Paraconsistent logic

Resumo: Neste artigo analisamos as críticas apresentadas por George Berkeley, em The analyst (1734), ao método das fluxões e à inconsistência intrínseca à noção de infinitésimo do cálculo diferencial e integral, introduzido por Isaac Newton. Procuramos mostrar que as críticas de Berkeley não eram de todo infundadas, uma vez que foram necessários quase duzentos anos para que viesse a ser introduzida por Karl Weierstrass a definição rigorosa de limite, que propiciou uma solução para o problema dos infinitésimos. São mencionadas ainda duas outras teorias contemporâneas, com abordagens distintas para a solução da questão do infinitésimo: a análise não-standard de Abraham Robinson e o cálculo diferencial paraconsistente proposto por Newton da Costa. Apesar de serem citados alguns autores importantes para o desenvolvimento do cálculo, este artigo não se propõe a analisar suas obras e não pretende apresentar uma história do cálculo diferencial e integral. Palavras-chave: George Berkeley, método das fluxões, infinitésimo, cálculo diferencial paraconsistente.

Resumen El propósito de este artículo es mostrar el modo en que Borges, en su relato Tigres azules, desarrolla varias ideas interesantes en torno a la lógica, el lenguaje y las matemáticas. El punto de partida es el escepti-cismo borgeano sobre la capacidad de la lógica clásica para abordar el problema de la verdad y los mundos imposibles. Se defiende que ob-jetos inconsistentes e, incluso, autocontradictorios no necesariamente dan lugar a mundos física o lógicamente imposibles. Se sugiere que las paradojas que Borges construye en el cuento no son paradojas de autorreferencia, sino variantes de la paradoja de Moore y de las para-dojas de implicación material, por lo que para superarlas se requiere un enfoque no-clásico que neutralice ECQ, esto es, un sistema de lógica paraconsistente. Abstract Tigres azules of J. L. Borges. Logic, Truth and Impossible Worlds. The aim of this text is show how Borges, in his story Tigres azules, develops several interesting ideas about logic, language and mathematics. The starting point is Borgesian skepticism about the ability of classical logic to take the problem of truth and impossible worlds. It is defended that inconsistent and even self-contradictory objects do not necessarily give rise to physically or logically impossible worlds. It is suggested that the paradoxes that Borges constructs in the story

The Judiciary uses information technology through database that distributes information to stakeholders, using
the Internet, about the progress of processes or the content of judicial decisions and other administrative matters
of interests of justice. Undoubtedly, this use of technology was a breakthrough, but did not result in sufficient
acceleration in the services of the Judiciary, which could suffer big impact with the use of intelligent systems for
the services of the notary and the judge. It should be considered that distribute information loaded into a database
is a different thing than smart, because that can make decisions. Intelligent system is one that employs artificial
intelligence techniques that enable the machine to perform tasks requiring intelligence. Among these techniques
can cite: Case Based Reasoning, Intelligent Agents, Expert Systems and others. Thus, a lawsuit would be
managed by an intelligent system, eliminating the use of some treatments of manual and intellectual personnel
office and the judge. Thus, it would be possible to combine the use of internet via Artificial Intelligence to meet
demand without compromising the judicial outcome of these decisions? The thesis of the unity of fair solution
can be used in a particular direction, which will be one of the bases of the hypothesis to be launched in this
article on the technique of self-rule as precedent reason: the idea of the unity of the fair solution don’t
necessarialy admites the real existence of only one answer to every legal issue, but can be postulated by all legal
reasoning that is to be correct and without contradictions. The purpose of this article is to present some ideas for
reflection, taking into account the mechanisms of artificial intelligence and the use of information technology to help the judiciary to overcome some of the shortcomings of the current system: the steady increase in the volume of cases and decisions of divergent Courts.

