Being a pragmatic and not a referential approach tosemantics, the dialogical formulation ofparaconsistency allows the following semantic idea tobe expressed within a semi-formal system: In anargumentation it sometimes makes sense to distinguishbetween the contradiction of one of the argumentationpartners with himself (internal contradiction) and thecontradiction between the partners (externalcontradiction). The idea is that externalcontradiction may involve different semantic contextsin which, say A and ¬A have been asserted.The dialogical approach suggests a way of studying thedynamic process of contradictions through which thetwo (...) contexts evolve for the sake of argumentation intoone system containing both contexts.More technically, we show a new, dialogical, way tobuild paraconsistent systems for propositional andfirst-order logic with classical and intuitionisticfeatures (i.e. paraconsistency both with and withouttertium non-datur) and present theircorresponding tableaux. (shrink)

The present paper concerns a technical study of PRIEST'S logic of paradox [Pri 79], We prove that this logic has no proper paraconsistent strengthening. It is also proved that the mentioned logic is the largest paraconsistent one satisfaying TARSKI'S conditions for the classical conjunction and disjunction together with DE MORGAN'S laws for negation. Finally, we obtain for the logic of paradox an algebraic completeness result related to Kleene lattices.

The paper concerns interpretations of the paraconsistent logic LP which model theories properly containing all the sentences of first order arithmetic. The paper demonstrates the existence of such models and provides a complete taxonomy of the finite ones.

The issue of what consequences to draw from the existence of non-classical logical systems has been the subject of an interesting debate across a diversity of fields. In this paper the matter of alternative logics is considered with reference to a specific belief system and its propositions :the Azande are said to maintain beliefs about witchcraft which, when expressed propositionally, appear to be inconsistent. When the Azande have been presented with such inconsistencies, they either fail to see them as such (...) or else accept them as non-problematical. Is our knowledge of logical truths a relative and culturally determined phenomenon, or is there some (transcendent) criterion that allows us to adjudicate between alternative logical systems? The authors propose an approach for resolving disputes about the status of Azande reasoning which assumes a paraconsistent framework, thus providing a new perspective on this debate. (shrink)

This paper offers an inconsistent model of motion perception. It was prompted by work on inconsistent motion due to Hegel and, following him, Priest. But the paper skirts Hegel's full scale idealism, by proposing that the inconsistency is with the cognitive contents of motion perception. The paper draws on work in the psychology of perception, and in the theory of inconsistency. I begin by noting the prima facie argument that temporal change threatens inconsistency, and canvassing ways in which this might (...) be avoided. The orthodox reply to the prima facie argument is that one and the same thing can have different incompatible properties at different times. This is plausible when applied to motion in the physical world. However, applied to cognition, it can be seen that the phenomenon of phi or beta, combined with the mechanism of a Reichardt detector, lends support to one key step in the prima facie argument, namely re-identification over time. The inconsistency of the model is then seen to follow from application of the Fade Principle. Advantages of the model are stressed: including a simple explanation of the debilitating condition of akinetopsia, that is, motion blindness; and a suggestion as to how to account for the “moving now.”. (shrink)

This paper begins an analysis of the real line using an inconsistency-tolerant (paraconsistent) logic. We show that basic field and compactness properties hold, by way of novel proofs that make no use of consistency-reliant inferences; some techniques from constructive analysis are used instead. While no inconsistencies are found in the algebraic operations on the real number field, prospects for other non-trivializing contradictions are left open.

This paper first offers a standard modal extension of dialetheic logics that respect the normal semantics for negation and conjunction, in an attempt to adequately model absolutism, the thesis that there are true contradictions at metaphysically possible worlds. It is shown, however, that the modal extension has unsavoury consequences for both absolutism and dialetheism. While the logic commits the absolutist to dialetheism, it commits the dialetheist to the impossibility of the actual world. A new modal logic AV is then proposed (...) which avoids these unsavoury consequences by invalidating the interdefinability rules for the modal operators with the use of two valuation relations. However, while using AV carries no significant cost for the absolutist, the same isn't true for the dialetheist. Although using AV allows her to avoid the consequence that the actual world is an impossible world, it does so only on the condition that the dialetheist admits that she cannot give a dialetheic solution to all self-refe.. (shrink)

(No abstract is available for this citation).

