One of Da Costa’s motives when he constructed the paraconsistent logic C! was to dualise the negation of intuitionistic logic. In this paper I explore a different way of going about this task. A logic is defined by taking the Kripke semantics for intuitionistic logic, and dualising the truth conditions for negation. Various properties of the logic are established, including its relation to C!. Tableau and natural deduction systems for the logic are produced, as are appropriate algebraic structures. The paper (...) then investigates dualising the intuitionistic conditional in the same way. This establishes various connections between the logic, and a logic called in the literature ‘Brouwerian logic’ or ‘closed-set logic’. (shrink)

Journal Name: Apeiron Issue: Ahead of print.

This paper continues the work of Priest and Sylvan in Simplified Semantics for Basic Relevant Logics, a paper on the simplified semantics of relevant logics, such as B⁺ and B. We show that the simplified semantics can also be used for a large number of extensions of the positive base logic B⁺, and then add the dualising '*' operator to model negation. This semantics is then used to give conservative extension results for Boolean negation.

One of the few points of agreement to be found in mainstream responses to the logical and semantic problems generated by vagueness is the view that if any modification of classical logic and semantics is required at all then it will only be such as to admit underdetermined reference and truth-value gaps. Logics of vagueness including many valued logics, fuzzy logics, and supervaluation logics all provide responses in accord with this view. The thought that an adequate response might require the (...) recognition of cases of overdetermination and truth value gluts has few supporters. This imbalance lacks justification. As it happens, Jaskowski's paraconsistent discussive logic-a logic which admits truth value gluts-can be defended by reflecting on similarities between it and the popular supervaluationist analysis of vagueness already in the philosophical literature. A simple dualisation of supervaluation semantics results in a paraconsistent logic of vagueness based on what has been termed subvaluational semantics. (shrink)

Płonka sums consist of an algebraic construction similar, in some sense, to direct limits, which allows to represent classes of algebras defined by means of regular identities. Recently, Płonka sums have been connected to logic, as they provide algebraic semantics to logics obtained by imposing a syntactic filter to given logics. In this paper, I present a very general topological duality for classes of algebras admitting a Płonka sum representation in terms of dualisable algebras.

In a paper published in 1939, Ernest Nagel described the role that projective duality had played in the reformulation of mathematical understanding through the turn of the nineteenth century, claiming that the discovery of the principle of duality had freed mathematicians from the belief that their task was to describe intuitive elements. While instances of duality in mathematics have increased enormously through the twentieth century, philosophers since Nagel have paid little attention to the phenomenon. In this paper I will argue (...) that a reassessment is overdue. Something beyond doubt is that category theory has an enormous amount to say on the subject, for example, in terms of arrow reversal, dualising objects and adjunctions. These developments have coincided with changes in our understanding of identity and structure within mathematics. While it transpires that physicists have employed the term ‘duality’ in ways which do not always coincide with those of mathematicians, analysis of the latter should still prove very useful to philosophers of physics. Consequently, category theory presents itself as an extremely important language for the philosophy of physics. (shrink)

There is a robust tendency within the contemporary feminist mainstream to argue against and ultimately reject the so-called ‘dualising or dualist philosophy’ since it is the supportive paradigm background for any gender discrimination originated from the hegemonic sovereignty of masculinity over femininity. In this paper, having dived deeper into the feminist critical depiction of the logical binarist foundation on which the dualising philosophy is said to be well-grounded, I will proceed to portray and examine a sequence of doctrines that feminist (...) philosophers have developed to shed light on the fact that the hegemonic sovereignty of masculinity over femininity has been theoretically initiated from the logical disjunction 'p or not p' to masculinity essentialism. Finally, I will end by pointing to a tension between underlying assumptions of the feminist sequentialist approach and what feminists themselves claim to adhere to as the highest ideal: non-naturalising gender differences. (shrink)