The present work is devoted to the exploration of some formal possibilities suggesting, since some years, the possibility to elaborate a new, whole geometry, relative to the concept of “opposition”. The latter concept is very important and vast (as for its possible applications), both for philosophy and science and it admits since more than two thousand years a standard logical theory, Aristotle’s “opposition theory”, whose culminating formal point is the so called “square of opposition”. In some sense, the whole present enterprise consists in discovering and ordering geometrically an infinite amount of “avatars” of this traditional square structure (also called “logical square” or “Aristotle’s square”). The results obtained here go even beyond the most optimistic previous expectations, for it turns out that such a geometry exists indeed and offers to science many new conceptual insights and formal tools. Its main algorithms are the notion of “logical bi-simplex of dimension m” (which allows “opposition” to become “n-opposition”) and, beyond it, the notions of “Aristotelian pq-semantics” and “Aristotelian pq-lattice” (which allow opposition to become p-valued and, more generally, much more fine-grained): the former is a game-theoretical device for generating “opposition kinds”, the latter gives the structure of the “opposition frameworks” containing and ordering the opposition kinds. With these formal means, the notion of opposition reaches a conceptual clarity never possible before. The naturalness of the theory seems to be maximal with respect to the object it deals with, making this geometry the new standard for dealing scientifically with opposition phenomena. One question, however, philosophical and epistemological, may seem embarrassing with it: this new, successful theory exhibits fundamental logical structures which are shown to be intrinsically geometrical: the theory, in fact, relies on notions like those of “simplex”, of “n-dimensional central symmetry” and the like. Now, despite some appearances (that is, the existence, from time to time, of logics using some minor spatial or geometrical features), this fact is rather revolutionary. It joins an ancient and still unresolved debate over the essence of mathematics and rationality, opposing, for instance, Plato’s foundation of philosophy and science through Euclidean geometry and Aristotle’s alternative foundation of philosophy and science through logic. The geometry of opposition shows, shockingly, that the logical square, the heart of Aristotle’s transcendental, anti-Platonic strategy is in fact a Platonic formal jungle, containing geometrical-logical hyper-polyhedra going into infinite. Moreover, this fact of discovering a lot of geometry inside the very heart of logic, is also linked to a contemporary, raging, important debate between the partisans of “logic-inspired philosophy” (for short, the analytic philosophers and the cognitive scientists) and those, mathematics-inspired, who begin to claim more and more that logic is intrinsically unable to formalise, alone, the concept of “concept” (the key ingredient of philosophy), which in fact requires rather geometry, for displaying its natural “conceptual spaces” (Gärdenfors). So, we put forward some philosophical reflections over the aforementioned debate and its deep relations with questions about the nature of concepts. As a general epistemological result, we claim that the geometrical theory of oppositions reveals, by contrast, the danger implicit in equating “formal structures” to “symbolic calculi” (i.e. non-geometrical logic), as does the paradigm of analytic philosophy. We propose instead to take newly in consideration, inspired by the geometry of logic, the alternative paradigm of “structuralism”, for in it the notion of “structure” is much more general (being not reduced to logic alone) and leaves room to formalisations systematically missed by the “pure partisans” of “pure logic”.

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 (partiality) and truth value gluts (overdeterminedness). 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.

French philosopher Gilles Deleuze wrote two 'logic' books: Francis Bacon: The Logic of Sensation and The Logic of Sense. However, in neither of these books nor in any other works does Deleuze articulate in a formal way the features of the logic he employs. He certainly does not use classical logic. And the best options for the non-classical logic that he may be implementing are: fuzzy, intuitionist, and many-valued. These are applicable to his concepts of heterogeneous composition and becoming, affirmative synthetic disjunction, and powers of the false.

In The Logic of Gilles Deleuze: Basic Principles, Corry Shores examines the applicability of three non-classical logics to Deleuze's philosophy, by building from the philosophical and logical writings of Graham Priest, the world's leading proponent of dialetheism. Through so doing, Shores argues that Deleuze's logic is best understood as a dialetheic, paraconsistent, many-valued logic.

Jan Łukasiewicz’ work on Aristotle and the principle of non-contradiction, originally published in 1910, presents one of the first and truly pioneering investigation into the logical and metaphysical foundations of this principle.

Łukasiewicz applies the newly developed analytic tools of mathematical logic to Aristotle’s seminal defense of the principle in Book IV of the Metaphysics and aims to show that the principle is not nearly as secure as the generally accepted, mostly uncritical, and often dogmatic belief in its universal truth would have it.

This introduction to examines the historical and philosophical background of Łukasiewicz’ work, its relation to Hegel 's metaphysical logic which gave rise to the debate about Die logische Frage (the question of logic), and Łukasiewicz’ attempt to develop a first Non-Aristotelian logic, a logic that does not require or endorse Aristotle's principle of non-contradiction.

An interpretation of Wittgenstein’s much criticized remarks on Gödel’s First Incompleteness Theorem is provided in the light of paraconsistent arithmetic: in taking Gödel’s proof as a paradoxical derivation, Wittgenstein was drawing the consequences of his deliberate rejection of the standard distinction between theory and metatheory. The reasoning behind the proof of the truth of the Gödel sentence is then performed within the formal system itself, which turns out to be inconsistent. It is shown that the features of paraconsistent arithmetics match with some intuitions underlying Wittgenstein’s philosophy of mathematics, such as its strict finitism and the insistence on the decidability of any mathematical question.

