This introduction explains the logic of bringing together the perspectives of Hyman Minsky and Karl Polanyi to analyze processes of financialization. Although Minsky and Polanyi have very different intellectual trajectories, there are important complementarities in their approaches. The introduction also explains the focus of the three papers in this special section written by Kurtuluş Gemici, Lucas Kirkpatrick, and David Woodruff.

It is widely accepted that there is a clear sense in which the first-order paraconsistent constructive logic with strong negation of Almukdad and Nelson, QN4, is more constructive than intuitionistic first-order logic, QInt. While QInt and QN4 both possess the disjunction property and the existence property as characteristics of constructiveness (or constructivity), QInt lacks certain features of constructiveness enjoyed by QN4, namely the constructible falsity property and the dual of the existence property. This paper deals with the constructiveness of the (...) contra-classical, connexive, paraconsistent, and contradictory non-trivial first-order logic QC, which is a connexive variant of QN4. It is shown that there is a sense in which QC is even more constructive than QN4. The argument focuses on a problem that is mirror-inverted to Raymond Smullyan’s drinker paradox, namely the invalidity of what will be called the drinker truism and its dual in QN4 (and QInt), and on a version of the Brouwer-Heyting-Kolmogorov interpretation of the logical operations that treats proofs and disproofs on a par. The validity of the drinker truism and its dual together with the greater constructiveness of QC in comparison to QN4 may serve as further motivation for the study of connexive logics and suggests that constructive logic is connexive and contradictory (the latter understood as being negation inconsistent). (shrink)

It is usually accepted in the literature that negation is a contradictory-forming operator and that two statements are contradictories if and only if it is logically impossible for both to be true and logically impossible for both to be false. These two premises have been used by Hartley Slater [Slater, 1995] to argue that paraconsistent negation is not a “real” negation because a sentence and its paraconsistent negation can be true together. In this paper we claim that a counterpart (...) of Slater´s argument can be directed against the negation operator of classical logic. Carnap’s discovery that there are models of classical propositional logic with non-standard or non-normal interpretations of the connectives will be used to build such an argument. One such non-normal valuation which can be added to the set of classically admissible valuations without altering the set of theorems or the set of valid consequences assigns true to every well-formed formula and, therefore, assigns a designated value to every formula and its negation. We ponder the consequences of these arguments for the claims that paraconsistent negations are not genuine negations and that the negation of classical logic is a contradictory-forming operator. (shrink)

The application of paraconsistent logics to theological contradictions is a fascinating move. Jc Beall’s (J Anal Theol, 7(1): 400–439, 2019) paper entitled ‘Christ—A Contradiction: A Defense of ‘Contradictory Christology’ is a notable example. Beall proposes a solution to the fundamental problem of Christology. His solution aims at making the case, and defending the viability of, what he has termed, ‘Contradictory Christology’. There are at least two essential components of Beall’s ‘Contradictory Christology’. These include the dogmatic statements (...) of Chalcedon and FDE logic. The first is the theological contradiction in question. The second is the type of paraconsistent logic. Both components are integral to a contradictory theology in general. I argue that there can be no such thing as an Islamic contradictory theology. I make the case by establishing two points. These points correspond to both integral components of a contradictory theology in general. The first is that an Islamic theological contradiction does not entail an actual (logical) contradiction. The second is that FDE logic, including alternative sub-classical systems of logic, are not adequate in tolerating an Islamic theological contradiction. (shrink)

Beall has given more or less convincing arguments to the effect that neither classical logic, nor K3, nor LP, nor S3 can play the role he expects from logic: to be the basement theory for all true theories, including true theology. However, he has not considered all the pertinent competitors, and he has not given any reassurance that he has not gone too low in the hierarchy of logics to find his desired “universal closure of all true theories”. In (...) this paper, I put forward those additional arguments to show the superiority of FDE with respect to logics that include a detachable conditional but that are very much like FDE otherwise. I also discuss the problem that theological consequence might not contrapose even if theological consequence is supposed to extend FDE consequence and the latter does contrapose. (shrink)

