is the Edinburgh LF( HHP91]). (shrink)

Given classical (2 valued) structures and and a homomorphism h of onto , it is shown how to construct a (non-degenerate) 3-valued counterpart of . Classical sentences that are true in are non-false in . Applications to number theory and type theory (with axiom of infinity) produce finite 3-valued models in which all classically true sentences of these theories are non-false. Connections to relevant logic give absolute consistency proofs for versions of these theories formulated in (...) class='Hi'>relevant logic (the proof for number theory was obtained earlier by R. K. Meyer and suggested the present abstract development). (shrink)

This book is a collection of contributions honouring Arnon Avron’s seminal work on the semantics and proof theory of non-classical logics. It includes presentations of advanced work by some of the most esteemed scholars working on semantic and proof-theoretical aspects of computer science logic. Topics in this book include frameworks for paraconsistent reasoning, foundations of relevance logics, analysis and characterizations of modal logics and fuzzy logics, hypersequent calculi and their properties, non-deterministic semantics, algebraic structures for (...) many-valued logics, and representations of the mechanization of mathematics. Avron’s foundational and pioneering contributions have been widely acknowledged and adopted by the scientific community. His research interests are very broad, spanning over proof theory, automated reasoning, non-classical logics, foundations of mathematics, and applications of logic in computer science and artificial intelligence. This is clearly reflected by the diversity of topics discussed in the chapters included in this book, all of which directly relate to Avron’s past and present works. This book is of interest to computer scientists and scholars of formal logic. (shrink)

There is an intimate connection between proofs of the natural deduction systems and typed lambda calculus. It is well-known that in simply typed lambda calculus, the notion of formulae-as-types makes it possible to find fine structure of the implicational fragment of intuitionistic logic, i.e., relevant logic, BCK-logic and linear logic. In this paper, we investigate three classical substructural logics (GL, GLc, GLw) of Gentzen's sequent calculus consisting of implication and negation, which contain some (...) of the right structural rules. In terms of Parigot's -calculus with proper restrictions, we introduce a proof term assignment to these classical substructural logics. According to these notions, we can classify the -terms into four categories. It is proved that well-typed GLx--terms correspond to GLx proofs, and that a GLx--term has a principal type if stratified where x is nil, c, w or cw. Moreover, we investigate embeddings of classical substructural logics into the corresponding intuitionistic substructural logics. It is proved that the Gödel-style translations of GLx--terms are embeddings preserving substructural logics. As by-products, it is obtained that an inhabitation problem is decidable and well-typed GLx--terms are strongly normalizable. (shrink)

The aim of this paper is to provide a proof-theoretic characterization of relevant logics including fusion and fission connectives, as well as Ackermann’s truth constant. We achieve this by employing the well-established methodology of labelled sequent calculi. After having introduced several systems, we will conduct a detailed proof-theoretic analysis, show a cut-admissibility theorem, and establish soundness and completeness. The paper ends with a discussion that contextualizes our current work within the broader landscape of the proof (...) theory of relevant logics. (shrink)

This monograph presents a general theory of weakly implicative logics, a family covering a vast number of non-classical logics studied in the literature, concentrating mainly on the abstract study of the relationship between logics and their algebraic semantics. It can also serve as an introduction to algebraic logic, both propositional and first-order, with special attention paid to the role of implication, lattice and residuated connectives, and generalized disjunctions. Based on their recent work, the authors develop a powerful (...) uniform framework for the study of non-classical logics. In a self-contained and didactic style, starting from very elementary notions, they build a general theory with a substantial number of abstract results. The theory is then applied to obtain numerous results for prominent families of logics and their algebraic counterparts, in particular for superintuitionistic, modal, substructural, fuzzy, and relevant logics. The book may be of interest to a wide audience, especially students and scholars in the fields of mathematics, philosophy, computer science, or related areas, looking for an introduction to a general theory of non-classical logics and their algebraic semantics. (shrink)

The volume deals with the history of logic, the question of the nature of logic, the relation of logic and mathematics, modal or alternative logics (many-valued, relevant, paraconsistent logics) and their relations, including translatability, to classical logic in the Fregean and Russellian sense, and, more generally, the aim or aims of philosophy of logic and mathematics. Also explored are several problems concerning the concept of definition, non-designating terms, the interdependence of quantifiers, (...) and the idea of an assertion sign. The contributions concerned with Wittgenstein's investigations into the philosophy of logic and mathematics pursue issues relating to logical necessity, the undeniability of the law of the excluded middle, and the source of self-evidence, often characterized in the literature as the "rule-following considerations". Additionally, they examine Wittgenstein's attitudes towards the very idea of set-theory as a possible foundation for arithmetic. The volume also includes a number of contributions on specific issues concerning Wittgenstein's views on moral and religious judgements. (shrink)