A dialetheia is a sentence, A, such that both it and its negation, ¬A, are true (we shall talk of sentences throughout this entry; but one could run the definition in terms of propositions, statements, or whatever one takes as her favourite truth-bearer: this would make little difference in the context). Assuming the fairly uncontroversial view that falsity just is the truth of negation, it can equally be claimed that a dialetheia is a sentence which is both true and false...

In this paper I describe how several notions and constructions in topos logic can be dualized, giving rise to complement-toposes with their paraconsistent internal logic, instead of the usual standard toposes with their intuitionistic logic.

Foucault famously questioned the humor, and possibly deceit, at the heart of Deleuze’s philosophy by asserting its sincerity and seriousness. In this paper I address how humor and sincerity, and thereby truth and falsity, are philosophical concepts at the core of Deleuze’s thought. Using Deleuze’s work Difference and Repetition as the focal object of my interrogation, I seek to show how truth and falsity are structured by difference and repetitions. Deleuze’s philosophy of pure difference does not merely try to critique traditional notions of truth, but attempts to define a new model of truth and falsity that bears both difference and repetitions from the previous models in the history of philosophy.

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 (partiality) and truth value gluts (overdeterminedness). Simple and natural semantic rules for negation and the conditional operator are formulated based on an incompatibility relation and a 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 intuitionis-tic 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.

Paper presented at the congress Nikolai Vasiliev's Logical Legacy and Modern Logic
October 24-25, 2012, Moscow, Russia
to appear in D.Zaitsev (ed), Nikolai Vasiliev's Logical Legacy and Modern Logic, Springer, Dordrecht, 2015.

In this paper we examine up to which point Modern logic can be qualified as
non-Aristotelian. After clarifying the difference between logic as reasoning and
logic as a theory of reasoning, we compare syllogistic with propositional and
first-order logic. We touch the question of formal validity, variable and
mathematization and we point out that Gentzen’s cut-elimination theorem can
be seen as the rejection of the central mechanism of syllogistic – the cut-rule
has been first conceived as a modus Barbara by Hertz. We then examine the
non-Aristotelian aspect of some non-classical logics, in particular
paraconsistent logic. We argue that a paraconsistent negation can be seen as
neo-Aristotelian since it corresponds to the notion of subcontrary in Boethius’
square of opposition. We end by examining if the comparison promoted by
Vasiliev between non-Aristotelian logic and non-Euclidian geometry makes
sense.

Este trabajo se centra en la discusión sobre la disyunción de Gödel, así como las posiciones antimecanicistas que surgen de los teoremas de Gödel, en el escenario actual de la pluralidad de la lógica. El trabajo está motivado por dos ideas, en primer lugar, que lo que prueban las posiciones antimecanicistas de Lucas y Penrose no es la afirmación de que la mente humana no puede ser capturada por ninguna máquina, sino que no puede ser capturada por el tipo de cálculo que lleva una máquina de Turing . Segundo, aunque los resultados limitantes de Gödel son de gran importancia en el campo de la aritmética clásica, cuando pensamos en la mente matemática humana, las paradojas e inconsistencias entran en escena y nos motivan a cambiar la lógica. Por lo tanto, sostengo que la ruta paraconsistente es aceptable en esta discusión.

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.

This article is focused on answering the question to what extent one can be a sceptic. Sextus Empiricus’s Outlines of Scepticism serves as a guide. In section 1, it is investigated whether three logical laws have a certain foundation or are subject to doubt. In section 2, Sextus’s way to deal with these laws is examined; the question arises how dogmatic his approach is. After that, a possible ‘reply’ by Sextus to the criticism receives attention. Section 3 is concentrated on a possible alternative to Sextus’s approach. Besides logical laws, some important methods are concerned.

Esta investigación ontológica aborda lo relativo al existir: qué es el existir; qué es aquello de lo que cabe decir que es o existe; si hay sentidos irreducibles de ese verbo `existir'; qué sea el no-existir -lo cual conduce al
problema de si el existir se da por grados; si cabría la posibilidad de que no existiera nada; cómo se relaciona la existencia con la referencialidad lingüística; cuáles sean las leyes generales que rigen todo lo existente. Este último problema se desglosa en el estudio de varios principios ontológicos: los de inteligibilidad, razón suficiente, identidad de los indiscernibles, interdependencia, gradualidad y otros. Esos siete problemas se estudian con las técnicas de la filosofía analítica y desde un punto de vista dialéctico, o sea el de la afirmación de que hay verdades mutuamente contradictorias que estriban en la parcial coexistencia de propiedades opuestas en el mismo sujeto, cuando se dan por grados.

