In this paper I urge friends of truth-value gaps and truth-value gluts – proponents of paracomplete and paraconsistent logics – to consider theories not merely as sets of sentences, but as pairs of sets of sentences, or what I call ‘bitheories,’ which keep track not only of what holds according to the theory, but also what fails to hold according to the theory. I explain the connection between bitheories, sequents, and the speech acts of assertion and denial. I illustrate the (...) usefulness of bitheories by showing how they make available a technique for characterising different theories while abstracting away from logical vocabulary such as connectives or quanti- fiers, thereby making theoretical commitments independent of the choice of this or that particular non-classical logic. (shrink)

This book explores the nature and significance of Pyrrhonism, the most prominent and influential form of skepticism in Western philosophy. Not only did Pyrrhonism play an important part in the philosophical scene of the Hellenistic and Imperial age, but it also had a tremendous impact on Renaissance and modern philosophy and continues to be a topic of lively discussion among both scholars of ancient philosophy and epistemologists. The focus and inspiration of the book is the brand of Pyrrhonism expounded in (...) the extant works of Sextus Empiricus. Its aim is twofold: to offer a critical interpretation of some of the central aspects of Sextus’s skeptical outlook and to examine certain debates in contemporary philosophy from a neo-Pyrrhonian perspective. The first part explores the aim of skeptical inquiry, the defining features of Pyrrhonian argumentation, the epistemic challenge posed by the Modes of Agrippa, and the Pyrrhonist’s stance on the requirements of rationality. The second part focuses on present-day discussions of the epistemic significance of disagreement, the limits of self-knowledge, and the nature of rationality. The book will appeal to researchers and graduate students interested in skepticism. (shrink)

A logic is called 'paraconsistent' if it rejects the rule called 'ex contradictione quodlibet', according to which any conclusion follows from inconsistent premises. While logicians have proposed many technically developed paraconsistent logical systems and contemporary philosophers like Graham Priest have advanced the view that some contradictions can be true, and advocated a paraconsistent logic to deal with them, until recent times these systems have been little understood by philosophers. This book presents a comprehensive overview on paraconsistent logical systems to change (...) this situation. The book includes almost every major author currently working in the field. The papers are on the cutting edge of the literature some of which discuss current debates and others present important new ideas. The editors have avoided papers about technical details of paraconsistent logic, but instead concentrated upon works that discuss more 'big picture' ideas. Different treatments of paradoxes takes centre stage in many of the papers, but also there are several papers on how to interpret paraconistent logic and some on how it can be applied to philosophy of mathematics, the philosophy of language, and metaphysics. (shrink)

In early Buddhist logic, it was standard to assume that for any state of a ff airs there were four possibilities: that it held, that it did not, both, or neither. This is the catuskoti (or tetralemma). Classical logicians have had a hard time mak­ing sense of this, but it makes perfectly good sense in the se­mantics of various paraconsistent logics, such as First Degree Entailment. Matters are more complicated for later Buddhist thinkers, such as Nagarjuna, who appear to suggest (...) that none of these options , or more than one, may hold. The point of this paper is to examine the matter, including the formal logical machinery that may be appropriate. (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)

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

Notions such as śūnyatā, catuṣkoṭi, and Indra's net, which figure prominently in Buddhist philosophy, are difficult to readily accommodate within our ordinary thinking about everyday objects. Famous Buddhist scholar Nāgārjuna considered two levels of reality: one called conventional reality, and the other ultimate reality. Within this framework, śūnyatā refers to the claim that at the ultimate level objects are devoid of essence or "intrinsic properties", but are interdependent by virtue of their relations to other objects. Catuṣkoṭi refers to the claim (...) that four truth values, including contradiction, are admissible in reasoning. Indra's net refers to the claim that every part of a whole is reflective of the whole. Here we present category theoretic constructions that are reminiscent of these Buddhist concepts. The universal mapping property definition of mathematical objects, wherein objects of a universe of discourse are defined not in terms of their content, but in terms of their relations to all objects of the universe is reminiscent of śūnyatā. The objective logic of perception, with perception modeled as [a category of] two sequential processes (sensation followed by interpretation), and with its truth value object of four truth values, is reminiscent of the Buddhist logic of catuṣkoṭi. The category of categories, wherein every category has a subcategory of sets with zero structure within which every category can be modeled, is reminiscent of Indra's net. Our thorough elaboration of the parallels between Buddhist philosophy and category theory can facilitate better understanding of Buddhist philosophy, and bring out the broader philosophical import of category theory beyond mathematics. (shrink)