Resolution is an effective deduction procedure for classical logic. There is no similar "resolution" system for non-classical logics (though there are various automated deduction systems). The paper presents resolution systems for intuistionistic predicate logic as well as for modal and temporal logics within the framework of labelled deductive systems. Whereas in classical predicate logic resolution is applied to literals, in our system resolution is applied to L(abelled) R(epresentation) S(tructures). Proofs are discovered by a refutation (...) procedure defined on LRSs, that imposes a hierarchy on clause sets of such structures together with an inheritance discipline. This is a form of Theory Resolution. For intuitionistic logic these structures are called I(ntuitionistic) R(epresentation) S(tructures). Their hierarchical structure allows the restriction of unification of individual variables and/or constants without using Skolem functions. This structures must therefore be preserved when we consider other (non-modal) logics. Variations between different logics are captured by fine tuning of the inheritance properties of the hierarchy. For modal and temporal logics IRS's are extended to structures that represent worlds and/or times. This enables us to consider all kinds of combined logics. (shrink)

This paper uses Avron’s algebraic semantics for the logic RMI to model some ideas in the philosophy of science. Avron’s relevant disjunctive structures are each partitioned into relevance domains. Each relevance domain is a boolean algebra. I employ this semantics to act as a formal framework to represent what Nancy Cartwright calls the “dappled world”. On the dappled world hypothesis, local scientific theories each represent restricted aspects and regions of the universe. I use relevance domains to represent (...) the domains of each of these local theories and I provide a formalisation of the salient relationships between so-called fundamental theories and local theories. I also examine ways in which the paraconsistent nature of RMI can be used to deal with inconsistencies within and between theories adopted by scientists. The paper ends with some suggestions about updating RDS given changes in the theories that science adopts. (shrink)

A logic is called 'paraconsistent' if it rejects the rule called 'ex contradictione quodlibet', according to which any conclusion follows from inconsistent premises. While logicians have proposed many technically developed paraconsistent logical systems and contemporary philosophers like Graham Priest have advanced the view that some contradictions can be true, and advocated a paraconsistent logic to deal with them, until recent times these systems have been little understood by philosophers. This book presents a comprehensive overview (...) on paraconsistent logical systems to change this situation. The book includes almost every major author currently working in the field. The papers are on the cutting edge of the literature some of which discuss current debates and others present important new ideas. The editors have avoided papers about technical details of paraconsistent logic, but instead concentrated upon works that discuss more 'big picture' ideas. Different treatments of paradoxes takes centre stage in many of the papers, but also there are several papers on how to interpret paraconistent logic and some on how it can be applied to philosophy of mathematics, the philosophy of language, and metaphysics. (shrink)

This chapter focuses on the development of Gerhard Gentzen's structural proof theory and its connections with intuitionism. The latter is important in proof theory for several reasons. First, the methods of Hilbert's old proof theory were limited to the “finitistic” ones. These methods proved to be insufficient, and they were extended by infinitistic principles that were still intuitionistically meaningful. It is a general tendency in proof theory to try to use weak principles. A second reason for (...) the importance of intuitionism for proof theory is that the proof-theoretical study of intuitionistic logic has become a prominent part of logic, with applications in computer science. (shrink)

Proofs and countermodels are the two sides of completeness proofs, but, in general, failure to find one does not automatically give the other. The limitation is encountered also for decidable non-classical logics in traditional completeness proofs based on Henkin’s method of maximal consistent sets of formulas. A method is presented that makes it possible to establish completeness in a direct way: For any given sequent either a proof in the given logical system or a countermodel in the (...) corresponding frame class is found. The method is a synthesis of a generation of calculi with internalized relational semantics, a Tait–Schütte–Takeuti style completeness proof, and procedures to finitize the countermodel construction. Finitizations for intuitionistic propositional logic are obtained through the search for a minimal derivation, through pruning of infinite branches in search trees by means of a suitable syntactic counterpart of semantic filtration, or through a proof-theoretic embedding into an appropriate provability logic. A number of examples illustrates the method, its subtleties, challenges, and present scope. (shrink)

Depending on what one means by the main connective of logic, the -if..., then... -, several systems of logic result: classic and modal logics, intuitionistic logic or relevance logic. This book presents the underlying ideas, the syntax and the semantics of these logics. Soundness and completeness are shown constructively and in a uniform way. Attention is paid to the interdisciplinary role of logic: its embedding in the foundations of mathematics and its intimate connection with (...) philosophy, in particular the philosophy of language. Set theory is presented both as a conditio sine qua non for logic and as a interesting exact ontology. The study of infinite sets yields perplexing results. Formalization of informal number theory results in formal number theory; Godel's incompleteness is treated. At appropriate places attention is paid to paradoxes, intuitionism, conditionals, the historical development of logic, to logic programming and automated theorem proving for classical logic.". (shrink)

