Timothy Williamson has maintained that the applicability of classical mathematics in science raises a problem for the endorsement, in non-mathematical domains, of a wide range of non-classical logics. We show that this is false.

Elimination of quantifiers is shown to fail dramatically for a group of well‐known mathematical theories (classically enjoying the property) against a wide range of relevant logical backgrounds. Furthermore, it is suggested that only by moving to more extensional underlying logics can we get the property back.

Gert H. Muller The growth of the number of publications in almost all scientific areas, as in the area of (mathematical) logic, is taken as a sign of our scientifically minded culture, but it also has a terrifying aspect. In addition, given the rapidly growing sophistica tion, specialization and hence subdivision of logic, researchers, students and teachers may have a hard time getting an overview of the existing literature, partic ularly if they do not have an extensive library available in (...) their neighbourhood: they simply do not even know what to ask for! More specifically, if someone vaguely knows that something vaguely connected with his interests exists some where in the literature, he may not be able to find it even by searching through the publications scattered in the review journals. Answering this challenge was and is the central motivation for compiling this Bibliography. The Bibliography comprises (presently) the following six volumes (listed with the corresponding Editors): I. Classical Logic W. Rautenberg II. Non-classical Logics W. Rautenberg III. Model Theory H. -D. Ebbinghaus IV. Recursion Theory P. G. Hinman V. Set Theory A. R. Blass VI. Proof Theory; Constructive Mathematics J. E. Kister; D. van Dalen & A. S. Troelstra. (shrink)

Provability, Computability and Reflection.

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)

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)

In this paper we develop what we can describe as a “dual two-sided” cut-free sequent calculus system for the non-classical logics of truth lp, k3, stt and a non-reflexive logic ts which is, arguably, more elegant than the three-sided sequent calculus developed by Ripley for the same logics. Its elegance stems from how it employs more or less the standard sequent calculus rules for the various connectives and truth, and the fact that it offers a rather neat connection between (...) derivable sequents and validity in comparison to the calculus developed by Ripley. (shrink)

The Knower paradox purports to place surprising a priori limitations on what we can know. According to orthodoxy, it shows that we need to abandon one of three plausible and widely-held ideas: that knowledge is factive, that we can know that knowledge is factive, and that we can use logical/mathematical reasoning to extend our knowledge via very weak single-premise closure principles. I argue that classical logic, not any of these epistemic principles, is the culprit. I develop a consistent theory (...) validating all these principles by combining Hartry Field's theory of truth with a modal enrichment developed for a different purpose by Michael Caie. The only casualty is classical logic: the theory avoids paradox by using a weaker-than-classical K3 logic. I then assess the philosophical merits of this approach. I argue that, unlike the traditional semantic paradoxes involving extensional notions like truth, its plausibility depends on the way in which sentences are referred to--whether in natural languages via direct sentential reference, or in mathematical theories via indirect sentential reference by Gödel coding. In particular, I argue that from the perspective of natural language, my non-classical treatment of knowledge as a predicate is plausible, while from the perspective of mathematical theories, its plausibility depends on unresolved questions about the limits of our idealized deductive capacities. (shrink)

In this paper, we use algebra-valued models to study cardinal numbers in a class of non-classical set theories. The algebra-valued models of these non-classical set theories validate the Axiom of Choice, if the ground model validates it. Though the models are non-classical, the foundations of cardinal numbers in these models are similar to those in classical set theory. For example, we show that mathematical induction, Cantor’s theorem, and the Schröder–Bernstein theorem hold in these models. We also (...) study a few basic properties of cardinal arithmetic. In addition, the generalized continuum hypothesis is proved to be independent of these non-classical set theories. (shrink)

The paper is devoted to the contributions of Helena Rasiowa to the theory of non-classical negation. The main results of Rasiowa in this area concerns–constructive logic with strong (Nelson) negation.

The Church-Turing Thesis is widely regarded as true, because of evidence that there is only one genuine notion of computation. By contrast, there are nowadays many different formal logics, and different corresponding foundational frameworks. Which ones can deliver a theory of computability? This question sets up a difficult challenge: the meanings of basic mathematical terms are not stable across frameworks. While it is easy to compare what different frameworks say, it is not so easy to compare what they mean. We (...) argue for some minimal conditions that must be met if two frameworks are to be compared; if frameworks are radical enough, comparison becomes hopeless. Our aim is to clarify the dialectical situation in this bourgeoning area of research, shedding light on the nature of non-classical logic and the notion of computation alike. (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)

This is the first introductory textbook on non-classical propositional logics.