This article proposes to interpret Nāgārjuna’s Madhyamaka philosophy as a flexible philosophical position rather than a fixed position within the metaphysical landscape. Nāgārjuna’s enigmatic mode of reasoning complicates efforts to definitively delineate his position in respect to the status of reality. Instead, it proves more worthwhile to consider his philosophical stance as dynamic, flexible, non-static, and adaptive relative to his philosophical opponents. Presenting Nāgārjuna as occupying a clearly defined metaphysical position fails to capture the full richness and complexity of the (...) Madhyamaka philosophy, and goes precisely against the Mādhyamika’s spirit and method. Nāgārjuna constantly undermines the assertions of his adversaries, expressing no fixed thesis of his own, thereby embodying the fluidity of the middle way. I contend that Nāgārjuna’s adeptness at articulating the middle way stems from his subversive-styled rhetorical strategies coupled with his application of the prasaṅga method. Furthermore, this interpretation aligns more closely with Buddhist practice in general and sheds light on the pedagogical dimensions of the Madhyamaka philosophy. (shrink)

This paper reconsiders the notion that Nagarjuna and Wittgenstein’s Tractatus may only be seen as comparable under a shared ineffability thesis, that is, the idea that reality is impossible to describe in sensible discourse. Historically, Nagarjuna and the early Wittgenstein have both been widely construed as offering either metaphysical theories or attempts to refute all such theories. Instead, by employing an interpretive framework based on a ‘resolute’ reading of the Tractatus, I suggest we see their philosophical affinity in terms of (...) a shared conception of philosophical method without proposing theses. In doing so, this offers us a new way to understand Nagarjuna’s characteristic claims both to have ‘no views’ and refusal to accept that things exist ‘inherently’ or with ‘essence’. Therefore, instead of either a view about the nature of a mind-independent ‘ultimate reality’ or a thesis concerning the rejection of such a domain, I propose that we understand Nagarjuna’s primary aim as ‘therapeutic’, that is, concerned with the dissolution of philosophical problems. However, this ‘therapy’ should neither be confined to the psychotherapeutic metaphor nor should it be taken to imply a private enlightenment only available to philosophers. Instead, for Nagarjuna and Wittgenstein, philosophical problems are cast as a source of disquiet for all of us; what their work offers is a soteriology, a means towards our salvation. (shrink)

In a recent article in this journal, Giuseppe Ferraro mounted a sustained attack on the semantic interpretation of the Madhyamaka doctrine of emptiness, an interpretation that has been championed by the authors. The present paper is their reply to that attack.

Some time ago I advanced on the pages of this journal a critique of the interpretation given by Jay L. Garfield and Mark Siderits (hereafter GS) of Nāgārjuna’s doctrine of the two truths (Ferraro, J Indian Philos 41(2):195–219, 2013.1); to my article the two authors responded with a ‘defense of the semantic interpretation’ of the Madhyamaka doctrine of emptiness (GS, J Indian Philos 41(6):655–664, 2013). Their reply, however, could not consider my personal understanding of Nāgārjuna’s notions of śūnyatā and dve (...) satye. My interpretation of these notions was in fact absent in the paper GS responded, and was only related in a second paper of mine (Ferraro, J Indian Philos 41(5):563–590, 2013.2), still unpublished at the time of GS’s drafting of their ‘defense’. Therefore, in the present rejoinder, I found it appropriate, rather than responding point by point to the passages of GS’s reply, to advance a comparison between the semantic interpretation and my ‘realist antimetaphysical’ understanding of Nāgārjuna’s thought. (shrink)

I provide an interpretation of Wittgenstein's much criticized remarks on Gödel's First Incompleteness Theorem in the light of paraconsistent arithmetics: in taking Gödel's proof as a paradoxical derivation, Wittgenstein was right, given 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. I show that the models of paraconsistent arithmetics (obtained via the Meyer-Mortensen (...) collapsing filter on the standard model) match with many intuitions underlying Wittgensteins philosophy of mathematics, such as its strict finitism and the insistence on the decidability of any meaningful mathematical question. (shrink)

We here attempt to address certain criticisms of the philosophical import of the so-called Brazilian approach to paraconsistency by providing some epistemic elucidations of the whole enterprise of the logics of formal inconsistency. In the course of this discussion, we substantiate the view that difficulties in reasoning under contradictions in both the Buddhist and the Aristotelian traditions can be accommodated within the precepts of the Brazilian school of paraconsistency.