This book is dedicated to V.A. Yankov’s seminal contributions to the theory of propositional logics. His papers, published in the 1960s, are highly cited even today. The Yankov characteristic formulas have become a very useful tool in propositional, modal and algebraic logic. The papers contributed to this book provide the new results on different generalizations and applications of characteristic formulas in propositional, modal and algebraic logics. In particular, an exposition of Yankov’s results and their applications in algebraic logic, (...) the theory of admissible rules and refutation systems is included in the book. In addition, the reader can find the studies on splitting and join-splitting in intermediate propositional logics that are based on Yankov-type formulas which are closely related to canonical formulas, and the study of properties of predicate extensions of non-classical propositional logics. The book also contains an exposition of Yankov’s revolutionary approach to constructive proof theory. The editors also include Yankov’s contributions to history and philosophy of mathematics and foundations of mathematics, as well as an examination of his original interpretation of history of Greek philosophy and mathematics. (shrink)

In this paper we extend to non-classical set theories the standard strategy of proving independence using Boolean-valued models. This extension is provided by means of a new technique that, combining algebras (by taking their product), is able to provide product-algebra-valued models of set theories. In this paper we also provide applications of this new technique by showing that: (1) we can import the classical independence results to non-classical set theory (as an example we prove the independence of (...) $\mathsf {CH}$ ); and (2) we can provide new independence results. We end by discussing the role of non-classical algebra-valued models for the debate between universists and multiversists and by arguing that non-classical models should be included as legitimate members of the multiverse. (shrink)

The subject of Labelled Non-Classical Logics is the development and investigation of a framework for the modular and uniform presentation and implementation of non-classical logics, in particular modal and relevance logics. Logics are presented as labelled deduction systems, which are proved to be sound and complete with respect to the corresponding Kripke-style semantics. We investigate the proof theory of our systems, and show them to possess structural properties such as normalization and the subformula property, which we exploit (...) not only to establish advantages and limitations of our approach with respect to related ones, but also to give, by means of a substructural analysis, a new proof-theoretic method for investigating decidability and complexity of (some of) the logics we consider. All of our deduction systems have been implemented in the generic theorem prover Isabelle, thus providing a simple and natural environment for interactive proof development. Labelled Non-Classical Logics is essential reading for researchers and practitioners interested in the theory and applications of non-classical logics. (shrink)

This article presents modal versions of resource-conscious logics. We concentrate on extensions of variants of linear logic with one minimal non-normal modality. In earlier work, where we investigated agency in multi-agent systems, we have shown that the results scale up to logics with multiple non-minimal modalities. Here, we start with the language of propositional intuitionistic linear logic without the additive disjunction, to which we add a modality. We provide an interpretation of this language on a class of (...) Kripke resource models extended with a neighbourhood function: modal Kripke resource models. We propose a Hilbert-style axiomatisation and a Gentzen-style sequent calculus. We show that the proof theories are sound and complete with respect to the class of modal Kripke resource models. We show that the sequent calculus admits cut elimination and that proof-search is in PSPACE. We then show how to extend the results when non-commutative connectives are added to the language. Finally, we put the l.. (shrink)

Free logics are a family of first-order logics which came about as a result of examining the existence assumptions of classical logic (Hintikka _The Journal of Philosophy_, _56_, 125–137 1959 ; Lambert _Notre Dame Journal of Formal Logic_, _8_, 133–144 1967, 1997, 2001 ). What those assumptions are varies, but the central ones are that (i) the domain of interpretation is not empty, (ii) every name denotes exactly one object in the domain and (iii) the quantifiers have existential (...) import. Free logics reject the claim that names need to denote in (ii). _Positive_ free logic concedes that some atomic formulas containing non-denoting names (including self-identity) are true, _negative_ free logic treats them as uniformly false, and _neutral_ free logic as taking a third value. There has been a renewed interest in analyzing proof theory of free logic in recent years, based on intuitionistic logic in Maffezioli and Orlandelli (_Bulletin of the Section of Logic_, _48_(2), 137–158 2019 ) as well as classical logic in Pavlović and Gratzl (_Journal of Philosophical Logic_, _50_, 117–148 2021 ), there for the positive and negative variants. While the latter streamlines the presentation of free logics and offers a more unified approach to the variants under consideration, it does not cover neutral free logic, since there is some lack of both clear formal intuitions on the semantic status of formulas with empty names, as well as a satisfying account of the conditional in this context. We discuss extending the results to this third major variant of free logics. We present a series of G3 sequent calculi adapted from Fjellstad (_Studia Logica_, _105_(1), 93–119 2017, _Journal of Applied Non-Classical Logics_, _30_(3), 272–289 2020 ), which possess all the desired structural properties of a good proof system, including admissibility of contraction and all versions of the cut rule. At the same time, we maintain the unified approach to free logics and moreover argue that greater clarity of intuitions is achieved once neutral free logic is conceptualized as consisting of two sub-varieties. (shrink)

