Mark Balaguer argues for full blooded platonism (FBP), and argues that FBP alone can solve Benacerraf's familiar epistemic challenge. I note that if FBP really can solve Benacerraf's epistemic challenge, then FBP is not alone in its capacity so to solve; RFBP—really full blooded platonism—can do the trick just as well, where RFBP differs from FBP by allowing entities from inconsistent mathematics. I also argue briefly that there is positive reason for endorsing RFBP.

The view that contradictions cannot be true has been part of accepted philosophical theory since at least the time of Aristotle. In this regard, it is almost unique in the history of philosophy. Only in the last forty years has the view been systematically challenged with the advent of dialetheism. Since Graham Priest introduced dialetheism as a solution to certain self-referential paradoxes, the possibility of true contradictions has been a live issue in the philosophy of logic. Yet, despite the arguments (...) advanced by dialetheists, many logicians and philosophers still hold the opinion that contradictions cannot be true. -/- Rather than advocating the truth of certain contradictions, this thesis offers a different challenge to the classical logician. By showing that it can be philosophically coherent to propose that true contradictions are metaphysically possible, the thesis suggests that the classical logician must do more than she currently has to justify her confidence in the impossibility of true contradictions. Simply fighting off the dialetheist’s putative examples of true contradictions at the actual world isn’t enough to justify the classical logician’s conclusion that true contradictions are impossible. -/- To aid the thesis dialectically, we introduce a new position, absolutism, which hypothesises that it’s metaphysically possible for at least one contradiction to be true, contrasting with the dialetheic hypothesis that some contradictions are true in the actual world. We demonstrate that absolutism can be given a philosophically coherent interpretation, an appropriate logic, and that certain criticisms are completely toothless against absolutism. The challenge put to the classical logician is then: On what logical or philosophical grounds can we rule out the metaphysical possibility of true contradictions? (shrink)

Dialetheism is the view that some contradictions are true. Resorting to either metaphysical dialetheism or semantic dialetheism may seem like an appropriate resolve to certain theological contradictions. At least for those who concede to theological contradictions, and take dialetheism seriously. However, I demonstrate that neither of these types of dialetheism would serve to be amenable in resolving an Islamic theological contradiction. This is a theological contradiction that I refer to as ‘the paradox of an unknowable and ineffable God’. As a (...) result of this, I propose an alternative type of dialetheism which aims to resolve the paradox of an unknowable and ineffable God. I call this type of dialetheism, ‘mystical dialetheism’. (shrink)

This paper describes a modal conception of sets, according to which sets are 'potential' with respect to their members. A modal theory is developed, which invokes a naive comprehension axiom schema, modified by adding `forward looking' and `backward looking' modal operators. We show that this `bi-modal' naive set theory can prove modalized interpretations of several ZFC axioms, including the axiom of infinity. We also show that the theory is consistent by providing an S5 Kripke model. The paper concludes with some (...) discussion of the nature of the modalities involved, drawing comparisons with noneism, the view that there are some non-existent objects. (shrink)

What follows is a taxonomy of arguments that regresses of inferential justification are vicious. They fall out into four general classes: conceptual arguments from incompleteness, conceptual arguments from arbitrariness, ought-implies-can arguments from human quantitative incapacities, and ought-implies can arguments from human qualitative incapacities. They fail with a developed theory of "infinitism" consistent with valuational pluralism and modest epistemic foundationalism.

As far as ‘modern’ logical theories of vagueness are concerned, a main distinction can be drawn between ‘semantical’ ones and ‘pragmatical’ ones. The latter are defended here, because they tend to retake into account important contextual dimensions of the problem abandoned by the former. Their inchoate condition seems not alarming, since they are of surprisingly recent date. This, however, could very well be an accidental explanation. That is, the true reason for it might sooner or later turn out to be (...) bearing exactly on the fundamental human limitations, when it comes to theorizing, that these approaches are urging us to appreciate. (shrink)

On the basis of a wide range of historical examples various features of axioms are discussed in relation to their use in mathematical practice. A very general framework for this discussion is provided, and it is argued that axioms can play many roles in mathematics and that viewing them as self-evident truths does not do justice to the ways in which mathematicians employ axioms. Possible origins of axioms and criteria for choosing axioms are also examined. The distinctions introduced aim at (...) clarifying discussions in philosophy of mathematics and contributing towards a more refined view of mathematical practice. (shrink)

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)

Philosophers have tried to explain how science finds the truth by using new developments in logic to study scientific language and inference. R. G. Collingwood argued that only a logic of problems could take context into account. He was ignored, but the need to reconcile secure meanings with changes in context and meanings was seen by Karl Popper, W. v. O. Quine, and Mario Bunge. Jagdish Hattiangadi uses problems to reconcile the need for security with that for growth. But he (...) mistakenly insists that all problems are mere contradictions and artificially separates rigid from flexible aspects of meanings. In order to resolve the conflict we must (1) replace the quest for rigid terms with techniques for improvement, (2) use plausible arguments to uncover confused meanings, (3) use frameworks to choose problems and to regulate meanings, and (4) employ a bootstrap approach that uses frameworks to improve meanings and refined meanings to improve frameworks. (shrink)