The article is about one of the vital problem for analytic philosophy which is how to define truth value for sentences which include their own truth predicate. The aim of the article is to determine Saul Kripke’s approach to widen epistemological truth to create a systemic model of truth. Despite a lot of work on the subject, the theme of truth is no less relevant to modern philosophy. With the help of S. Kripke’s article “Outline of the Theory of Truth” (...) and R. L. Kirkham’s work “Theories of Truth: A Critical Introduction” the author tried to represent a non-classical approach which is now known as Saul Kripke’s truth value gaps theory. Considering his solving of the Liar Paradox, as well as the Strenghened Liar, it is obvious that there could be the circumstances (or facts) that make sentences have no truth value. In that case the solving of the paradox attracts Kleene’s non-classical three-valued logic which contains the third way and the solution falls into the gap between truth and falsity. Hence there are no relevant and full satisfactory ways to solve the Liar’s Paradox by the language-levels ap- proach. The task consists in an extension of the approach to natural languages and elimination of an ad hoc element as well. Besides that, the language-levels approach refers to the speaker that means that only the last one is allowed to define the hierarchy of the levels and truth value of sentence, but often he does not know what level he is speaking of. As Kripke stated, contingent facts can lead to a paradox which isn’t allowed for a logical point of view. However usage of mathematical apparatus allows the theory in question to be universalized and moreover paves the way to futher epistemological develop- ment of the problem in question. As Kripke’s theory is formalized and presented as a functional series, it gives access to a system model of truth whose elements would be epistemological theories (concepts) of truth. Such a model, for instance, confirms the validity of the deflationary concept of truth. Refs 7. (shrink)

Algebraic approach to study of classical and non-classical logical calculi was developed and systematically presented by Helena Rasiowa in [48], [47]. It is very fruitful in investigation of non-classical logics because it makes possible to study large families of logics in an uniform way. In such research one can replace logics with suitable classes of algebras and apply powerful machinery of universal algebra. In this paper we present an overview of results on interpolation and definability in modal (...) and positive logics,and also in extensions of Johansson's minimal logic. All these logics are strongly complete under algebraic semantics. It allows to combine syntactic methods with studying varieties of algebras and to flnd algebraic equivalents for interpolation and related properties. Moreover, we give exhaustive solution to interpolation and some related problems for many families of propositional logics and calculi. (shrink)

We investigate mathematical structures that provide natural semantics for families of (quantified) non-classical logics featuring special unary connectives, known as recovery operators, that allow us to ‘recover’ the properties of classical logic in a controlled manner. These structures are known as topological Boolean algebras, which are Boolean algebras extended with additional operations subject to specific conditions of a topological nature. In this study, we focus on the paradigmatic case of negation. We demonstrate how these algebras are well-suited to (...) provide a semantics for some families of paraconsistent Logics of Formal Inconsistency and paracomplete Logics of Formal Undeterminedness. These logics feature recovery operators used to earmark propositions that behave ‘classically’ when interacting with non-classical negations. Unlike traditional semantical investigations, which are carried out in natural language (extended with mathematical shorthand), our formal meta-language is a system of higher-order logic (HOL) for which automated reasoning tools exist. In our approach, topological Boolean algebras are encoded as algebras of sets via their Stone-type representation. We use our higher-order meta-logic to define and interrelate several transformations on unary set operations, which naturally give rise to a topological cube of opposition. Additionally, our approach enables a uniform characterization of propositional, first-order and higher-order quantification, including restrictions to constant and varying domains. With this work, we aim to make a case for the utilization of automated theorem proving technology for conducting computer-supported research in non-classical logics. All the results presented in this paper have been formally verified, and in many cases obtained, using the Isabelle/HOL proof assistant. (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)

Provability, Computability and Reflection.