The recent literature on Nāgārjuna’s catuṣkoṭi centres around Jay Garfield’s and Graham Priest’s interpretation. It is an open discussion to what extent their interpretation is an adequate model of the logic for the catuskoti, and the Mūla-madhyamaka-kārikā. Priest and Garfield try to make sense of the contradictions within the catuskoti by appeal to a series of lattices – orderings of truth-values, supposed to model the path to enlightenment. They use Anderson & Belnaps's framework of First Degree Entailment. Cotnoir has argued (...) that the lattices of Priest and Garfield cannot ground the logic of the catuskoti. The concern is simple: on the one hand, FDE brings with it the failure of classical principles such as modus ponens. On the other hand, we frequently encounter Nāgārjuna using classical principles in other arguments in the MMK. There is a problem of validity. If FDE is Nāgārjuna’s logic of choice, he is facing what is commonly called the classical recapture problem: how to make sense of cases where classical principles like modus pones are valid? One cannot just add principles like modus pones as assumptions, because in the background paraconsistent logic this does not rule out their negations. In this essay, I shall explore and critically evaluate Cotnoir’s proposal. In detail, I shall reveal that his framework suffers collapse of the kotis. Taking Cotnoir’s concerns seriously, I shall suggest a formulation of the catuskoti in classical Boolean Algebra, extended by the notion of an external negation as an illocutionary act. I will focus on purely formal considerations, leaving doctrinal matters to the scholarly discourse – as far as this is possible. (shrink)

The creation of paraconsistent logics have expanded the boundaries of formal logic by introducing coherent systems that tolerate contradictions without triviality. Thanks to their novel approach and rigorous formalization they have already found many applications in computer science, linguistics and mathematics. As a natural next step, some philosophers have also tried to answer the question if human everyday reasoning could be accurately modelled with paraconsistent logics. The purpose of this article is to argue against the notion that human reasoning is (...) paraconsistent. Numerous findings in the area of cognitive psychology and cognitive neuroscience go against the hypothesis that humans tolerate contradictions in their inferences. Humans experience severe stress and confusion when confronted with contradictions. Experiments on the ways in which humans process contradictions point out that the first thing humans do is remove or modify one of the contradictory statements. From an evolutionary perspective, contradiction is useless and even more dangerous than lack of information because it takes up resources to process. Furthermore, it appears that when logicians, anthropologists or psychologists provide examples of contradictions in human culture and behaviour, their examples very rarely take the form of: \\). Instead, they are often conditional statements, probabilistic judgments, metaphors or seemingly incompatible beliefs. At different points in time humans are definitely able to hold contradictory beliefs, but within one reasoning leading to a particular behaviour, contradiction is never tolerated. (shrink)

According to dialetheism, some contradictions are true. In a recent paper, Aaron Cotnoir has suggested that theists who are also dialetheists can resolve the paradox of the stone by accepting a contradiction, and arguing that God both can and can't make the stone. However, Zach Weber has replied that dialetheism is no help for avoiding one of the most serious problems for theism, namely the problem of evil. In this paper, I argue the situation is even worse than this for (...) dialetheist theists, since one motivation for dialetheism closes off what otherwise might be a loophole in a classic version of the problem of evil. (shrink)

The Chinese Mādhyamikas Seng Zhao 374–414, Jizang 549–623, and Zhiyi 538–597 try to demonstrate that the linguistic strategies in the textual transmission of the Buddha’s teaching give us access to a sense of “liberation” which reaches beyond language. For them, this ineffable sense is what constitutes the dharma in the shape of sūtra and śāstra. Liberation is considered the constitutive but hidden “root” of all the teachings transmitted via the canonical word, those again account for the Buddha’s “traces” guiding back (...) to the meaning of that root. The practitioner’s comprehension of the dharma must embrace this circular mutuality of “root and traces,” reconciling speech with silence. The article discusses two philosophical aspects: Chinese Madhyamaka semantics and pragmatics exploring the various signifying functions in our use of language, and the hermeneutical approach that the Chinese masters develop in their exegetical works to unravel such a sense of “liberation” in reference to the textual transmission of Indian Mahāyāna thought. (shrink)

Substructural pluralism about the meaning of logical connectives is best understood as the view that natural language connectives have all (and only) the properties conferred by classical logic, but that particular occurrences of these connectives cannot simultaneously exhibit all these properties. This is just a more sophisticated way of saying that while natural language connectives are ambiguous, they are not so in the way classical logic intends them to be. Since this view is usually framed as a means to resolve (...) paradoxes, little attention is paid to the logical properties of the ambiguous connectives themselves. The present paper sets out to fill this gap. First, I argue that substructural logicians should care about these connectives; next, I describe a consequence relation between a set of ambiguous premises and an ambiguous conclusion, and review the logical properties of ambiguous connectives; and finally, I highlight how ambiguous connectives can explain our intuitions about logical rivalry. (shrink)