PALABRAS-CLAVE:
    ontología, existencia, ser, no-ser, inexistencia, grados de ser, grados de realidad, nada, referencialidad lingüística, principios ontológicos, inteligibilidad, razón suficiente, identidad de los indiscernibles, interdependencia, gradualidad, filosofía analítica, dialéctica, verdades contradictorias,  verdad parcial, coexistencia de opuestos, grados de verdad.


Abstract:
    Dialectical ontology foundations.
This ontological research takes up the question of existence: what is to exist; what does exist; whether there are irreducible senses of `existence'; what is non-existence, which raises the problem of whether existence comes in degrees; if there could be nothing; how existence relates with linguistic referentiality; what are the most general laws governing whatever exists. The latter problem breaks down into several ontological principles: those of intelligibility, sufficient reason, identity of indiscernibles, interdependence, graduality and others.
Those seven problems are gone into through the techniques belonging to analytical philosophy and from a dialectical view-point, which means the assertion of mutually contradictory truths, resulting from the partial coexistence of opposite properties in the same subject, when they come in degrees.

Keywords:
ontological research, existence, non-existence, degrees of reality, nothingness, linguistic referentiality, ontological principles, intelligibility of being,
sufficient reason, identity of indiscernibles, interdependence, graduality, analytical philosophy, dialectics, contradictory truths, partial truth, coexistence of opposites, degrees of truth.

This essay builds on my ongoing research on spiritual practices as foundation of an experimental ontology in the wake of the phenomenological tradition of the 20th century. In order to exemplify the categorical and logical foundations of this practice, it focuses on Nicholas of Cusa’s trinitarian ‚non-aliud’ speculations in his late Trialogue Directio speculantis (1462). More precisely, it shows how Cusa’s experimental practice of thinking shaped his conjectures on the ‘non-otherness’ of God and recalls the logical foundations of his claim that his trinitarian definition of the ‘non-other’ represents as a ‘definition which defines itself and all things’. The logic of definition that justifies this claim is incompatible with the representationalist Scotist tradition of the late Middle Ages and the subsequent idea of a ‘formal logics’ such as in Kant and Frege. Hence, it comes as no surprise that Cusa’s ‘definition of everything’ appears as esoteric to modern readers. Yet, on hindsight we can see why it is superior to the tradition of modern philosophy both continental and analytic. In contrast to the ‘analytic’ project of developing a ‚theory of everything‘ that meets the axiomatic standards of a ‘formal logic’, and in contrast to ‘critical’ accounts of the conditions of the (im-)possibility to do, such as in Schelling and Hegel, Cusa builds on an uncompromisingly realistic, participatory ontology. This has far reaching theological implications: While modern attempts to develop a univocal concept of God became irresistibly entangled in false dichotomies, such as the dichotomy between theism and various types of monism (including pantheism and panentheism), Cusa’s participatory ontology avoids the enforced alternatives of formal ontologies. The critical point of his related Guide to Contemplation comes into view, if we pay attention to the performative features of his ‘definition of everything’: It points to a reality that cannot be be ‘seen’ unless we meet it in the right spiritual attitude. This enables to see how Cusa’s early considerations about the ‘name of Jesus’ shaped his philosophical ontology, and how his analogical ontology undergirded a spiritual practice that enables us to ‘see God in everything’.

Nicephorus Blemmydes' symmetric formula "through each other" proposed instead of the Filioque in the 1250s belongs to the Cappadocian tradition of Triadology. This conclusion is not demonstrable from a habitual (in patrology) philological analysis but can be reached through a logical analysis. The implied theory of numbers is sketched.

Morgan Thomas has recently shown that a family of naïve set theories based on Asenjo-Priest's "logic of paradox" (LP) and other related logics suffer from serious expressive limitations. At first sight these results could seem a serious drawback for the prospects of inconsistent mathematics, since these naïve set theories either lack even the most elementary concepts needed to express basic notions of classical mathematics or are nearly trivial. I will argue that these results are not a doom for LP-mathematics, but rather suggest a different, non-foundational role of set theory in non-classical mathematics.