When faced with the charge that a given concept of God is contradictory, the standard move among philosophers and theologians has been to try to explain away the contradiction and show that the concept of God in question is consistent. This has to do, of course, with the Law of Non-Contradiction (LNC). Another option, which has recently generated interest among logicians and analytic philosophers of religion, is to reject such a move as unnecessary and defend what might be called (...) the contradictory God thesis. To be sure, something close to that can be found in philosophers such as Pseudo-Dionysius, Thomas Aquinas and Nicolaus de Cusa. However, it is only recently that this approach has gained momentum, certainly driven by the contemporary advance of dialetheism and glut theoretic approaches in general, and paraconsistent logic. Needless to say, a standard move among defenders of the contradictory God thesis is to challenge the LNC. The argumentation, however, is seldomly framed in conceptual terms. Instead, it is mostly framed in ontological terms, as God being a contradictory entity. From this perspective, the contradictory God thesis is the thesis that God is a contradictory object. My goal in this paper is to provide a conceptual assessment of the discussion surrounding the contradictory God thesis. To achieve this, I make use of a general and hopefully non-controversial meta-theory of concepts and adopt a semantic approach rather than a metaphysical one. Within this framework, I address the following questions: What are the different ways we can understand the contradictory God thesis? What grounds are there for rejecting a contradictory concept of God? What standard moves are available to defend oneself from such criticisms and how do they relate to the LNC? What challenges do they present? As a secondary goal, the paper introduces a novel defense of the contradictory God thesis, drawing on a specific interpretation of the theology of the 16th-century Indian thinker Jīva Gosvāmī, which avoids the need to challenge the LNC. I term this approach ‘non-dialetheic mystical contradictory theism.’. (shrink)

There are two overarching aims of the five collated papers that make up my thesis. The first is to demonstrate that making sense of an ineffable Islamic God in virtue of classical logic and various truth theories (under the purview of analytic philosophy) motivates a theological contradiction. The second is to offer a solution to this problem. I spend a substantial part of my thesis establishing the first of these aims. The reason for this is twofold. Firstly, it is to (...) illustrate the incompatibility between an ineffable God of Islam and various modes of logical and metaphysical inquiry that fall under the purview of analytic philosophy. Although, it becomes increasingly evident that we cannot philosophically make sense of an absolute ineffable God, my inquiry still bears relevance. It offers a comprehensive insight into the logical and metaphysical perspectives that are responsible for motivating the theological contradiction in question. Secondly, fleshing out the various logical and metaphysical perspectives helps lay the theoretical groundwork for the solution. It is not my aim to establish the ineffability of God within the Islamic tradition. That is, I do not engage with Islamic theology beyond referring to, and teasing out, an ineffable view of God from selected Islamic theological sources. The primary focus of my work is to establish that theological contradictions are motivated when assessing them against certain (analytic) philosophical modes. This brings me to the second aim of my thesis, namely, the solution. After having established how an Islamic theological contradiction is motivated, it begs a solution. Prior to offering my solution, I evaluate the recent work on Christian contradictory theology by Jc Beall (2019, 2021). Beall’s proposed solution to the fundamental problem of Christology is what he calls ‘Contradictory Christology’. Although this may seem like a plausible solution for an Islamic theological contradiction, I argue to the contrary. Finally, I propose my own solution to the problem. I call this ‘Islamic Mystical Dialetheism’. (shrink)

The application of paraconsistent logics to theological contradictions is a fascinating move. Jc Beall’s (J Anal Theol, 7(1): 400–439, 2019) paper entitled ‘Christ—A Contradiction: A Defense of ‘Contradictory Christology’ is a notable example. Beall proposes a solution to the fundamental problem of Christology. His solution aims at making the case, and defending the viability of, what he has termed, ‘Contradictory Christology’. There are at least two essential components of Beall’s ‘Contradictory Christology’. These include the dogmatic statements (...) of Chalcedon and FDE logic. The first is the theological contradiction in question. The second is the type of paraconsistent logic. Both components are integral to a contradictory theology in general. I argue that there can be no such thing as an Islamic contradictory theology. I make the case by establishing two points. These points correspond to both integral components of a contradictory theology in general. The first is that an Islamic theological contradiction does not entail an actual (logical) contradiction. The second is that FDE logic, including alternative sub-classical systems of logic, are not adequate in tolerating an Islamic theological contradiction. (shrink)

