Non-Classical Logic

Classical recapture

by Andrew Aberdein

In Prospettive della logica e della filosofia della scienza, V. Fano, M. Stanzione, & G. Tarozzi, edd. (Catanzaro: Rubettino, 2001), pp. 11–18.

The recapture relationship is an important element to any understanding of the connexion between different systems of... more The recapture relationship is an important element to any understanding of the connexion between different systems of logic. Loosely speaking, one system of logic recaptures another if it is possible to specify a subsystem of the former system which exhibits the same patterns of inference as the latter system. In particular if a relationship of this kind can be shown to exist between a non-classical logic and classical logic, the non-classical system is said to exhibit classical recapture. This has been invoked by several proponents of non-classical logics to argue that their system retains classical logic as a limit case, and is therefore a methodologically progressive successor to classical logic. In this paper I advance and defend a new and more precise account of recapture and the character of its reception by the proponents of the recapturing system. I then indicate some of the applications of classical recapture which this account makes possible.

The Philosophy of Alternative Logics

by Andrew Aberdein

Co-authored with Stephen Read. In Development of modern logic, L. Haaparanta, ed. (Oxford: Oxford University Press, 2009), pp. 613-723.

This chapter focuses on alternative logics. It discusses a hierarchy of logical reform. It presents case studies that... more This chapter focuses on alternative logics. It discusses a hierarchy of logical reform. It presents case studies that illustrate particular aspects of the logical revisionism discussed in the chapter. The first case study is of intuitionistic logic. The second case study turns to quantum logic, a system proposed on empirical grounds as a resolution of the antinomies of quantum mechanics. The third case study is concerned with systems of relevance logic, which have been the subject of an especially detailed reform program. Finally, the fourth case study is paraconsistent logic, perhaps the most controversial of serious proposals.

Keywords: classical logic, logical theory, intuitionistic logic, quantum logic, relevance logic, paraconsistent logic

Misyurov D.A. Dialectical formulas based on the binary notation as the development formulas // Credo New. 2012. №2

by Dmitry Misyurov

The article suggests dialectical formulas based on the binary notation as the development formulas: formula with... more The article suggests dialectical formulas based on the binary notation as the development formulas: formula with dominant and the non-dominant elements; universal formula; formula with symbolic weight of elements; tautological formula. For example, it suggests an opportunity to use the dialectical formulas for modeling and artificial intelligence creation, etc.

"Redes, lógicas no clásicas y neuronas. De los límites de la matematización más allá de la Física"

by Vicente Caballero de la Torre

En el presente artículo se exponen las líneas maestras de la "Teoría de Grafos" y aquellos problemas de... more En el presente artículo se exponen las líneas maestras de la "Teoría de Grafos" y aquellos problemas de corte formal que la misma teoría muestra como modelo para explicar el funcionamiento del cerebro. Las redes -concepto que dicha teoría intenta sistematizar y comprender matemáticamente- son de sumo interés para cualquiera que pretenda arrojar una cierta luz sobre el perfil que en la actualidad están tomando fenómenos tan diversos y actuales como el terrorismo, la cibernética y, por supuesto, las últimas investigaciones neurocientíficas.

Contradictions in Dōgen

by Koji Tanaka

This is my contribution to the discussion forum on Contradictions in Buddhism to be published in Philosophy East & West, 2013.

In their article ‘The Way of the Dialetheist: Contradictions in Buddhism’, Deguchi, Garfield and Priest argue that... more In their article ‘The Way of the Dialetheist: Contradictions in Buddhism’, Deguchi, Garfield and Priest argue that some (though not all) of the contradictions that appear in Buddhist texts should be accepted. An examination of their argument depends on what sort(s) of negation is (are) used in the texts. In order to see apparently contradictory statements as affirmations of true contradictions, we must assume that ‘not' (or its variance) is used as a contradiction forming operator. In this paper, I examine the conception of negation(s) that is (are) salient in the writings of Dōgen and argue that he would not agree that his sentences are to be considered, and accepted, as contradictory.

Harmony, Normality and Stability

by Nils Kürbis

As the title says: my account of proof-theoretic harmony, normality and stability!

Gentzen mentions that it should be possible to specify a function that maps introduction rules onto elimination rules... more Gentzen mentions that it should be possible to specify a function that maps introduction rules onto elimination rules in systems of natural deduction. This paper specifies such a function. I specify two kinds of rules, one in which it is more natural to assume an introduction rule to be given and elimination rules are determined from it, and another kind in which it is an elimination rule which is given and the introduction rules are determined from it. The process also works the other way round, so that it doesn't really matter which rules are supposed to be given first. The process is very general and applies to a large class of logics. The paper begins with a discussion of the philosophical importance of this in connection with the notion of harmony. I discuss Dummett's ideas on harmony and stability, which is supposed to be stronger than harmony. Dummett suggests that normalisability is a formal criterion of harmony. However, he seems to aim at something else, and this criterion does not give an independent formally precise notion of stability. I propose formally precise definitions of harmony and stability, which are distinct from normalisability. My aim is not exegetical, and according to my definitions, classical as well as intuitionist logics count as governed by stable (and hence harmonious) rules of inference.