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)

"This report investigates the question of the universality of classical logic. The approach is to show that an almost arbitrary logical system can be translated reasonably intuitively and almost automatically into classical logic. The path leading to this result goes through the analysis of what is reasonable logic, how to find semantics for it, how to build a labelled deductive system (LDS) for it, how to translate a LDS into classical logic and how to automate the process (...) using SCAN. This report relies on other papers, published and/or to be published as explained in the acknowledgements. (shrink)

Identification of distinct items is a basic technique in mathematics. However, identification suffers from a certain weakness of resolve in that it is (classically) accompanied by dropping the original disidentification, which causes a loss of information about the theory which sources the identity. This article proposes an alternative, namely keeping the disidentification along with the identification. This produces an inconsistent theory which is generally an extension of the source theory. The concept of a Dunn–Meyer extension is defined to study (...) these properties. It is seen that this technique is sensitive to the choice of background logic, particularly RM3 as opposed to closed-set logic. By employing the Routley functor, a best-choice logic is found for this construction. (shrink)

We investigate two large families of logics, differing from each other by the treatment of negation. The logics in one of them are obtained from the positive fragment of classical logic (with or without a propositional constant ff for “the false”) by adding various standard Gentzen-type rules for negation. The logics in the other family are similarly obtained from LJ+, the positive fragment of intuitionistic logic (again, with or without ff). For all the systems, we provide simple semantics which (...) is based on non-deterministic four-valued or three-valued structures, and prove soundness and completeness for all of them. We show that the role of each rule is to reduce the degree of non-determinism in the corresponding systems. We also show that all the systems considered are decidable, and our semantics can be used for the corresponding decision procedures. Most of the extensions of LJ+ (with or without ff) are shown to be conservative over the underlying logic, and it is determined which of them are not. (shrink)

In this paper I propose a new approach to the foundation of mathematics: non-monotonic set theory. I present two completely different methods to develop set theories based on adaptive logics. For both theories there is a finitistic non-triviality proof and both theories contain (a subtle version of) the comprehension axiom schema. The first theory contains only a maximal selection of instances of the comprehension schema that do not lead to inconsistencies. The second allows for all the instances, also the (...) inconsistent ones, but restricts the conclusions one can draw from them in order to avoid triviality. The theories have enough expressive power to form a justification/explication for most of the established results of classical mathematics. They are therefore not limited by Gödel’s incompleteness theorems. This remarkable result is possible because of the non-recursive character of the final proofs of theorems of non-monotonic theories. I shall argue that, precisely because of the computational complexity of these final proofs, we cannot claim that non-monotonic theories are ideal foundations for mathematics. Nevertheless, thanks to their strength, first order language and the recursive dynamic (defeasible) proofs of theorems of the theory, the non-monotonic theories form (what I call) interesting pragmatic foundations. (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)

This article addresses the issue of compositionality of mental representations from the perspective of a foundational framework for cognitive science. The dynamical cognition framework is inspired partially by connectionism and partially by the persistence of the problem of relevance within classical computational cognitive science. It treats cognition in terms of the mathematics of dynamical systems: total occurrent cognitive states are mathematically/structurally realized as points in a high-dimensional dynamical system, and these mathematical points are physically realized by total-activation states (...) of a neural network with specific connection weights. The framework repudiates the classicist assumption that cognitive-state transitions conform to a tractably computable transition function over cognitive states. Computational Theory of Mind states that the causal role of a mental representation is syntactically determined, but this idea of syntactic determination of causal role is ambiguous. (shrink)

This book explores some of Kit Fine's outstanding contributions to logic, philosophy of language, philosophy of mathematics, and metaphysics, among others. Contributing authors address in-depth issues about truthmaker semantics, counterfactual conditionals, grounding, vagueness, non-classical consequence relations, and arbitrary objects, offering critical reflections and novel research contributions. Each chapter is accompanied by an extensive commentary, in which Kit Fine offers detailed responses to the ideas and themes raised by the contributors. The book includes a brief autobiography and exhaustive list (...) of his publications to this date. This book is of interest to logicians of all stripes and to analytic philosophers more generally. (shrink)