Recent works in theology—including Beall (The Contradictory Christ and Divine Contradiction) and essays in Rutledge (Paradox and Contradiction in Theology)—have articulated or defended contradictory or contradictory-like theologies. Given the use of contemporary formal logic in setting out these theologies, a natural question concerning the existence of historical precedents to contradictory or contradictory-like theologies has arisen. In this article we present a short, accessible overview of contradictory and contradictory-like theologies alongside an attempt to address (...) the question of historical precedents for such views. (shrink)

This chapter contains sections titled: Introduction Relevant Logic Paraconsistent Logic Dialetheism Boolean Negation The Logical Choice Conclusion.

This book is the first in the field of paraconsistency to offer a comprehensive overview of the subject, including connections to other logics and applications in information processing, linguistics, reasoning and argumentation, and philosophy of science. It is recommended reading for anyone interested in the question of reasoning and argumentation in the presence of contradictions, in semantics, in the paradoxes of set theory and in the puzzling properties of negation in logic programming. Paraconsistent logic comprises a major logical theory (...) and offers the broadest possible perspective on the debate of negation in logic and philosophy. It is a powerful tool for reasoning under contradictoriness as it investigates logic systems in which contradictory information does not lead to arbitrary conclusions. Reasoning under contradictions constitutes one of most important and creative achievements in contemporary logic, with deep roots in philosophical questions involving negation and consistency This book offers an invaluable introduction to a topic of central importance in logic and philosophy. It discusses the history of paraconsistent logic; language, negation, contradiction, consistency and inconsistency; logics of formal inconsistency and the main paraconsistent propositional systems; many-valued companions, possible-translations semantics and non-deterministic semantics; paraconsistent modal logics; first-order paraconsistent logics; applications to information processing, databases and quantum computation; and applications to deontic paradoxes, connections to Eastern thought and to dialogical reasoning. (shrink)

Prominent philosophers have argued that contradictions contain either too much or too little information to be useful. We dispute this with what we call the “Paradox of the Two Firefighters.” Suppose you are awakened in your hotel room by a fire alarm. You open the door. You see three possible ways out: left, right, straight ahead. You see two firefighters. One says there is exactly one safe route and it is to your left. The other says there is exactly one (...) safe route and it is to your right. While the two firemen are giving you contradictory information, they are also both giving you the perhaps useful information that there is a safe way out and it is not straight ahead. We give two analyses. The first uses the “Opinion Tetrahedron,” introduced by Dunn as a generalization of Audun Jøsang’s “Opinion Triangle.” The Opinion Tetrahedron in effect embeds the values of the “Belnap-Dunn 4-valued Logic” into a context of subjective probability generalized to allow for degrees of belief, disbelief, and two kinds of uncertainty—that in which the reasoner has too little information and that in which the reasoner has too much information. Jøsang had only a single value for uncertainty. We also present an alternative solution, again based on subjective probability but of a more standard type. This solution builds upon “linear opinion pooling.” Kiefer had already developed apparatus for assessing risk using expert opinion, and this influences the second solution. Finally, we discuss how these solutions might apply to “Big Data” and the World Wide Web. (shrink)