Closure is the idea that what is true about a theory of truth should be true in it. Commitment to closure under truth motivates non-classical logic; commitment to closure under validity leads to substructural logic. These moves can be thought of as responses to revenge problems. With a focus on truth in mathematics, I will consider whether a noncontractive approach faces a similar revenge problem with respect to closure under provability, and argue that if a noncontractive theory is to be (...) genuinely closed, then it must be free of all contraction, even in the metatheory. (shrink)

In this paper I investigate the relation between physics and metaphysics in Plato’s participation theory. I show that the logic shoring up Plato’s metaphysics in paraconsistent, as had been suggested already by Graham Priest. The transformation of the paradoxical One-and-Many of the pre-Socratics into a paraconsistent Great-and-Small bridges the abyss between archaic rationality and the world of classical logic based ultimately on the principle of contradiction. Indeed, language is an organ of perception, not simply a means of communication. J. Jaynes, (...) Origin of Consciousness. (shrink)

In this paper I argue that inconsistencies in scientific theories may arise from the type of causality relation they—tacitly or explicitly—embody. All these seemingly different causality relations can be subsumed under a general strategy developed to defeat the paradoxes which inevitably occur in our experience of the real. With respect to this, scientific theories are just a subclass of the larger class of metaphysical theories, construed as theories that attempt to explain a (part of) the world consistently. All metaphysical theories (...) share a common structural backbone specificially designed to defeat paradoxes, their often wildly diverging ontological claims notwithstanding. This common structure shapes the procedures which govern the invention of ideas in the context of such theories, by codifying some onto-logical a priori assumptions regarding the consistency of reality into its bare conceptual framework. Causality plays a key rôle here, because it implies conservation of identity, itself a far from simple notion. It imposes strong demands on the universalising power of the theories concerned. These demands are often met by the introduction of a metalevel which encompasses the notions of ‘system’ and ‘lawful behaviour’. In classical mechanics, the division between universal and particular leaves its traces in the separate treatment of cinematics and dynamics. The fundamental backbone’s specific gestalt thus functions as a theory’s individual signature and paves the way to a comparative historical approach towards their study. An important part of my paper therefore explores the strong connections between paradoxes as they appear and are dealt with in ancient philosophy and their re-appearance in early modern natural philosophy and science. This analysis is applied to the mechanical theories of Newton and Leibniz, with some surprising results. (shrink)

In this paper we look at the suitability of modern interval-based temporal logic for modeling John Buridan’s treatment of tensed sentences in his Sophismata. Building on the paper, we develop Buridan’s analysis of temporal logic, paying particular attention to his notions of negation and the absolute/relative nature of the future and the past.We introduce a number of standard modern propositional interval temporal logics to illustrate where Buridan’s interval-based temporal analysis differs from the standard modern approaches. We give formal proofs of (...) some claims in, and sketch how the standard modern systems could be defined in terms of Buridan’s proposals, showing that his logic can be taken as more basic. In diesem Aufsatz betrachten wir das Geeignetsein einer modernen intervallbasierten temporalen Logik für die Modellierung von John Buridans Behandlung zeitabhängiger Sätze in seinen Sophismata. Aufbauend auf den Aufsatz stellen wir Buridans Analyse der temporalen Logik dar, mit besonderer Berücksichtigung seiner Begriffe der Negation und der absoluten/relativen Natur der Zukunft und der Vergangenheit. Wir führen einige standardmäßige moderne intervallbasierte temporale Aussagenlogiken ein, um zu illustrieren, wo Buridans intervallbasierte temporale Analyse von den standardmäßigen modernen Zugangsweisen abweicht. Wir geben formale Beweise einiger Behauptungen in an und skizzieren, wie die standardmäßigen modernen Systeme im Sinne der Vorschläge Buridans definiert werden könnten; wir zeigen damit, dass seine Logik als die grundlegendere aufgefasst werden kann. (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<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)

In some logics, anything whatsoever follows from a contradiction; call these logics explosive. Paraconsistent logics are logics that are not explosive. Paraconsistent logics have a long and fruitful history, and no doubt a long and fruitful future. To give some sense of the situation, I’ll spend Section 1 exploring exactly what it takes for a logic to be paraconsistent. It will emerge that there is considerable open texture to the idea. In Section 2, I’ll give some examples of techniques for (...) developing paraconsistent logics. In Section 3, I’ll discuss what seem to me to be some promising applications of certain paraconsistent logics. In fact, however, I don’t think there’s all that much to the concept ‘paraconsistent’ itself; the collection of paraconsistent logics is far too heterogenous to be very productively dealt with under a single label. Perhaps that will emerge as we go. (shrink)

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)

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)

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.

(No abstract is available for this citation).