In this paper we present a new hierarchy of analytical tableaux systems TNDC n, 1≤n<ω, for da Costa's hierarchy of propositional paraconsistent logics Cn, 1≤n<ω. In our tableaux formulation, we introduce da Costa's “ball” operator “o”, the generalized operators “k” and “(k)”, for 1≤k, and the negations “~k”, for k≥1, as primitive operators, differently to what has been done in the literature, where these operators are usually defined operators. We prove a version of Cut Rule for the TNDC n, 1≤n<ω, (...) and also prove that these systems are logically equivalent to the corresponding systems Cn, 1≤n<ω. The systems TNDC n constitute completely automated theorem proving systems for the systems of da Costa's hierarchy Cn, 1≤n<ω. (shrink)

What is the rational response when confronted with a set of propositions each of which we have some reason to accept, and yet which taken together form an inconsistent class? This was, in a nutshell, the problem addressed by the Jaina logicians of classical India, and the solution they gave is, I think, of great interest, both for what it tells us about the relationship between rationality and consistency, and for what we can learn about the logical basis of philosophical (...) pluralism. The Jainas claim that we can continue to reason in spite of the presence of inconsistencies, and indeed construct a many-valued logical system tailored to the purpose. My aim in this paper is to offer a new interpretation of that system and to try to draw out some of its philosophical implications. (shrink)

ABSTRACT The concept of paraconsistent logical consequence is usually negatively defined as a validity semantics in which not every sentences is deducible or in which inferential explosion does not occur. Paraconsistency has been negatively characterized in this way because paraconsistent logics have been designed specifically to avoid the trivialization of deductive inference entailed by the classical paradoxes of material implication for applications in a system that tolerates syntactical contradictions. The effect of the negative characterization of paraconsistency has been to encourage (...) an unsystematic development of distinct versions of paraconsistent logic. It has also contributed to a restricted overview of the full range of paraconsistent formalisms. After reviewing a standard negative characterization of paraconsistency and commenting on its limitations, I propose a positive characterization that makes possible a constructivistic description of the complete spectrum of distinct families of paraconsistent logics, and the description of a new type of maximally relevant paraconsistency, which I argue compares favorably with previously identified categories of paraconsistent logics. (shrink)

This paper raises the question under what circumstances a plurality forms a set, parallel to the Special Composition Question for mereology. The range of answers that have been proposed in the literature are surveyed and criticised. I argue that there is good reason to reject both the view that pluralities never form sets and the view that pluralities always form sets. Instead, we need to affirm restricted set formation. Casting doubt on the availability of any informative principle which will settle (...) which pluralities form sets, the paper concludes by affirming a naturalistic approach to the philosophy of set theory. (shrink)

In 1880, when Oliver Wendell Holmes (later to be a Justice of the U.S. Supreme Court) criticized the logical theology of law articulated by Christopher Columbus Langdell (the first Dean of Harvard Law School), neither Holmes nor Langdell was aware of the revolution in logic that had begun, the year before, with Frege's Begriffsschrift. But there is an important element of truth in Holmes's insistence that a legal system cannot be adequately understood as a system of axioms and corollaries; and (...) this element of truth is not obviated by the more powerful logical techniques that are now available. (shrink)

Jaina philosophy provides a very distinctive account of logic, based on the theory of ?sevenfold predication?. This paper provides a modern formalisation of the logic, using the techniques of many-valued and modal logic. The formalisation is applied, in turn, to some of the more problematic aspects of Jaina philosophy, especially its relativism.

On the face of it, normative conflicts are commonplace. Yet standard deontic logic declares them to be logically impossible. That prompts the question, What are the proper principles of normative reasoning if such conflicts are possible? This paper examines several alternatives that have been proposed for a logic of 'ought' that can accommodate normative conflicts, and finds all of them unsatisfactory as measured against three criteria of adequacy. It then introduces a new logic that does meet all three criteria, and (...) so allows for the possibility of genuine normative conflicts. (shrink)

This paper presents a range of new triviality proofs pertaining to naïve truth theory formulated in paraconsistent relevant logics. It is shown that excluded middle together with various permutation principles such as A → (B → C)⊩B → (A → C) trivialize naïve truth theory. The paper also provides some new triviality proofs which utilize the axioms ((A → B)∧ (B → C)) → (A → C) and (A → ¬A) → ¬A, the fusion connective and the Ackermann constant. An (...) overview over various ways to formulate Leibniz’s law in non-classical logics and two new triviality proofs for naïve set theory are also provided. (shrink)