This article explores the contradictory nature of the ghost in Hamlet and shows how Shakespeare seeks to manipulate the reader’s response in Hamlet by using contradictions and ambiguities. The article also explores the ways in which the reader responds to these contradictions and reconstructs a palpable world in the impalpable world of the text. These contradictions compel the reader to participate in the composition of the text and make him keep changing his own approach to the work with the (...) result that the more he reads the play, the deeper he finds himself entrenched in contradictions. As he fails to grasp the logic of events, the reader relates his own world to the text instead of relating the events of the text to his world and recreates his own world. Therefore, he can easily detach himself from the text and let his imagination run loose as the play proves too vague for him to comprehend. In reading Hamlet, the imagination runs wild and travels far beyond the text to an extent where the reader perceives things, which stand not within but utterly outside the text. Eventually, the reality achieved by the reader in the course of reading the play is only the reality which dwells in the innermost recesses of his mind. (shrink)

I argue that Giorgio Agamben employs two, contradictory theories of the state in his works. Earlier works, such as "The Coming Community" and "Means without End", suggest that the state today functions as an aspect of the society of the spectacle where spectacle is the logical extension of the commodity form under late capitalism. This part of Agamben's work attributes a determined character to the state and a determining power to the economic forces of capitalism that conditions particular forms (...) of the state. Later work, such as "Homo Sacer: Sovereign Power and Bare Life" and " State of Exception", are preoccupied with the logic of juridical sovereignty and the increased frequency of states of emergency. This part of Agamben's work attributes a determining strength to the state under current conditions. Although his earlier work provides a more coherent narrative of how it is possible to move from contemporary society to ideal community, it does not provide the theory of political action necessary to overcome the power of the state he describes when he theorizes the state in "Homo Sacer" and " State of Exception". None of the three possibilities of political action present in his later works provides passage beyond state sovereignty without violating his philosophical commitments. (shrink)

In this chapter, I grant Jc Beall’s assertion that the best understanding of the doctrine of the incarnation posits that Christ is a contradictory being, in the sense that it has him satisfying complementary pairs of predicates. I also argue, however, that by attending to a distinction between predicate negation and sentence negation, this view can be upheld without positing any classical logical contradictions. I argue that the resulting Christological view has several advantages over Beall’s: It is more conservative (...) about avoiding contradiction, exhibits greater logical neutrality, and comports better with Christian tradition. (shrink)

Both I and Belnap, motivated the "Belnap-Dunn 4-valued Logic" by talk of the reasoner being simply "told true" (T) and simply "told false" (F), which leaves the options of being neither "told true" nor "told false" (N), and being both "told true" and "told false" (B). Belnap motivated these notions by consideration of unstructured databases that allow for negative information as well as positive information (even when they conflict). We now experience this on a daily basis with the Web. But (...) the 4-valued logic is deductive in nature, and its matrix is discrete: there are just four values. In this paper I investigate embedding the 4-valued logic into a context of probability. Jøsang's Subjective Logic introduced uncertainty to allow for degrees of belief, disbelief, and uncertainty. We extend this so as to allow for two kinds of uncertainty— that in which the reasoner has too little information (ignorance) and that in which the reasoner has too much information (conflicted). Jøsang's "Opinion Triangle" becomes an "Opinion Tetrahedron" and the 4-values can be seen as its vertices. I make/prove various observations concerning the relation of non-classical "probability" to non-classical logic. (shrink)

This book argues that the standard (orthodox) doctrine of incarnation (of "God enfleshed") is best understood along glut-theoretic lines: the incarnate God is a contradictory being. Example: because God, the Christ figure is all-knowing; but because human, ignorant. And so on. Standard theological theory in the tradition recognizes the apparent contradiction in its core doctrines; Beall argues that the appearance should be accepted as veridical.