This article introduces, studies, and applies a new system of logic which is called ‘HYPE’. In HYPE, formulas are evaluated at states that may exhibit truth value gaps and truth value gluts. Simple and natural semantic rules for negation and the conditional operator are formulated based on an incompatibility relation and a partial fusion operation on states. The semantics is worked out in formal and philosophical detail, and a sound and complete axiomatization is provided both for the propositional and (...) the predicate logic of the system. The propositional logic of HYPE is shown to contain first-degree entailment, to have the Finite Model Property, to be decidable, to have the Disjunction Property, and to extend intuitionistic propositional logic conservatively when intuitionistic negation is defined appropriately by HYPE’s logical connectives. Furthermore, HYPE’s first-order logic is a conservative extension of intuitionistic logic with the Constant Domain Axiom, when intuitionistic negation is again defined appropriately. The system allows for simple model constructions and intuitive Euler-Venn-like diagrams, and its logical structure matches structures well-known from ordinary mathematics, such as from optimization theory, combinatorics, and graph theory. HYPE may also be used as a general logical framework in which different systems of logic can be studied, compared, and combined. In particular, HYPE is found to relate in interesting ways to classical logic and various systems of relevance and paraconsistent logic, many-valued logic, and truthmaker semantics. On the philosophical side, if used as a logic for theories of type-free truth, HYPE is shown to address semantic paradoxes such as the Liar Paradox by extending non-classical fixed-point interpretations of truth by a conditional as well-behaved as that of intuitionistic logic. Finally, HYPE may be used as a background system for modal operators that create hyperintensional contexts, though the details of this application need to be left to follow-up work. (shrink)

This work contributes to the theory of judgement aggregation by discussing a number of significant non-classical logics. After adapting the standard framework of judgement aggregation to cope with non-classical logics, we discuss in particular results for the case of Intuitionistic Logic, the Lambek calculus, Linear Logic and Relevant Logics. The motivation for studying judgement aggregation in non-classical logics is that they offer a number of modelling choices to represent agents’ reasoning in aggregation problems. (...) By studying judgement aggregation in logics that are weaker than classical logic, we investigate whether some well-known impossibility results, that were tailored for classical logic, still apply to those weak systems. (shrink)

Sequent calculi constitute an interesting and important category of proof systems. They are much less known than axiomatic systems or natural deduction systems are, and they are much less known than they should be. Sequent calculi were designed as a theoretical framework for investigations of logical consequence, and they live up to the expectations completely as an abundant source of meta-logical results. The goal of this book is to provide a fairly comprehensive view of sequent calculi -- (...) including a wide range of variations. The focus is on sequent calculi for various non-classical logics, from intuitionistic logic to relevance logic, through linear and modal logics. A particular version of sequent calculi, the so-called consecution calculi, have seen important new developments in the last decade or so. The invention of new consecution calculi for various relevance logics allowed the last major open problem in the area of relevance logic to be solved positively: pure ticket entailment is decidable. An exposition of this result is included in chapter 9 together with further new decidability results (for less famous systems). A series of other results that were obtained by J. M. Dunn and me, or by me in the last decade or so, are also presented in various places in the book. Some of these results are slightly improved in their current presentation. Obviously, many calculi and several important theorems are not new. They are included here to ensure the completeness of the picture; their original formulations may be found in the referenced publications. This book contains very little about semantics, in general, and about the semantics of non-classical logic in particular. (shrink)

In this paper we give a brief treatment of a theory of proofs for a system of Full Intuitionistic Linear Logic. This system is distinct from Classical Linear Logic, but unlike the standard Intuitionistic Linear Logic of Girard and Lafont includes the multiplicative disjunction par. This connective does have an entirely natural interpretation in a variety of categorical models of Intuitionistic Linear Logic. The main proof-theoretic problem arises from the observation of (...) Schellinx that cut elimination fails outright for an intuitive formulation of Full Intuitionistic Linear Logic; the nub of the problem is the interaction between par and linear implication. We present here a term assignment system which gives an interpretation of proofs as some kind of non-deterministic function. In this way we find a system which does enjoy cut elimination. The system is a direct result of an analysis of the categorical semantics, though we make an effort to present the system as if it were purely a proof-theoretic construction. (shrink)

Greek, Indian and Arabic Logic marks the initial appearance of the multi-volume Handbook of the History of Logic. Additional volumes will be published when ready, rather than in strict chronological order. Soon to appear are The Rise of Modern Logic: From Leibniz to Frege. Also in preparation are Logic From Russell to Gödel, The Emergence of Classical Logic, Logic and the Modalities in the Twentieth Century, and The Many-Valued and Non-Monotonic Turn in (...) class='Hi'>Logic. Further volumes will follow, including Mediaeval and Renaissance Logic and Logic: A History of its Central. In designing the Handbook of the History of Logic, the Editors have taken the view that the history of logic holds more than an antiquarian interest, and that a knowledge of logic's rich and sophisticated development is, in various respects, relevant to the research programmes of the present day. Ancient logic is no exception. The present volume attests to the distant origins of some of modern logic's most important features, such as can be found in the claim by the authors of the chapter on Aristotle's early logic that, from its infancy, the theory of the syllogism is an example of an intuitionistic, non-monotonic, relevantly paraconsistent logic. Similarly, in addition to its comparative earliness, what is striking about the best of the Megarian and Stoic traditions is their sophistication and originality. Logic is an indispensably important pivot of the Western intellectual tradition. But, as the chapters on Indian and Arabic logic make clear, logic's parentage extends more widely than any direct line from the Greek city states. It is hardly surprising, therefore, that for centuries logic has been an unfetteredly international enterprise, whose research programmes reach to every corner of the learned world. Like its companion volumes, Greek, Indian and Arabic Logic is the result of a design that gives to its distinguished authors as much space as would be needed to produce highly authoritative chapters, rich in detail and interpretative reach. The aim of the Editors is to have placed before the relevant intellectual communities a research tool of indispensable value. Together with the other volumes, Greek, Indian and Arabic Logic, will be essential reading for everyone with a curiosity about logic's long development, especially researchers, graduate and senior undergraduate students in logic in all its forms, argumentation theory, AI and computer science, cognitive psychology and neuroscience, linguistics, forensics, philosophy and the history of philosophy, and the history of ideas. (shrink)