The main goal of this paper is to explain the link between the algebraic models and the Kripke-style models for certain classes of propositional non-classical logics. We consider logics that are sound and complete with respect to varieties of distributive lattices with certain classes of well-behaved operators for which a Priestley-style duality holds, and present a way of constructing topological and non-topological Kripke-style models for these types of logics. Moreover, we show that, under certain additional assumptions on the variety (...) of the algerabic models of the given logics, soundness and completeness with respect to these classes of Kripke-style models follows by using entirely algebraical arguments from the soundness and completeness of the logic with respect to its algebraic models. (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 book is just what its title says: an introduction to nonclassical logic. And it is a very good one. Given the extensive interest in nonclassical logics, in various parts of the philosophical scene, it is a welcome addition to the corpus. Typical courses in logic, at all levels and in both philosophy departments and mathematics departments, focus exclusively on classical logic. Most instructors, and some textbooks, give some mention to some nonclassical systems, but usually few details are (...) provided. (shrink)

We present a novel approach to represent ecological systems using reaction networks, and show how a particular framework called chemical organization theory sheds new light on the longstanding complexity–stability debate. Namely, COT provides a novel conceptual landscape plenty of analytic tools to explore the interplay between structure and stability of ecological systems. Given a large set of species and their interactions, COT identifies, in a computationally feasible way, each and every sub-collection of species that is closed and self-maintaining. These sub-collections, (...) called organizations, correspond to the groups of species that can survive together in the long-term. Thus, the set of organizations contains all the stable regimes that can possibly happen in the dynamics of the ecological system. From here, we propose to conceive the notion of stability from the properties of the organizations, and thus apply the vast knowledge on the stability of reaction networks to the complexity–stability debate. As an example of the potential of COT to introduce new mathematical tools, we show that the set of organizations can be equipped with suitable joint and meet operators, and that for certain ecological systems the organizational structure is a non-boolean lattice, providing in this way an unexpected connection between logico-algebraic structures, popular in the foundations of quantum theory, and ecology. (shrink)

We present a novel approach to represent ecological systems using reaction networks, and show how a particular framework called chemical organization theory sheds new light on the longstanding complexity–stability debate. Namely, COT provides a novel conceptual landscape plenty of analytic tools to explore the interplay between structure and stability of ecological systems. Given a large set of species and their interactions, COT identifies, in a computationally feasible way, each and every sub-collection of species that is closed and self-maintaining. These sub-collections, (...) called organizations, correspond to the groups of species that can survive together in the long-term. Thus, the set of organizations contains all the stable regimes that can possibly happen in the dynamics of the ecological system. From here, we propose to conceive the notion of stability from the properties of the organizations, and thus apply the vast knowledge on the stability of reaction networks to the complexity–stability debate. As an example of the potential of COT to introduce new mathematical tools, we show that the set of organizations can be equipped with suitable joint and meet operators, and that for certain ecological systems the organizational structure is a non-boolean lattice, providing in this way an unexpected connection between logico-algebraic structures, popular in the foundations of quantum theory, and ecology. (shrink)

The logicD-J of the weak exluded middle with constant domains is proved to be incomplete with respect to Kripke semantics, by introducing models in presheaves on an arbitrary category. Additional incompleteness results are obtained for the modal systems with nested domains extendingQ-S4.1.

This paper reviews the impact of Rasiowa's well-known book on the evolution of algebraic logic during the last thirty or forty years. It starts with some comments on the importance and influence of this book, highlighting some of the reasons for this influence, and some of its key points, mathematically speaking, concerning the general theory of algebraic logic, a theory nowadays called Abstract Algebraic Logic. Then, a consideration of the diverse ways in which these key points can be generalized allows (...) us to survey some issues in the development of the field in the last twenty to thirty years. The last part of the paper reviews some recent lines of research that in some way transcend Rasiowa's approach. I hope in this way to give the reader a general view of Rasiowa's key position in the evolution of Algebraic Logic during the twentieth century. (shrink)

My purpose here is purely historical. It is not an attempt to resolve the question as to whether Russell did or did not countenance nonclassical logics, and if so, which nonclassical logics, and still less to demonstrate whether he himself contributed, in any manner, to the development of nonclassical logic. Rather, I want merely to explore and insofar as possible document, whether, and to what extent, if any, Russell interacted with the various, either the various candidates or their, ideas that (...) Dejnožka and others have proposed as potentially influential in Russell’s intellectual reactions to nonclassical logic or to the philosophical concepts that might contribute to his reactions to nonclassical logics. (shrink)