The pluralist sheds the more traditional ideas of truth and ontology. This is dangerous, because it threatens instability of the theory. To lend stability to his philosophy, the pluralist trades truth and ontology for rigour and other ‘fixtures’. Fixtures are the steady goal posts. They are the parts of a theory that stay fixed across a pair of theories, and allow us to make translations and comparisons. They can ultimately be moved, but we tend to keep them fixed temporarily. Apart (...) from considering rigour of proof as a fixture, I discuss fixed models, invariant notions and fixed information about objects across theories. There are other fixtures, but it is enough to start with these. (shrink)

The article examines a fundamental problem in classical Jaina philosophy, namely, the ontological status of dead matter in the hylozoistic and at the same time dualistic Jaina worldview. This question is of particular interest in view of the widespread contemporary Jaina practice of venerating bone relics and stūpas of prominent saints. The main argument proposed in this article is, that, from a classical doctrinal point of view, bone relics of renowned ascetics are valuable for Jainas, if at all, because of (...) their unique physical attributes, rather than the presumed presence of the deceased in the remains as posited in much of the extant literature on relic worship across cultures. The specific focus of the article are Jaina and non-Jaina explanations of the qualities of special matter in terms of karmic and natural processes of transformation. (shrink)

This paper reflects on metametaphysics and as such develops a metametameta-physical view: that quietist metametaphysics requires dialetheism, and in turn a paraconsistent logic. I demonstrate this using Carnap’s metametaphysical position in his 'Empiricism, Semantics and Ontology' as an example, with regard to how it exhibits self-reference and results in inconsistency. I show how applying Carnap’s position to itself produces a dilemma, both horns of which lead to a contradiction. Such inconsistency commonly arises from meta-theories with global scope, as the 'meta' (...) approach aims to transcend the scope of that which it is theorizing about, whilst the global nature will place itself back within the scope of that which it is theorizing about, which together result in the theory referring to itself whilst refuting itself. I argue that any global metametaphysical theory that draws a limit to thought will face self-reference problems leading to contradictory realms. My conclusion is conditional: If we want to meta-philosophize in such a way and treat quietist meta-theories as being true, then we need to be dialetheist and utilize a paraconsistent logic in order to accommodate the contradictions that result from such theorizing. (shrink)

Does there exist any equivalence between the notions of inconsistency and consequence in paraconsistent logics as is present in the classical two valued logic? This is the key issue of this paper. Starting with a language where negation ( ${\neg}$ ) is the only connective, two sets of axioms for consequence and inconsistency of paraconsistent logics are presented. During this study two points have come out. The first one is that the notion of inconsistency of paraconsistent logics turns out to (...) be a formula-dependent notion and the second one is that the characterization (i.e. equivalence) appears to be pertinent to a class of paraconsistent logics which have double negation property. (shrink)

I prove that the three basic propositions of Vasiliev's paraconsistent logic have a semantic interpretation by means of the intuitionist logic. The interpèretation is confirmed by amens of the da Costa's model of Vasiliev's paraconsistent logic.

In this paper we present a new hierarchy of analytical tableaux systems TNDC n, 1≤n (...) and also prove that these systems are logically equivalent to the corresponding systems Cn, 1≤nshrink)

Richard Sylvan (né Routley) was one of Australasia's most prolific and systematic philosophers. Though known for his innovative work in logic and metaphysics, the astonishing breadth of his philosophical endeavours included almost all reaches of philosophy. Taking the view that very basic assumptions of mainstream philosophy were fundamentally mistaken, he sought radical change across a wide range of theories. However, his view of the centrality of logic and recognition of the possibilities opened up by logical innovation in the fundamental areas (...) of metaphysics resulted in his working primarily in these two, closely connected fields. It is this work in logic and metaphysics that is the main focus of what follows. (shrink)

The so-called paradoxes of material implication have motivated the development of many non-classical logics over the years, such as relevance logics, paraconsistent logics, fuzzy logics and so on. In this note, we investigate some of these paradoxes and classify them, over minimal logic. We provide proofs of equivalence and semantic models separating the paradoxes where appropriate. A number of equivalent groups arise, all of which collapse with unrestricted use of double negation elimination. Interestingly, the principle ex falso quodlibet, and several (...) weaker principles, turn out to be distinguishable, giving perhaps supporting motivation for adopting minimal logic as the ambient logic for reasoning in the possible presence of inconsistency. (shrink)