Goal Directed Proof Theory presents a uniform and coherent methodology for automated deduction in non-classical logics, the relevance of which to computer science is now widely acknowledged. The methodology is based on goal-directed provability. It is a generalization of the logic programming style of deduction, and it is particularly favourable for proof search. The methodology is applied for the first time in a uniform way to a wide range of non-classical systems, covering intuitionistic, intermediate, (...) modal and substructural logics. The book can also be used as an introduction to these logical systems form a procedural perspective. Readership: Computer scientists, mathematicians and philosophers, and anyone interested in the automation of reasoning based on non-classical logics. The book is suitable for self study, its only prerequisite being some elementary knowledge of logic and proof theory. (shrink)

The present article shows that there are consistent and decidable manyvalued systems of propositional logic which satisfy two or all the three criteria for non-trivial inconsistent theories by da Costa . The weaker one of these paraconsistent system is also able to avoid a series of paradoxes which come up when classical logic is applied to empirical sciences. These paraconsistent systems are based on a 6-valued system of propositional logic for avoiding difficulties in several (...) domains of empirical science ). (shrink)

This chapter contains sections titled: Non‐Sequiturs are Bad The Real Use of Premises Implication From Proof Theory to Semantics Adding Conjunction The Problem of Disjunction Routley and Meyer's Ternary Relation Rules for Disjunction The Semantics of Negation Rules for Negation Disjunctive Syllogism Logics Stronger than R Logics Weaker than R Relevant Logics and Natural Language Conditionals Theory of Properties Summary.

This paper gives an account of Anderson and Belnap’s selection criteria for an adequate theory of entailment. The criteria are grouped into three categories: criteria pertaining to modality, those pertaining to relevance, and those related to expressive strength. The leitmotif of both this paper and its prequel is the relevant legitimacy of disjunctive syllogism. Relevant logics are commonly held to be paraconsistent logics. It is shown in this paper, however, that both E and R can be extended (...) to explosive logics which satisfy all of Anderson and Belnap’s selection criteria, provided the truth-constant known as the Ackermann constant is available. One of the selection criteria related to expressive strength is having an “enthymematic” conditional for which a deduction theorem holds. I argue that this allows for a new interpretation of Anderson and Belnap’s take on logical consequence, namely as committing them to pluralism about logical consequence. (shrink)

We provide a direct method for proving Craig interpolation for a range of modal and intuitionistic logics, including those containing a "converse" modality. We demonstrate this method for classical tense logic, its extensions with path axioms, and for bi-intuitionistic logic. These logics do not have straightforward formalisations in the traditional Gentzen-style sequent calculus, but have all been shown to have cut-free nested sequent calculi. The proof of the interpolation theorem uses these calculi and is (...) purely syntactic, without resorting to embeddings, semantic arguments, or interpreted connectives external to the underlying logical language. A novel feature of our proof includes an orthogonality condition for defining duality between interpolants. (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 (...) class='Hi'>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)

Without directly addressing the Demarcation Problem for logic—the problem of distinguishing logical vocabulary from others—we focus on distinctive aspects of logical vocabulary in pursuit of a second goal in the philosophy of logic, namely, proposing criteria for the justification of logical rules. Our preferred approach has three components. Two of these are effectively Belnap’s, but with a twist. We agree with Belnap’s response to Prior’s challenge to inferentialist characterisations of the meanings of logical (...) constants. Belnap argued that for a logical constant to exist, its rules must be conservative over a previously given consequence relation and guarantee the uniqueness of the constant. The twist is that we require logical vocabulary to be provably conservative, not over a previously given formal consequence relation, but over a previously given meaningful vocabulary of a language. Uniqueness is also a feature defined in terms of provability: if two syntactically distinguished expressions are governed by the same rules, then formulas with them as main operators are interderivable. Belnap’s criteria are not only those for the existence of a logical constant, but more: they are what distinguishes logical vocabulary from all other expressions. It is the defining mark of a logical constant that it is provably conservative over the fragment of the language which excludes it and that its rules guarantee its uniqueness. The third component is the topic neutrality of logic. The provable conservativeness of logic over a previously given vocabulary of a language is motivated, in part, by appeal to molecularism in the theory of meaning. Molecularity is a feature endorsed by the theory of meaning as a whole, so it does not distinguish logical constants from other expressions. The same is true for conservativeness. We argue that molecularity, and hence conservativeness, are implicit presuppositions of speakers’ use of language: the addition of new vocabulary to a language is presupposed to be conservative. But this presupposition is one that we should be able reflectively to endorse (such as when attempting a rational reconstruction of the use of the vocabulary). Thus, where conservativeness is found to fail, this defect should be remedied and the use of the vocabulary corrected (as in classical negation) or excised altogether (as in pejoratives). We go on to characterise a notion of topic neutrality, which we argue applies to logical vocabulary. We then note that reflective endorsement of the conservativeness of a topic neutral vocabulary requires a proof that the vocabulary is conservative relative to any base vocabulary. Thus we require that logical vocabulary be demonstrably conservative. Allying this requirement, distinctive of logic, to a general consideration about the commitments of assertion yields a mode of justification for logical constants akin to some conceptions of Harmony, i.e., to the idea that the consequences of assertion of a logical complex need to be warranted by the grounds and ought to be the strongest consequences warranted by the grounds. Intuitionistic logic acquires a somewhat familiar justification but emanating from a new motivation. (shrink)