Context: Seventeen years ago Francisco Varela introduced neurophenomenology. He proposed the integration of phenomenological approaches to first-person experience – in the tradition of Husserl, Heidegger and Merleau-Ponty – with a neuro-dynamical, scientific approach to the study of the situated brain and body. Problem: It is time for a re-appraisal of this field. Has neurophenomenology already contributed to the sciences of the mind? If so, how? How should it best do so in future? Additionally, can neurophenomenology really help to resolve or (...) dissolve the “hard problem” of the relation between mind and body, as Varela claimed? Method: The papers in this special issue arose out of a conference organised by the Consciousness and Experiential Psychology Section of the British Psychological Society in Bristol, UK, in September 2012. We have invited a representative sample of the speakers at that conference to present their work here. Results: Various papers argue that the first-person methods of phenomenology are distinct from, and more robust than, the failed “introspectionist” methods of early modern psychology. The “elicitation interview” emerges as a successful and widely adopted method to have emerged from this field. Phenomenological techniques are already being successfully applied to neuroscientific problems. Various specific proposals for new techniques and applications are made. Implications: It is time to take neurophenomenology seriously. It has proven its worth, and it is ripe with the potential for further immediate, successful applications. Constructivist content: Varela’s key aim was to develop a non-dualising approach to the science of consciousness. The papers in this special issue look at the philosophical and practical details of successfully putting such an approach into practice. (shrink)

Problem: The paper’s main focus is on the question of whether Mitterer’s non-dualising philosophy is able to show a way out of the antagonistic opposition of fact and fiction, realism and constructivism. In addition, since Mitterer’s philosophy has hardly been discussed so far in historiography and theory of history, I also examine the question of whether his approach can provide new theoretical insights in these disciplines. Method: I follow a close reading of Mitterer’s texts and relate them to the propositions (...) of a variety of publications from the field of theory of history. Results: Mitterer’s arguments show, on the one hand, how to expel the idea of creation from our historical thinking as there is no authenticity waiting behind our world of descriptions. On the other hand, they make clear that historiography no longer has to decide between realism and relativism. Rather, it is a relationist approach that shows a way to research the many entanglements and complexities between past realities and their descriptions. A different way to write history becomes visible: a history of realities. Implications: The discussion of Mitterer’s non-dualism not only shows how historical research might benefit from this philosophy (in leaving behind some dualistic foundations of the discipline or the search for historical origins) but also how non-dualism might profit from the insights of history (with regard to power, processuality, and temporality. (shrink)

This paper focuses on natural dualities for varieties of bilattice-based algebras. Such varieties have been widely studied as semantic models in situations where information is incomplete or inconsistent. The most popular tool for studying bilattices-based algebras is product representation. The authors recently set up a widely applicable algebraic framework which enabled product representations over a base variety to be derived in a uniform and categorical manner. By combining this methodology with that of natural duality theory, we demonstrate how to build (...) a natural duality for any bilattice-based variety which has a suitable product representation over a dualisable base variety. This procedure allows us systematically to present economical natural dualities for many bilattice-based varieties, for most of which no dual representation has previously been given. Among our results we highlight that for bilattices with a generalised conflation operation. Here both the associated product representation and the duality are new. Finally we outline analogous procedures for pre-bilattice-based algebras. (shrink)

Cet article a pour objet l’analyse de trois types de théorisations de la signification basées sur un modèle binaire du signe. Celles de Frege, Husserl et Saussure. Relevant d’un même paradigme, les deux premières sont confrontées en tant que s’y développent deux conceptions opposées de la signification – extensionnelle chez Frege, intensionnelle chez Husserl – contribuant à la mise en place, selon des perspectives opposées, de la dualisation de la syntaxe et de la sémantique. Relativement à cette conséquence, leur paradigme (...) est ensuite opposé à celui de la sémiologie structurale de Saussure dans laquelle l’interaction de la syntaxe et de la sémantique est internalisée au niveau du système qu’intègrent les signes comme éléments. De sorte que ce que l’on vise au terme de cet article, c’est à mettre au jour une explication possible de la raison de la divergence entre deux types de conceptions de la sémantique qui organiseront le développement de courants concurrents tout au long de l’histoire de la linguistique du 20e siècle, l’une formelle et mathématisée, l’autre structurale. (shrink)

After indicating why this is needed, the paper proves a non-triviality result for paraconsistent theory containing arithmetic, naive truth and denotation predicates, and descriptions. The result is obtained by dualising a construction of Kroon. Its most notable feature is that there is a trivial object- one that has every property.