Logic: the Basics is an accessible introduction to the core philosophy topic of standard logic. Focussing on traditional Classical Logic the book deals with topics such as mathematical preliminaries, propositional logic, monadic quantified logic, polyadic quantified logic, and English and standard ‘symbolic transitions’. With exercises and sample answers throughout this thoroughly revised new edition not only comprehensively covers the core topics at introductory level but also gives the reader an idea of how they can take their knowledge further and the (...) philosophical questions around logic. -/- Logic: the Basics is essential reading for first-year undergraduate philosophy students on standard introductory logic courses. (shrink)

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. (shrink)

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.

Knowledge Representation and Reasoning (KR&R) is based on the idea that propositional content can be rigorously represented in formal languages long the province of logic, in such a way that these representations can be productively reasoned over by humans and machines; and that this reasoning can be used to produce knowledge-based systems (KBSs). As such, KR&R is a discipline conventionally regarded to range across parts of artificial intelligence (AI), computer science, and especially logic. This standard view of KR&R’s participating fields (...) is correct — but dangerously incomplete. The view is incomplete because, as we explain herein, sophisticated KR&R must rely heavily upon philosophy. Encapsulated, the reason is actually quite straightforward: Sophisticated KR&R must include the representation of not only simple properties, but also concepts that are routine in the formal sciences (theoretical computer science, mathematics, logic, game theory, etc.), and everyday socio-cognitive concepts like mendacity, deception, betrayal, and evil. Because in KR&R the representation of such concepts must be rigorous in order to enable machine reasoning (e.g., machine-generated and machine-checked proofs that a is lying to b) over them, philosophy, devoted as it is in no small part to supplying analyses of such concepts, is a crucial partner in the overall enterprise. To put the point another way: When the knowledge to be represented is such as to require lengthy formulas in expressive formal languages for that representation, philosophy must be involved in the game. In addition, insofar as the advance of KR&R must allow formalisms and processes for representing and reasoning over visual propositional content, philosophy will be a key contributor into the future. (shrink)

Niklas Luhmann verwendet in seiner soziologischen Systemtheorie offenbar etwas, das er den Büchern des englischen Mathematikers George Spencer Brown entnimmt. Dessen Formenkalkül ist für Luhmann, wie Günther Schulte treffend bemerkt, “Mädchen für alles, mit dem er nicht nur in der Lage ist Teezukochen, sondern auch Auto oder Straßenbahn zu fahren”. Der erste Blick in Spencer Browns Laws of Form vermittelt einen anderen Eindruck: nichts scheinen sie mit soziologischer Systemtheorie zu tun zu haben. Der vorliegende Text bearbeitet hieran anknüpfend eine recht (...) bescheidene Frage, die sich gleichwohl jedem Luhmann-Leser schon einmal gestellt haben dürfte: Was wollen die Laws of Form und was will Luhmann mit ihnen? Als Antwort ergibt sich, nach Zurückverfolgung der relevanten Fußnoten, eine gute und eine schlechte Nachricht. Die schlechte Nachricht ist, daß die Lektüre der Laws of Form offenbar niemandem wirklich weiterhelfen kann, auch Luhmann selbst nicht. Die gute ist folglich, daß dem Luhmann-Leser die Notwendigkeit erspart bleibt, einen so dunklen, weil sparsamen Kalkül zu verstehen. Das meiste nämlich, was Luhmann den Laws of Form angeblich entnimmt, steht auf den zweiten Blick nicht darin. Er wird es also ohnehin durch andere Texte begründen müssen. (shrink)