We propose a distinction between two different concepts of time that play a role in physics: geometric time and creative time. The former is the time of deterministic physics and merely parametrizes a given evolution. The latter is instead characterized by real change, i.e. novel information that gets created when a non-necessary event becomes determined in a fundamentally indeterministic physics. This allows us to give a naturalistic characterization of the present as the moment that separates the potential future from the (...) determined past. We discuss how these two concepts find natural applications in classical and intuitionistic mathematics, respectively, and in classical and multivalued tensed logic, as well as how they relate to the well-known A- and B-theories in the philosophy of time. (shrink)

We construct an extension P of the standard language of classical propositional logic by adjoining to the alphabet of a new category of logical-pragmatic signs. The well formed formulas of are calledradical formulas (rfs) of P;rfs preceded by theassertion sign constituteelementary assertive formulas of P, which can be connected together by means of thepragmatic connectives N, K, A, C, E, so as to obtain the set of all theassertive formulas (afs). Everyrf of P is endowed with (...) atruth value defined classically, and everyaf is endowed with ajustification value, defined in terms of the intuitive notion of proof and depending on the truth values of its radical subformulas. In this framework, we define the notion ofpragmatic validity in P and yield a list of criteria of pragmatic validity which hold under the assumption that only classical metalinguistic procedures of proof be accepted. We translate the classical propositional calculus (CPC) and the intuitionistic propositional calculus (IPC) into the assertive part of P and show that this translation allows us to interpret Intuitionistic Logic as an axiomatic theory of the constructive proof concept rather than an alternative to Classical Logic. Finally, we show that our framework provides a suitable background for discussing classical problems in the philosophy of logic. (shrink)

We give a proof-theoretic as well as a semantic characterization of a logic in the signature with conjunction, disjunction, negation, and the universal and existential quantifiers that we suggest has a certain fundamental status. We present a Fitch-style natural deduction system for the logic that contains only the introduction and elimination rules for the logical constants. From this starting point, if one adds the rule that Fitch called Reiteration, one obtains a proof system for (...) class='Hi'>intuitionistic logic in the given signature; if instead of adding Reiteration, one adds the rule of Reductio ad Absurdum, one obtains a proof system for orthologic; by adding both Reiteration and Reductio, one obtains a proof system for classical logic. Arguably neither Reiteration nor Reductio is as intimately related to the meaning of the connectives as the introduction and elimination rules are, so the base logic we identify serves as a more fundamental starting point and common ground between proponents of intuitionistic logic, orthologic, and classical logic. The algebraic semantics for the logic we motivate proof-theoretically is based on bounded lattices equipped with what has been called a weak pseudocomplementation. We show that such lattice expansions are representable using a set together with a reflexive binary relation satisfying a simple first-order condition, which yields an elegant relational semantics for the logic. This builds on our previous study of representations of lattices with negations, which we extend and specialize for several types of negation in addition to weak pseudocomplementation; in an appendix, we further extend this representation to lattices with implications. Finally, we discuss adding to our logic a conditional obeying only introduction and elimination rules, interpreted as a modality using a family of accessibility relations. (shrink)

This book discusses the philosophical work of Décio Krause. Non-individuality, as a new metaphysical category, was thought to be strongly supported by quantum mechanics. No one did more to promote this idea than the Brazilian philosopher Décio Krause, whose works on the metaphysics and logic of non-individuality are now widely regarded as part of the consolidated literature on the subject. This volume brings together chapters elaborating on the ideas put forward and defended by Krause, developing them in many different (...) directions, commenting on aspects not completely developed so far, and, more importantly, critically addressing their current formulations and defenses by Krause himself. Given that Krause’s ideas do connect directly and indirectly with a wide array of subjects, such as the philosophy of quantum mechanics, more broadly understood, the philosophy of logic and logical philosophy, non-classical logics, metaphysics, and ontology, this volume contains important material for the research on logic and foundations of science, broadly understood. All the invited contributors have already worked with the ideas developed by Décio (some of them still work with them), being also distinct authors and extremely relevant in their areas of expertise. The volume is aimed at philosophers, including those of physics and quantum mechanics. (shrink)