The main goal of this paper is to explain the link between the algebraic and the Kripke-style models for certain classes of propositional logics. We start by presenting a Priestley-type duality for distributive lattices endowed with a general class of well-behaved operators. We then show that finitely-generated varieties of distributive lattices with operators are closed under canonical embedding algebras. The results are used in the second part of the paper to construct topological and non-topological Kripke-style models for logics that are (...) sound and complete with respect to varieties of distributive lattices with operators in the above-mentioned classes. (shrink)

Gert H. Muller The growth of the number of publications in almost all scientific areas, as in the area of logic, is taken as a sign of our scientifically minded culture, but it also has a terrifying aspect. In addition, given the rapidly growing sophistica tion, specialization and hence subdivision of logic, researchers, students and teachers may have a hard time getting an overview of the existing literature, partic ularly if they do not have an extensive library available in their (...) neighbourhood: they simply do not even know what to ask for! More specifically, if someone vaguely knows that something vaguely connected with his interests exists some where in the literature, he may not be able to find it even by searching through the publications scattered in the review journals. Answering this challenge was and is the central motivation for compiling this Bibliography. The Bibliography comprises the following six volumes : I. Classical Logic W. Rautenberg II. Non-classical Logics W. Rautenberg III. Model Theory H. -D. Ebbinghaus IV. Recursion Theory P. G. Hinman V. Set Theory A. R. Blass VI. Proof Theory; Constructive Mathematics J. E. Kister; D. van Dalen & A. S. Troelstra. (shrink)

In so-called Kripke-type models, each sentence is assigned either to true or to false at each possible world. In this setting, every possible world has the two-valued Boolean algebra as the set of truth values. Instead, we take a collection of algebras each of which is attached to a world as the set of truth values at the world, and obtain an extended semantics based on the traditional Kripke-type semantics, which we call here the algebraic Kripke semantics. We introduce algebraic (...) Kripke sheaf semantics for super-intuitionistic and modal predicate logics, and discuss some basic properties. We can state the Gödel-McKinsey-Tarski translation theorem within this semantics. Further, we show new results on super-intuitionistic predicate logics. We prove that there exists a continuum of super-intuitionistic predicate logics each of which has both of the disjunction and existence properties and moreover the same propositional fragment as the intuitionistic logic. (shrink)

An old and well-known objection to non-classical logics is that they are too weak; in particular, they cannot prove a number of important mathematical results. A promising strategy to deal with this objection consists in proving so-called recapture results. Roughly, these results show that classical logic can be used in mathematics and other unproblematic contexts. However, the strategy faces some potential problems. First, typical recapture results are formulated in a purely logical language, and do not generalize nicely (...) to languages containing the kind of vocabulary that usually motivates non-classical theories—for example, a language containing a naïve truth predicate. Second, proofs of recapture results typically employ classical principles that are not valid in the targeted non-classical system; hence, non-classical theorists do not seem entitled to those results. In this paper, we analyze these problems and provide solutions on behalf of non-classical theorists. To address the first problem, we provide a novel kind of recapture result, which generalizes nicely to a truth-theoretic language. As for the second problem, we argue that it relies on an ambiguity and that, once the ambiguity is removed, there are no reasons to think that non-classical logicians are not entitled to their recapture results. (shrink)

We introduce non-associative linear logic, which may be seen as the classical version of the non-associative Lambek calculus. We define its sequent calculus, its theory of proof-nets, for which we give a correctness criterion and a sequentialization theorem, and we show proof search in it is polynomial.

There exist several phenomena breaking the classical probability laws. The systems related to such phenomena are context-dependent, so that they are adaptive to other systems. In this paper, we present a new mathematical formalism to compute the joint probability distribution for two event-systems by using concepts of the adaptive dynamics and quantum information theory, e.g., quantum channels and liftings. In physics the basic example of the context-dependent phenomena is the famous double-slit experiment. Recently similar examples have been found in (...) biological and psychological sciences. Our approach is an extension of traditional quantum probability theory, and it is general enough to describe aforementioned contextual phenomena outside of quantum physics. (shrink)