Reaching Transparent Truth

by Pablo Cobreros

Joint work with Paul Egré, Dave Ripley and Robert van Rooij

This paper presents and defends a way to add a transparent truth predicate to classical logic, such that T<A>... more This paper presents and defends a way to add a transparent truth predicate to classical logic, such that T<A> and A are everywhere intersubstitutable, where all T-biconditionals hold, and where truth can be made compositional. A key feature of our framework, called STT (for Strict-Tolerant Truth), is that it supports a nontransitive relation of consequence. At the same time, it can be seen that the only failures of transitivity STT allows for arise in paradoxical cases

How to Synthesize a Paraconsistent Negation. The Case of CLuN

by Mariusz Urbanski

Urbański M. (2004). How to Synthesize a Paraconsistent Negation. The Case of CLuN, Logique et Analyse, 185-188, p. 319-333.

The aim of this paper is to apply synthetic tableaux method (STM) to the paraconsistent logic CLuN, developed by... more The aim of this paper is to apply synthetic tableaux method (STM) to the paraconsistent logic CLuN, developed by Diderik Batens. Soundness and completeness of STM with respect to CLuN semantics are proved. It is also shown how to interpret CLuN negation in terms of relations represented by the square of oppositions of traditional syllogistic.

Synthetic Tableaux for Łukasiewicz’s Calculus Ł3

by Mariusz Urbanski

Urbański M. (2002). Synthetic Tableaux for Łukasiewicz’s Calculus Ł3, Logique et Analyse, 177-178, p. 155-173.

In this paper synthetic tableaux for Łukasiewicz’s calculus Ł3 are presented in detail. Basic properties of synthetic... more In this paper synthetic tableaux for Łukasiewicz’s calculus Ł3 are presented in detail. Basic properties of synthetic tableaux are described as well as a systematic procedure for building a tableau for any given formula of Ł3.

What is Wrong with Classical Negation?

by Nils Kürbis

The focus of this paper are the meaning-theoretical arguments against classical logic that Dummett bases on... more The focus of this paper are the meaning-theoretical arguments against classical logic that Dummett bases on consideration about the meanings of negation. Using Dummettian principles, I shall outline three such arguments, of increasing strength, and show that they are unsuccessful by giving responses to each argument on behalf of the classical logician. What is crucial is that in responding to these arguments a classicist need not challenge any of the basic assumptions of Dummett’s outlook on the theory of meaning. In particular, I shall grant Dummett his general bias towards verificationism or justificationism, encapsulated in the slogan ‘meaning is use’. The second general assumption I see no need to question is Dummett’s particular breed of molecularism. Some of Dummett’s assumptions will have to be given up, if classical logic is to be vindicated in his meaning-theoretical framework. A major result of this paper will be that the meaning of negation cannot be defined by rules of inferences in the Dummettian framework.

Models of Possibilism and Trivialism

by Luis Estrada-González

Antepenultimate draft. Please refer to the published version to appear in Logic and Logical Philosophy. If you can read Spanish, read instead "Lecciones de la degeneración", the categorial twin of this paper: It's far better!

In this paper I probe the idea that neither possibilism nor trivialism could be ruled out on a purely logical basis. I... more In this paper I probe the idea that neither possibilism nor trivialism could be ruled out on a purely logical basis. I use the apparatus of relational structures used in the semantics for modal logics to engineer some models of possibilism and trivialism and I discuss a philosophical stance about logic, truth values and the meaning of connectives underlying such analysis.

The Manifold Concept of Logical Consequence: Tarski, Etchemendy and more

by Matteo Bianchetti

Draft only

I discuss Etchemendy's hidden assumptions that there is only one proper concept of logical consequence and that the... more I discuss Etchemendy's hidden assumptions that there is only one proper concept of logical consequence and that the main aim of logic is to investigate it. I show that the concept of logical consequence is essentially related to extra-logical ideas (from metaphysics, epistemology and concerning the nature and the aims of logic). I show that the concept of logical consequence is not central in Aristotle's works: his main problem is to show how we can justify a categorical sentence. I show that in the history of logic we find at least two great divisions of the considerations on the concept of logical consequence: following rules (e.g. Frege) and formal truth-preservation (e.g. Bolzano). I conclude with some reflections on this point.

Lecciones de la degeneración (O: ¿Qué hay de malo con la trivialidad?)

by Luis Estrada-González

Versión 4. He expandido la sección 3 y la parte de cómo se obtiene el trivialismo en un topos degenerado ahora es, creo, mucho más clara. Cualquier comentario es bienvenido.

Boolean Conservative Extension Results for some Modal Relevant Logics

by Koji Tanaka

Co-authored with Edwin Mares. Published in the Australasian Journal of Logic, 2010.