In this paper, I examine the possibility that recent developments in non-classical and paraconsistent logics might offer us a way to better understand the 'paradoxical', yet strikingly coherent nature of Cusanus' philosophical method for deriving 'names for God', particularly the 'three-step' form he proposes in De Docta Ignorantia - up to and including the ultimate 'paradoxical' element in his systematic account, the 'infinite-finite' Christ.
Building on previous research by Marko Ursic and Lorenzo Pena, I propose a four-valued logic NC which can serve as a model for illustrating Cusanus' method, as it is able to generate strikingly similar arguments to those Cusanus himself presents.
This approach provides some striking proof of the internal coherency of Cusanus' method and of the philosophical interest it should still present to the contemporary reader.

As a general theory of reasoning—and as a general theory of what holds true under every possible circumstance—logic is supposed to be ontologically neutral. It ought to have nothing to do with questions concerning what there is, or whether there is anything at all. It is for this reason that traditional Aristotelian logic, with its tacit existential presuppositions, was eventually deemed inadequate as a canon of pure logic. And it is for this reason that modern quantification theory, too, with its residue of existentially loaded theorems and patterns of inference, has been claimed to suffer from a defect of logical purity. The law of non-contradiction rules out certain circumstances as impossible—circumstances in which a statement is both true and false, or perhaps circumstances where something both is and is not the case. Is this to be regarded as a further ontological bias? If so, what does it mean to forego such a bias in the interest of greater neutrality—and ought we to do so?

Classical propositional logic can be characterized, indirectly , by means of a complementary formal system whose theorems are exactly those formulas that are not classical tautologies, i.e., contradictions and truth-functional contingencies. Since a formula is contingent if and only if its negation is also contingent, the system in question is paraconsistent. Hence classical propositional logic itself admits of a paraconsistent characterization, albeit "in the negative". More generally, any decidable logic with a syntactically incomplete proof theory allows for a paraconsistent characterization of its set of theorems. This, we note, has important bearing on the very nature of paraconsistency as standardly characterized.

Realist dialetheism is the view that there are contradictions in reality. One argument against this idea says that it is impossible because it has to make room for the possibility of a trivial reality, which is metaphysically impossible. Another argument against it says that the metaphysical structure of reality is such that it is impossible to have contradictions in it. I argue here that both arguments fail to establish the impossibility of realist dialetheism because they are based on a misconception about the notions of negation and contradiction which leads them, in the first case, to wrongly hold that dialetheism has to be compatible with trivialism, and, in the second case, to assume that the validity of the Aristotelian principle of non-contradiction prevents the existence of dialetheias.

En mi tesis doctoral defiendo un concepto de condicional derrotable para las normas similar,  aunque no idéntico,  al de C. Alchourrón y propongo un concepto de consecuencia falible que permite hacer inferencias no deductivas a partir de normas derrotables. Es un trabajo que tiene unos cuantos años y he modificado mis ideas en cuanto al desarrollo informal de la derrotabilidad (sobre todo porque ahora distingo la derrotabilidad de la ponderación y en un principio no hacía ninguna diferencia)  pero los sistemas formales siguen siendo, en mi opinión, propuestas interesantes en lógica no deductiva y en lógica deóntica. Acepto comentarios, críticas, sugerencia y en particular me interesa conocer el trabajo de los demás en los mismos tópicos y relacionados.

A logic for inconsistent mathematics must be strong enough to support reasoning in proofs, while weak enough to avoid paradoxes. We present a substructural logic intended to meet the needs of a working dialetheic mathematician—specifically, by adding a de Morgan negation to light linear logic, and extending the logic with a relevant conditional. The logic satisfies a deduction theorem. Soundness and completeness is established, showing in particular that contraction is invalidated. This opens the way for a robust naive set theory; we conclude by showing how the set theory provides a foundation for induction.

In recent editions of this journal, Jean-Yves Beziau [8] and Marcin Tkaczyk [42] have criticised a prominent dialetheic logic and common arguments for dialetheism, respectively. While Beziau argues that Priest's logic LP commits the dialetheist to trivialism, the thesis that all propositions are true, Tkaczyk maintains that the arguments traditionally proposed for dialetheism are faulty and ultimately that dialetheism should be rejected as self-refuting. This paper shows that both are mistaken in their contentions. Beziau's argument conflates truth-in-an-interpretation with truth simpliciter and Tkaczyk misconstrues the substance of dialetheic arguments. In the process of identifying these weaknesses of both arguments , the paper clarifies elements of both dialetheic logics and dialethe-ism which these discussions demonstrate are still misunderstood within the literature.