This volume offers a wide range of both reconstructions of Nikolai Vasiliev’s original logical ideas and their implementations in the modern logic and philosophy. A collection of works put together through the international workshop "Nikolai Vasiliev’s Logical Legacy and the Modern Logic," this book also covers foundations of logic in the light of Vasiliev’s contradictory ontology. Chapters range from a look at the Heuristic and Conceptual Background of Vasiliev's Imaginary Logic to Generalized Vasiliev-style Propositions. It includes works which cover (...) Imaginary and Non-Aristotelian Logics, Inconsistent Set Theory and the Expansion of Mathematical Thinking, Plurivalent Logic, and the Impact of Vasiliev's Imaginary Logic on Epistemic Logic. The Russian logician, Vasiliev, was widely recognized as one of the forerunners of modern non-classical logic. His "imaginary logic" developed in some of his work at the beginning of 20th century is often considered to be one of the first systems of paraconsistent and multi-valued logic. The novelty of his logical project has opened up prospects for modern logic as well as for non-classical science in general. This volume contains a selection of papers written by modern specialists in the field and deals with various aspects of Vasiliev's logical ideas. The logical legacy of Nikolai Vasiliev can serve as a promising source for developing an impressive range of philosophical interpretations, as it marries promising technical innovations with challenging philosophical insights. (shrink)

Logical pluralism is the view that there is more than one correct logic. This is not necessarily a controversial claim but in its most exciting formulations, pluralism extends to logics that have typically been considered rival accounts of logical consequence – to logics, that is, which adopt seemingly contradictory views about basic logical laws or argument forms. The logical pluralist challenges the philosophical orthodoxy that an argument is either deductively valid or invalid by claiming that there is (...) more than one way for an argument to be valid. In this book, Erik Stei defends logical monism, the view that there is exactly one correct logic. He provides a detailed analysis of different possible formulations of logical pluralism, and offers an original account of the plurality of correct logics that incorporates the benefits of both pluralist and monist approaches to logical consequence. His book will be valuable for a range of readers in the philosophy of logic. (shrink)

Belnap–Dunn’s relevance logic, \(\textsf{BD}\), was designed seeking a suitable logical device for dealing with multiple information sources which sometimes may provide inconsistent and/or incomplete pieces of information. \(\textsf{BD}\) is a four-valued logic which is both paraconsistent and paracomplete. On the other hand, De and Omori, while investigating what classical negation amounts to in a paracomplete and paraconsistent four-valued setting, proposed the expansion \(\textsf{BD2}\) of the four valued Belnap–Dunn logic by a classical negation. In this paper, we introduce a four-valued expansion (...) of BD called \({\textsf{BD}^\copyright }\), obtained by adding an unary connective \({\copyright }\,\ \) which is a consistency operator (in the sense of the Logics of Formal Inconsistency, _LFI_s). In addition, this operator is the unique one with the following features: it extends to \(\textsf{BD}\) the consistency operator of LFI1, a well-known three-valued _LFI_, still satisfying axiom _ciw_ (which states that any sentence is either consistent or contradictory), and allowing to define an undeterminedness operator (in the sense of Logic of Formal Undeterminedness, _LFU_s). Moreover, \({\textsf{BD}^\copyright }\) is maximal w.r.t. LFI1, and it is proved to be equivalent to BD2, up to signature. After presenting a natural Hilbert-style characterization of \({\textsf{BD}^\copyright }\) obtained by means of twist-structures semantics, we propose a first-order version of \({\textsf{BD}^\copyright }\) called \({\textsf{QBD}^\copyright }\), with semantics based on an appropriate notion of four-valued Tarskian-like structures called \(\textbf{4}\) -structures. We show that in \({\textsf{QBD}^\copyright }\), the existential and universal quantifiers are interdefinable in terms of the paracomplete and paraconsistent negation, and not by means of the classical negation. Finally, a Hilbert-style calculus for \({\textsf{QBD}^\copyright }\) is presented, proving the corresponding soundness and completeness theorems. (shrink)