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)

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.

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)

In this article, our aim is to take a step towards a full understanding of the notion of paraconsistency in the context of metainferential logics. Following the work initiated by Barrio et al. [2018], we will consider a metainferential logic to be paraconsistent whenever the metainferential version of Explosion is invalid. However, our contribution consists in modifying the definition of meta-Explosion by extending the standard framework and introducing a negation for inferences and metainferences. From this new perspective, Tarskian paraconsistent logics (...) such as LP will not turn out to be metainferentially paraconsistent, in contrast to, for instance, non-transitive logics like ST. Finally, we will end up by defining a logic which is metainferentially paraconsistent at every level, and discussing whether this logic is uniform through translations. (shrink)

In this first section we start with defining the notions of inconsistency and para-consistency, we give an example of an inconsistency and clarify what according to us is the basic problem with respect to the occurrence of inconsistencies. We are then in a position to state the aim of this paper.

We present the inconsistency-adaptive deontic logic DP r , a nonmonotonic logic for dealing with conflicts between normative statements. On the one hand, this logic does not lead to explosion in view of normative conflicts such as O A ∧ O ∼A, O A ∧ P ∼A or even O A ∧ ∼O A. On the other hand, DP r still verifies all intuitively reliable inferences valid in Standard Deontic Logic (SDL). DP r interprets a given premise set ‘as normally (...) as possible’ with respect to SDL. Whereas some SDL-rules are verified unconditionally by DP r , others are verified conditionally. The latter are applicable unless they rely on formulas that turn out to behave inconsistently in view of the premises. This dynamic process is mirrored by the proof theory of DP r. (shrink)

. This paper proposes a generalization of the theory of the process of explanation to include consistent as well as inconsistent situations. The generalization is strong, for example in the sense that, if the background theory and the initial conditions are consistent, it leads to precisely the same results as the theory from the lead paper (Halonen and Hintikka 2004). The paper presupposes (and refers to arguments for the view that) inconsistencies constitute problems and that scientists try to resolve them.

In [BAT 00b], the flat Rescher–Manor consequence relations — the Free, Strong, Argued, C-Based, andWeak consequence relation—were shown to be characterized by inconsistency-adaptive logics defined from the paraconsistent logic CLuN. This provided these consequence relations with a dynamic proof theory. In the present paper we show that the detour via an inconsistency-adaptive logic is not necessary. We present a direct dynamic proof theory, formulated in the language of Classical Logic, and prove its adequacy. The present paper contains the first direct (...) dynamic proof theory for consequence relations that are characterized by an adaptive logic. (shrink)

In this paper I reassess Floridi's solution to the Bar-Hillel-Carnap paradox (the information-yield of inconsistent propositions is maximal) by questioning the orthodox view that contradictions cannot be true. The main part of the paper is devoted to showing that the veridicality thesis (semantic information has to be true) is compatible with dialetheism (there are true contradictions), and that unless we accept the additional non-falsity thesis (information cannot be false) there is no reason to presuppose that there is no such thing (...) like contradictory information. (shrink)

Philosophy and Literature is an internationally renowned refereed journal founded by Denis Dutton at the University of Canterbury, Christchurch. It is now published by the Johns Hopkins University Press. Since its inception in 1976, Philosophy and Literature has been concerned with the relation between literary and philosophical studies, publishing articles on the philosophical interpretation of literature as well as the literary treatment of philosophy. Philosophy and Literature has sometimes been regarded as iconoclastic, in the sense that it repudiates academic pretensions, (...) insidious jargon and institutional vogue. Dutton, who remains the editor, still writes a regular column. A distinctive feature of Philosophy and Literature was the annual Bad Writing Contest, held from 1995–98, which sought to identify (and publish) the ‘most stylistically lamentable passages’ of academic prose, often to great amusement. (shrink)

Este é um conto sobre visão, pensamento e contradição, bem como sobre o papel que desempenham na primeira metade da Ciência da Lógica de Hegel. A Lógica começa com uma descida, nesse caso, uma queda do Ser ao Nada. Posteriormente, aproximadamente na metade de cada texto, há um certo paradoxo em que tudo está em jogo, a categoria da contradição. Nesse exato momento, o pensamento ao mesmo tempo falha e é renovado em um viés especulativo. Nessa seção, nos debruçamos sobre (...) algumas das interpretações influentes do uso hegeliano da contradição. Para isso, voltamo-nos a uma curiosa obra de arte, Pleiades, de James Turrell, como um exemplo estético dessa primeira queda. Progrediremos, então, através do texto com pensamento e visão como nosso duplo norteador em um passo relativamente rápido, sem desacelerar até encontrarmos a história de Hegel da contradição, onde mostro a natureza explosiva da contradição. Isso nos permitirá ver como Hegel invoca o poder da contradição para gerar a segunda metade da história da Ciência da Lógica. Eu começo com a descida do Ser ao Nada, o momento em que a escuridão emerge. (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)

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)