This paper is dedicated to Newton da Costa, who,among his many achievements, was the first toaim at dualising intuitionism in order to produce paraconsistent logics,the C-systems. This paper similarly dualises intuitionism to aparaconsistent logic, but the dual is a different logic, namely closed setlogic. We study the interaction between the properties of topologicalspaces, particularly separation properties, and logical theories on thosespaces. The paper begins with a brief survey of what is known about therelation between topology and modal logic, intuitionist logic (...) and paraconsistentlogic in respect of the incompleteness and inconsistency of theories.Necessary and sufficient conditions which relate the T1-property to theproperties of logical theories, are obtained. The result is then extendedto Hausdorff and Normal spaces. In the final section these methods areused to vary the modelling conditions for identity. (shrink)

This paper is dedicated to Newton da Costa, who,among his many achievements, was the first toaim at dualising intuitionism in order to produce paraconsistent logics,the C-systems. This paper similarly dualises intuitionism to aparaconsistent logic, but the dual is a different logic, namely closed setlogic. We study the interaction between the properties of topologicalspaces, particularly separation properties, and logical theories on thosespaces. The paper begins with a brief survey of what is known about therelation between topology and modal logic, intuitionist logic (...) and paraconsistentlogic in respect of the incompleteness and inconsistency of theories.Necessary and sufficient conditions which relate the T 1-property to theproperties of logical theories, are obtained. The result is then extendedto Hausdorff and Normal spaces. In the final section these methods areused to vary the modelling conditions for identity. (shrink)

Intuitionistic logic formalises the foundational ideas of L.E.J. Brouwer’s mathematical programme of intuitionism. It is one of the earliest non-classical logics, and the difference between classical and intuitionistic logic may be interpreted to lie in the law of the excluded middle, which asserts that either a proposition is true or its negation is true. This principle is deemed unacceptable from the constructive point of view, in whose understanding the law means that there is an effective procedure to determine the truth (...) of all propositions. This understanding of the distinction between the two logics supports the view that negation plays a vital role in the formulation of intuitionistic logic.Nonetheless, the formalisation of negation in intuitionistic logic has not been universally accepted, and many alternative accounts of negation have been proposed. Some seek to weaken or strengthen the negation, and others actively supporting negative inferences that are impossible with it.This thesis follows this tradition and investigates various aspects of negation in intuitionistic logic. Firstly, we look at a problem proposed by H. Ishihara, which asks how effectively one can conserve the deducibility of classical theorems into intuitionistic logic, by assuming atomic classes of non-constructive principles. The classes given in this section improve a previous class given by K. Ishii in two respects: (a) instead of a single class for the law of the excluded middle, two classes are given in terms of weaker principles, allowing a finer analysis and (b) the conservation now extends to a subsystem of intuitionistic logic called Glivenko’s logic. This section also discusses the extension of Ishihara’s problem to minimal logic.Secondly, we study the relationship between two frameworks for weak constructive negation, the approach of D. Vakarelov on one hand and the framework of subminimal negation by A. Colacito, D. de Jongh, and A. L. Vargas on the other hand. We capture a version of Vakarelov’s logic with the semantics of the latter framework, and clarify the relationship between the two semantics. This also provides proof-theoretic insights, which results in the formulation of a cut-free sequent calculus for the aforementioned system.Thirdly, we investigate the ways to unify the formalisations of some logics with contra-intuitionistic inferences. The enquiry concerns paraconsistent logics by R. Sylvan and A. B. Gordienko, as well as the logic of co-negation by G. Priest and of empirical negation by M. De and H. Omori. We take Sylvan’s system as basic, and formulate the frame conditions of the defining axioms of the other systems. The conditions are then used to obtain cut-free labelled sequent calculi for the systems.Finally, we consider L. Humberstone’s actuality operator for intuitionistic logic, which can be seen as the dualisation of a contra-intuitionistic negation. A compete axiomatisation of intuitionistic logic with actuality operator is given, and comparisons are made for some related operators.Abstract prepared by Satoru Niki.E-mail:[email protected]. (shrink)

Priest (2009) formulates a propositional logic which, by employing the worldsemantics for intuitionist logic, has the same positive part but dualises the negation, to produce a paraconsistent logic which it calls 'Da Costa Logic'. This paper extends matters to the first-order case. The paper establishes various connections between first order da Costa logic, da Costa's own Cω, and classical logic. Tableau and natural deductions systems are provided and proved sound and complete.