The question of ontological foundation has undergone a noteworthy revival in recent years: metaphysicians today quarrel about how exactly to understand the asymmetrical and hyperintensional relationship of grounding. One of the reasons for this revival is that the old quantificationalist meta-ontology inherited from Quine has been effectively criticised by leading philosophers favourable to a meta-ontology, the aim of which is to come to know “which facts/items ground (constitute the base of) which other facts/items”, thus to examine the relation of ontological (...) dependence between beings (e.g. chemical properties depend on physical properties, the economic situation on the behaviour of individuals etc.), i.e. to explore the hierarchical structure of reality. I shall not discuss here the relationship of grounding in itself, but make some historical-formal remarks on the properties of the ultimate ontological foundational item itself and its aporetic nature. To do so I explore various more or less exotic philosophical ecosystems in the following order: Heidegger (Seyn), Plato (ἕν), Wáng Bi (道, dào). On the way I shall propose a new interpretation both of certain hypotheses in the Parmenides and of the nature of the opposition between Wáng Bì and Guō Xiàng in regard to the logical grammar of the expression “nothing (無, wú)” (transl. J. Dudley). (shrink)

This paper explores early Australasian philosophy in some detail. Two approaches have dominated Western philosophy in Australia: idealism and materialism. Idealism was prevalent between the 1880s and the 1930s, but dissipated thereafter. Idealism in Australia often reflected Kantian themes, but it also reflected the revival of interest in Hegel through the work of ‘absolute idealists’ such as T. H. Green, F. H. Bradley, and Henry Jones. A number of the early New Zealand philosophers were also educated in the idealist tradition (...) and were influential in their communities, but produced relatively little. In Australia, materialism gained prominence through the work of John Anderson, who arrived in Australia in 1927, and continues to be influential. John Anderson had been a student of Henry Jones, who might therefore be said to have influenced both main strands of Australian philosophical thought. (shrink)

Philosophical conceptual analysis is an experimental method. Focusing on this helps to justify it from the skepticism of experimental philosophers who follow Weinberg, Nichols & Stich. To explore the experimental aspect of philosophical conceptual analysis, I consider a simpler instance of the same activity: everyday linguistic interpretation. I argue that this, too, is experimental in nature. And in both conceptual analysis and linguistic interpretation, the intuitions considered problematic by experimental philosophers are necessary but epistemically irrelevant. They are like variables introduced (...) into mathematical proofs which drop out before the solution. Or better, they are like the hypotheses that drive science, which do not themselves need to be true. In other words, it does not matter whether or not intuitions are accurate as descriptions of the natural kinds that undergird philosophical concepts; the aims of conceptual analysis can still be met. (shrink)

On the one hand, logic has (or ought to have) nothing to do with metaphysics; it ought to have nothing to do with questions concerning what there is, or whether there is anything at all. On the other hand, metaphysics can hardly get off the ground without the help of logical analysis; to be is to be a truth-maker, and the search for truth-makers requires that we lay open the logical structure of our language. So something’s gotta give: either logical (...) analysis is metaphysically biased, or metaphysicians must make up their mind before resorting to logical analysis. Or both. (shrink)

Over the past few decades, reasoning and rationality have been the focus of enormous interdisciplinary attention, attracting interest from philosophers, psychologists, economists, statisticians and anthropologists, among others. The widespread interest in the topic reflects the central status of reasoning in human affairs. But it also suggests that there are many different though related projects and tasks which need to be addressed if we are to attain a comprehensive understanding of reasoning.

This paper was presented at the 26th International Wittgenstein Symposium on “Knowledge and Belief”, Kirchberg am Wechsel, Austria (3rd - 10th August 2003). The focus of study is a dialogue situation in which one party holds that P while the other holds that ~P. A simple way to establish harmony between the parties in dialogue is to insist that each should include the other’s point of view. This is unsatisfactory because it results in an inconsistent set of beliefs. Clarity is (...) essential, therefore, on how to deal with inconsistency. In this paper, I will proceed in two steps. In the first, I will indicate how some attempts at trying to manage contradictions in terms of ontology remain unsatisfactory. In the second step, I will introduce and evaluate another way in terms of belief and acceptance. I make an original application of L.J. Cohen’s distinction between belief and acceptance, the former being a matter of feeling, the latter a matter of policy in reasoning. (shrink)

This presentation traces an historical root of the reductio ad absurdum mode of argumentation in Greek philosophy. I propose a new understanding of the liar paradox as an instance of this mode of argumentation. I show that the paradox was crea ted as part of a refutational argument in the controversy over the justification of realism and the realists concepts of truth and certainty. The paradox was part of the dialectical style of Greek scepticism, which was characterized, inter alia, by (...) the u se of the reductio ad absurdum. The paradox turns out to be a metaphysical and epistemological argument. (shrink)