This paper concerns the paraconsistent logic LPQ⊃,F and an application of it in the area of relational database theory. The notions of a relational database, a query applicable to a relational database, and a consistent answer to a query with respect to a possibly inconsistent relational database are considered from the perspective of this logic. This perspective enables among other things the definition of a consistent answer to a query with respect to a possibly inconsistent database without (...) resort to database repairs. In an earlier paper, LPQ⊃,F is presented with a sequent-style natural deduction proof system. In this paper, a sequent calculus proof system is presented instead because such proof systems are generally considered more suitable as the basis of proof search procedures than natural deduction proof systems and proof search procedures can serve as the core of algorithms for computing consistent answers to queries. (shrink)

Although the theory of definability had many important antecedents—such as the descriptive set theory initiated by the French semi-intuitionists in the early 1900s—the main ideas were first laid out in precise mathematical terms by Alfred Tarski beginning in 1929. We review here the basic notions of languages, explicit definability, and grammatical complexity, and emphasize common themes in the theories of definability for four important languages underlying, respectively, descriptive set theory, recursive function theory, classical pure logic, and finite-universe (...) class='Hi'>logic. We review the history of previous studies of the similarities and differences in the theories of definability of the first three of these four languages. A seminal role leading toward unification of the theories has been played by the separation principles introduced by Nikolai Luzin in 1927. Emphasizing analogies and driving toward further unification embracing finite-universe logic we concentrate on a simple example—the first and second separation principles for existential-universal first-order sentences . Using this as a test case for the fundamental problem of how to “finitize” arguments in classical pure logic to the finite-universe case, we are led to the analogous negative solution by using the theory of certain special graphs: a graph is - special for any positive integers m , n , p , q iff it is bipartite with m red points and n blue points and for every p -tuple of red points there is a blue point to which they are all connected . As an aside we introduce for further study a natural “Ramseyesque” increasing sequence A of positive integers, where A is the least positive integer n for which an -special graph exists. (shrink)

Priest and others have presented their “most telling” argument for paraconsistent logic: that only paraconsistent logics allow non-trivial inconsistent theories. This is a very prevalent argument; occurring as it does in the work of many relevant and more generally paraconsistent logicians. However this argument can be shown to be unsuccessful. There is a crucial ambiguity in the notion of non-triviality. Disambiguated the most telling reason for paraconsistent logics is either question-begging or mistaken. This highlights (...) an important confusion about the role of logic in our development of our theories of the world. Does logic chart good reasoning or our commitments? We also consider another abductive argument for paraconsistent logics which also is shown to fail. (shrink)

This book discusses the philosophical work of Décio Krause. Non-individuality, as a new metaphysical category, was thought to be strongly supported by quantum mechanics. No one did more to promote this idea than the Brazilian philosopher Décio Krause, whose works on the metaphysics and logic of non-individuality are now widely regarded as part of the consolidated literature on the subject. This volume brings together chapters elaborating on the ideas put forward and defended by Krause, developing them in many different (...) directions, commenting on aspects not completely developed so far, and, more importantly, critically addressing their current formulations and defenses by Krause himself. Given that Krause’s ideas do connect directly and indirectly with a wide array of subjects, such as the philosophy of quantum mechanics, more broadly understood, the philosophy of logic and logical philosophy, non-classical logics, metaphysics, and ontology, this volume contains important material for the research on logic and foundations of science, broadly understood. All the invited contributors have already worked with the ideas developed by Décio (some of them still work with them), being also distinct authors and extremely relevant in their areas of expertise. The volume is aimed at philosophers, including those of physics and quantum mechanics. (shrink)

The Companion Encyclopedia is the first comprehensive work to cover all the principal lines and themes of the history and philosophy of mathematics from ancient times up to the twentieth century. In 176 articles contributed by 160 authors of 18 nationalities, the work describes and analyzes the variety of theories, proofs, techniques, and cultural and practical applications of mathematics. The work's aim is to recover our mathematical heritage and show the importance of mathematics today by treating its interactions with (...) the related disciplines of physics, astronomy, engineering and philosophy. It also covers the history of higher education in mathematics and the growth of institutions and organizations connected with the development of the subject. Part 1 deals with mathematics in various ancient and non-Western cultures from antiquity up to medieval and Renaissance times. Part 2 treats developments in all the main areas of mathematics during the medieval and Renaissance periods up to and including the early 17th century. Parts 3-10 are divided into the main branches into which mathematics developed from the early 17th century onwards: calculus and mathematical analysis, logic and foundations, algebras, geometries, mechanics, mathematical physics and engineering, and probability and statistics. Parts 11-13 review the history of mathematics from an international perspective. The teaching of mathematics in higher education is examined in various countries, and mathematics in culture, art and society is covered. The Companion Encyclopedia features annotated bibliographies of both classic and contemporary sources; black and white illustrations, line figures and equations; biographies of major mathematicians and historians and philosophers of mathematics; a chronological table of main events in the developments of mathematics; and a fully integrated index of people, events and topics. (shrink)