This thesis is the attempt to find a logical model for, and trace the history of, the catuṣkoṭi as it developed in the Indo-Tibetan milieu and spread, via China, to Japan. After an introduction to the history and key-concepts of Buddhist philosophy, I will finish the first chapter with some methodological considerations about the general viability of comparative philosophy. Chapter §2 is devoted to a logical analysis of the catuṣkoṭi. Several attempts to model this fascinating piece of Buddhist philosophy with (...) the tools of classical logic shall be debunked. A paraconsistent alternative will be discussed but eventually dismissed. As a rejoinder, I shall propose a model for the catuṣkoṭi with the help of speech-acts. The remainder of this chapter will look at Chinese and Japanese forms of the catuṣkoṭi which I shall model in a quasi-recursive system. The third and final chapter will look at the Kyoto School's soku-hi dialectics which ties together the different threads of this essay. I will criticise an established, classical model of the soku-hi dialectics and offer an alternative with a second-order paraconsistent semantics. (shrink)

What might early Buddhist teachings offer neuroscience and how might neuroscience inform contemporary Buddhism? Both early Buddhist teachings and cognitive neuroscience suggest that the conditioning of our cognitive apparatus and brain plays a role in agency that may be either efficacious or non-efficacious. Both consider internal time to play a central role in the efficacy of agency. Buddhism offers an approach that promises to increase the efficacy of agency. This approach is found in five early Buddhist teachings that are re-interpreted (...) here with a view to explaining how they might be understood as a dynamic basis for ‘participatory will’ in the context of existing free will debates and the neuroscientific work of Patrick Haggard (et al.). These perspectives offer Buddhism and neuroscience a basis for informing each other as the shared themes of: (1) cognition is dynamic and complex/aggregate based, (2) being dynamic, cognition lacks a fixed basis of efficacy, and (3) efficacy of cognition may be achieved by an understanding of the concept of dynamic: as harmony and efficiency and by means of Buddha-warranted processes that involve internal time. (shrink)

Graham Priest in collaboration with J. Garfield and Y. Deguchi (henceforth: DGP) wrote several articles and responses arguing that the Buddhist philosopher Nagarjuna was a dialetheist thinker, i.e. that he not just identified and exposed certain contradictions but that he embraced it. These contradictions, according to DGP, always occur ``at the limits of thought'' i.e. when a certain view at the same time transcends the limit (``transcendence'') and is within that limit (``closure''). In Nagarjuna's case, these limital contradictions arise at (...) the boundary between ``conventional reality/truth'' (samvrti-sat/satya) and ``ultimate reality/truth'' (paramartha-sat/satya). Ultimate truth is that things lack intrinsic nature (svabhava}), i.e. that they are empty (sunya) of intrinsic nature. This emptiness is universal and it includes emptiness itself (emptiness of emptiness). But that means that being empty is intrinsic property of all things so it comes out that things both have (conventional truth) and lack (ultimate truth) intrinsic nature. This is ontological paradox. DGP identify also semantic and expressibility paradoxes in Nagarjuna. Although logically coherent and philosophically intriguing, I think that DGP's interpretation nevertheless overlooks a special kind of semantics that, presumably, works behind Nagarjuna's reasoning and that would be best described in terms of difference between first and second order statements, i.e. between terms referring to the world (primary system) and terms referring to the primary system (meta-system, comprising the ``meta'' concept of emptiness). Working in these two semantic levels N\={a}g\={a}rjuna, I believe, escapes contradictions --- and Nagarjuna is aware of them --- that arise ``at the limits of thought''. (shrink)

What might early Buddhist teachings offer neuroscience and how might neuroscience inform contemporary Buddhism? Both early Buddhist teachings and cognitive neuroscience suggest that the conditioning of our cognitive apparatus and brain plays a role in agency that may be either efficacious or non-efficacious. Both consider internal time to play a central role in the efficacy of agency. Buddhism offers an approach that promises to increase the efficacy of agency. This approach is found in five early Buddhist teachings that are re-interpreted (...) here with a view to explaining how they might be understood as a dynamic basis for ‘participatory will’ in the context of existing free will debates and the neuroscientific work of Patrick Haggard. These perspectives offer Buddhism and neuroscience a basis for informing each other as the shared themes of: cognition is dynamic and complex/aggregate based, being dynamic, cognition lacks a fixed basis of efficacy, and efficacy of cognition may be achieved by an understanding of the concept of dynamic: as harmony and efficiency and by means of Buddha-warranted processes that involve internal time. (shrink)