Do truth tables—the ordinary sort that we use to teach and explain basic propositional logic—require an assumption of consistency for their construction? In this essay we show that truth tables can be built in a consistency-­‐independent paraconsistent setting, without any appeal to classical logic. This is evidence for a more general claim: that when we write down the orthodox semantic clauses for a logic, whatever logic we presuppose in the background will be the logic that appears in the foreground. Rather than any one logic being privileged, then, on this count partisans across the logical spectrum are in relatively similar dialectical positions.

Alan Darley has implored Radical Orthodoxy to become " more radically Thomistic " by recovering the law of non-contradiction that has lately been lost by Milbank, Pickstock, and Hoff in a " Cusanian tributary from the Neoplatonic river ". He contends that this prohibition on all contradictions can be explained through Aristotle's indirect demonstration against any possibility of denying it. But he fails to recognize how this very indirect demonstration already operates as a viciously circular formalized dialectic. Plato developed this dialectic into the self-determination of Intellect; Plotinus suspended it from the divine hypostases; and Proclus systematized it as the creative hypercontrariety that imparts every determination. Thomas Aquinas inherited Proclus' creative hypercontrariety from Pseudo-Dionysius of Areopagite, Aristotle's Principle of Non-Contradiction from Avicenna, and sublated all contrariety into the divine impartation of individuated acts of existence. This impartation of existence into entities may have produced an aporia of existence: if existence is multiple then it is contrary to the individuality of every entity; but if existence is unitary then it may conceal the creative contrariety of participation. The indiscriminate application of the Principle of Non-Contradiction threatens to break the bonds of being and dissolve the Thomistic metaphysics of participation into post-Ockhamist nihilism. The further radicalization of Thomism may thus require the aporia of existence to be resolved through a Neoplatonic dialectic.

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.

We examine the relationship between the logics of nonsense of Bochvar and Halldén and the containment logics in the neighborhood of William Parry’s AI. We detail two strategies for manufacturing containment logics from nonsense logics—taking either connexive and paraconsistent fragments of such systems—and show how systems determined by these techniques have appeared as Frederick Johnson’s RC and Carlos Oller’s AL. In particular, we prove that Johnson’s system is precisely the intersection of Bochvar’s B_3 and Graham Priest’s non-symmetrized connexive logic and that Oller’s system lies just beneath the intersection of B_3 and Priest’s paraconsistent LP. We conclude by examining Oller’s system in more depth, giving it a characterization in terms of LP and showing that it plays the same role to Harry Deutsch’s paraconsistent containment logic S that Aleksandr Zinov'ev’s S_1 plays with respect to A I.

In his famous work on vagueness, Russell named “fallacy of verbalism” the fallacy that consists in mistaking the properties of words for the properties of things. In this paper, I examine two (clusters of) mainstream paraconsistent logical theories – the non-adjunctive and relevant approaches –, and show that, if they are given a strongly paraconsistent or dialetheic reading, the charge of committing the Russellian Fallacy can be raised against them in a sophisticated way, by appealing to the intuitive reading of their underlying semantics. The meaning of “intuitive reading” is clarified by exploiting a well-established distinction between pure and applied semantics. If the proposed arguments go through, the dialetheist or strong paraconsistentist faces the following Dilemma: either she must withdraw her claim to have exhibited true contradictions in a metaphysically robust sense – therefore, inconsistent objects and/or states of affairs that make those contradictions true; or she has to give up realism on truth, and embrace some form of anti-realistic (idealistic, or broadly constructivist) metaphysics. Sticking to the second horn of the Dilemma, though, appears to be promising: it could lead to a collapse of the very distinction, commonly held in the literature, between a weak and a strong form of paraconsistency – and this could be a welcome result for a dialetheist.

Is there a notion of contradiction – let us call it, for dramatic effect, “absolute” – making all contradictions, so understood, unacceptable also for dialetheists? It is argued in this paper that there is, and that spelling it out brings some theoretical benefits. First, it gives us a foothold on undisputed ground in the methodologically difficult debate on dialetheism. Second, we can use it to express, without begging questions, the disagreement between dialetheists and their rivals on the nature of truth. Third, dialetheism has an operator allowing it, against the opinion of many critics, to rule things out and manifest disagreement: for unlike other proposed exclusion-expressing-devices (for instance, the entailment of triviality), the operator used to formulate the notion of absolute contradiction appears to be immune both from crip-pling expressive limitations and from revenge paradoxes – pending a rigorous non-triviality proof for a formal dialetheic theory including it.