The paper essentially shows that the paraconsistent logicDR satisfies the depth relevance condition. The systemDR is an extension of the systemDK of [7] and the non-triviality of a dialectical set theory based onDR has been shown in [3]. The depth relevance condition is a strengthened relevance condition, taking the form: If DR- AB thenA andB share a variable at the same depth, where the depth of an occurrence of a subformulaB in a formulaA is roughly the number of nested (...) ''s required to reach the occurrence ofB inA. The method of proof is to show that a model structureM consisting of {M 0 , M1, ..., M}, where theM i s are all characterized by Meyer''s 6-valued matrices (c. f, [2]), satisfies the depth relevance condition. Then, it is shown thatM is a model structure for the systemDR. (shrink)

This book is the first in the field of paraconsistency to offer a comprehensive overview of the subject, including connections to other logics and applications in information processing, linguistics, reasoning and argumentation, and philosophy of science. It is recommended reading for anyone interested in the question of reasoning and argumentation in the presence of contradictions, in semantics, in the paradoxes of set theory and in the puzzling properties of negation in logic programming. Paraconsistent logic comprises a (...) major logical theory and offers the broadest possible perspective on the debate of negation in logic and philosophy. It is a powerful tool for reasoning under contradictoriness as it investigates logic systems in which contradictory information does not lead to arbitrary conclusions. Reasoning under contradictions constitutes one of most important and creative achievements in contemporary logic, with deep roots in philosophical questions involving negation and consistency This book offers an invaluable introduction to a topic of central importance in logic and philosophy. It discusses the history of paraconsistent logic; language, negation, contradiction, consistency and inconsistency; logics of formal inconsistency and the main paraconsistent propositional systems; many-valued companions, possible-translations semantics and non-deterministic semantics; paraconsistent modal logics; first-order paraconsistent logics; applications to information processing, databases and quantum computation; and applications to deontic paradoxes, connections to Eastern thought and to dialogical reasoning. (shrink)

Semantics connected to some information based metaphor are well-known in logic literature: a paradigmatic example is Kripke semantic for Intuitionistic Logic. In this paper we start from the concrete problem of providing suitable logic-algebraic models for the calculus of attribute dependencies in Formal Contexts with information gaps and we obtain an intuitive model based on the notion of passage of information showing that Kleene algebras, semi-simple Nelson algebras, three-valued ukasiewicz algebras and Post algebras of order three (...) are, in a sense, naturally and directly connected to partially defined information systems. In this way wecan provide for these logic-algebraic structures a raison dêetre different from the original motivations concerning, for instance, computability theory. (shrink)

This volume is a collection of essays in honour of Professor Mohammad Ardeshir. It examines topics which, in one way or another, are connected to the various aspects of his multidisciplinary research interests. Based on this criterion, the book is divided into three general categories. The first category includes papers on non-classical logics, including intuitionistic logic, constructive logic, basic logic, and substructural logic. The second category is made up of papers discussing issues in the (...) contemporary philosophy of mathematics and logic. The third category contains papers on Avicenna’s logic and philosophy. Mohammad Ardeshir is a full professor of mathematical logic at the Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran, where he has taught generations of students for around a quarter century. Mohammad Ardeshir is known in the first place for his prominent works in basic logic and constructive mathematics. His areas of interest are however much broader and include topics in intuitionistic philosophy of mathematics and Arabic philosophy of logic and mathematics. In addition to numerous research articles in leading international journals, Ardeshir is the author of a highly praised Persian textbook in mathematical logic. Partly through his writings and translations, the school of mathematical intuitionism was introduced to the Iranian academic community. (shrink)

We give proofs of the effective monotone interpolation property for the system of modal logic K, and others, and the system IL of intuitionistic propositional logic. Hence we obtain exponential lower bounds on the number of proof-lines in those systems. The main results have been given in [P. Hrubeš, Lower bounds for modal logics, Journal of Symbolic Logic 72 941–958; P. Hrubeš, A lower bound for intuitionistic logic, Annals of Pure and Applied (...) class='Hi'>Logic 146 72–90]; here, we give considerably simplified proofs, as well as some generalisations. (shrink)

We investigate the substitution Frege () proof system and its relationship to extended Frege () in the context of modal and superintuitionistic propositional logics. We show that is p-equivalent to tree-like , and we develop a “normal form” for -proofs. We establish connections between for a logic L, and for certain bimodal expansions of L.We then turn attention to specific families of modal and si logics. We prove p-equivalence of and for all extensions of , all tabular logics, (...) all logics of finite depth and width, and typical examples of logics of finite width and infinite depth. In most cases, we actually show an equivalence with the usual system for classical logic with respect to a naturally defined translation.On the other hand, we establish exponential speed-up of over for all modal and si logics of infinite branching, extending recent lower bounds by P. Hrubeš. We develop a model-theoretical characterization of maximal logics of infinite branching to prove this result. (shrink)