This paper shows that a collection of modal relevant logics are conservatively extended by the addition of Boolean... more This paper shows that a collection of modal relevant logics are conservatively extended by the addition of Boolean negation.

Guest Editors' Introduction

by Koji Tanaka

Co-authored with Francesco Berto, Edwin Mares and Francesco Paoli. Logic and Logical Philosophy: Special Issue on Paraconsistent Logic, Vol. 19, No. 1-2, 2010.

Making Sense of Paraconsistency: The Nature of Logic, Classical Logic and Paraconsistent Logic

by Koji Tanaka

Paraconsistency: Logic and Applications, Koji Tanaka, Francesco Berto, Edwin Mares and Francesco Paoli (eds.), Springer, forthcoming.

Max Cresswell and Hilary Putnam seem to hold the view, often shared by classical logicians, that paraconsistent logic... more Max Cresswell and Hilary Putnam seem to hold the view, often shared by classical logicians, that paraconsistent logic has not been made sense of, despite its well-developed mathematics. In this paper, I examine the nature of logic in order to understand what it means to make sense of logic. I then show that, just as one can make sense of non-normal modal logics (as Cresswell demonstrates), we can make ‘sense’ of paraconsistent logic. Finally, I turn the tables on classical logicians and ask what sense can be made of explosive reasoning. While I acknowledge a bias on this issue, it is not clear that even classical logicians can answer this question.

The Law of Non-Contradiction as a Metaphysical Principle

by Tuomas Tahko

The Australasian Journal of Logic, Vol. 7, pp. 32-47, 2009

The goals of this paper are two-fold: I wish to clarify the Aristotelian conception of the law of non-contradiction as... more The goals of this paper are two-fold: I wish to clarify the Aristotelian conception of the law of non-contradiction as a metaphysical rather than a semantic or logical principle, and to defend the truth of the principle in this sense. First I will explain what it in fact means that the law of non-contradiction is a metaphysical principle. The core idea is that the law of non-contradiction is a general principle derived from how things are in the world. For example, there are certain constraints as to what kind of properties an object can have, and especially: some of these properties are mutually exclusive. Given this characterisation, I will advance to examine what kind of challenges the law of noncontradiction faces—the main opponent here is Graham Priest. I will consider these challenges and conclude that they do not threaten the truth of the law of noncontradiction understood as a metaphysical principle.

Tolerant, Classical, Strict

by Pablo Cobreros

Joint work with Paul Egré, Dave Ripley and Robert van Rooij. To appear in Journal of Philosophical Logic. Available from Springer's 'online first' (open choice)

In this paper we investigate a semantics for first-order logic originally proposed by R. van Rooij to account for the... more In this paper we investigate a semantics for first-order logic originally proposed by R. van Rooij to account for the idea that vague predicates are tolerant, that is, for the principle that if x is P, then y should be P whenever y is similar enough to x. The semantics, which makes use of indifference relations to model similarity, rests on the interaction of three notions of truth: the classical notion, and two dual notions simultaneously defined in terms of it, which we call tolerant truth and stricttruth. We characterize the space of consequence relations definable in terms of those and discuss the kind of solution this gives to the sorites paradox. We discuss some applications of the framework to the pragmatics and psycholinguistics of vague predicates, in particular regarding judgments about borderline cases.

A note on Adams conditionals

by Luis Estrada-González

Co-authored with Paulina Raigosa. 'The Reasoner', December 2010.

The subjunctive conditional associated to an intensional disjunction is a bit more complex than usually thought, and... more The subjunctive conditional associated to an intensional disjunction is a bit more complex than usually thought, and it does not have the fatalist consequences attributed to it.

Differences between logics and meaning-variance: A categorial approach

by Luis Estrada-González

In Spanish. Antepenultimate draft of an article co-authored with Dr. Ivonne Pallares-Vega to appear in THEORIA. An International Journal for Theory, History and Foundations of Science.

We argue here that the meanings of logical connectives need not to differ in different logics. Based on the... more We argue here that the meanings of logical connectives need not to differ in different logics. Based on the category-theoretic treatment of the logical connectives, we argue against the well-known Quinean thesis that a difference between logics implies a difference in the meanings of connectives. We thus locate this change in the difference between certain objects rather than in the difference between the meaning of connectives. Finally, we try to show that the category-theoretic treatment of logical connectives is a form of semantic minimalism, according to which not all the usual semantic components are relevant in fixing the meaning of a connective.

En este artículo tratamos de hacer plausible la hipótesis de que las conectivas de diferentes lógicas no necesariamente difieren en significado. Utilizando el tratamiento categorista de las conectivas, argumentaremos contra la tesis quineana de que la diferencia de lógicas implica diferencia de significado entre sus conectivas, y ubicamos el cambio de tema en la diferencia de objetos más bien que en una tal diferencia de significado. Finalmente, intentamos mostrar que ese tratamiento categorista es una forma de minimalismo semántico, de acuerdo con el cual no todos los elementos semánticos usuales son relevantes para determinar el significado de las conectivas.