The dissertation has two parts, each dealing with a problem, namely: 1) What is the most adequate account of fuzziness -the so-called phenomenon of vagueness?, and 2) what is the most plausible solution to the sorites, or heap paradox? I will try to show that fuzzy properties are those which are gradual, amenable to be possessed in a greater or smaller extent. Acknowledgement of degrees in the instantiation of a property allows for a gradual transition from one opposite to the (...) other, each intermediate stage constituting an overlap in certain proportion of both contraries. Hence, degrees in the possession of a property give rise to simple contradictions. The reason why I have chosen those two questions is that they provide the main philosophical motivation for a particular brand of an infinite valued and paraconsistent logic. I will claim that Classical logic (CL) is not adequate to handle fuzzy situations, and, being deficient, is in need of being expanded to make room for degrees of truth and weak contradictions. One can hardly deny the importance of the debate, since what is ultimately at stake is what the limits of truth, rationality, intelligibility and possibility are. The main disciplines within which the research moves are the philosophy of language, philosophy of logic, and ontology. (shrink)

In this contribution, we argue that a philosophical clarification of the discussion of adult cognitive development in psychology is needed in order to get a clearer view of what is at stake in this debated phenomenon. On the one hand, we contend that rather than epistemological relativism, mature adult cognition should be described in terms of integration. Integration means understanding that people have different views with each other and from us, but we still need to respect them as people and (...) take their emotions into account. This does not mean simple acceptance of their views as true, as the descriptions of epistemological relativism would suggest. On the other hand, we argue that rather than calling for many-valued logic as some Piagetian views maintain, an adult way of thinking recognises that many conflicts are not logically exhaustive, i.e., that there are more than two solutions to a problem at hand. This often means confusion between two kinds of opposition: contrary and contradictory. While the latter is logically exhaustive in two-valued logic (e.g., good and not-good), the former is not (e.g., good and bad). Our suggestion is that youthful absolutist cognition tends to see conflicts in terms of contrary opposites and assuming them to be logically exhaustive (there is no other solution). A mature adult way of thinking, by contrast, sees that such dichotomies are not logically exhaustive and even in situations in which we have not been able to figure out the solution, it is still possible to find it. Finally, we explore the possibility of articulating an adult way of thinking which recognises that one’s own conceptions might not be insufficient or incorrect by a historical comparison to Ancient Skepticism. (shrink)

If we accept the idea that the value of a “logic” depends on its ability to fit the usual meaning of the “logical constants” which it formalises and the inferences we draw on their basis, we may ask if the meaning of “possible” is well captured by normal modal logic. We are faced with the following puzzle: if is meaningful and non contradictory is it not a logical truth that ? First, it is shown why there is no logical (...) law of the form , being a neutral wff, and then some philosophical conclusions are drawn: “possible” relative to truth must be distinguished from possible “relative” to meaning, and the former only is formalised in modal logic. (shrink)

This chapter focuses on one of the common fallacies in Western philosophy, 'kettle logic' (KL). KL is a fallacy in which multiple contradictory premises are presented to support a point. As such, it is logically impossible for all of the premises to be true. Formally, KL arguments are valid arguments, for it is impossible for contradictory premises to be true and the conclusion false. Sigmund Freud believes KL is employed in dreams quite often. Contradictory beliefs are commonly (...) offered, which give notice that one is in a dream. Freud's presentation of the tea kettle argument makes it clear that the man presents all of the premises as true. In real life and even in dreams, determining whether the arguer intends for all the considerations to be true, or whether additional information is missing, is much more difficult, but necessary in order to avoid the fallacy. (shrink)

Paradefinite logics are logics that can be used for handling contradictory or partial information. As such, paradefinite logics should be both paraconsistent and paracomplete. In this paper we consider the simplest semantic framework for introducing paradefinite logics. It consists of the four-valued matrices that expand the minimal matrix which is characteristic for first degree entailments: Dunn–Belnap matrix. We survey and study the expressive power and proof theory of the most important logics that can be (...) developed in this framework. (shrink)

We address the impingement of Artificial Intelligence (AI) on logic, by examining the requirements posed on logic by knowledge representation and reasoning issues which A I has addressed. We then outline some of AI's contributions, via Logic Programming, to more dynamic forms of logic, in order to deal with knowledge in flux, namely: incomplete and contradictory information; hypotheses making through abduction; argumentation; diagnosis and debugging; updating; and learning. Along the way we delve into implications for the philosophy of knowledge.

The working assumption of this paper is that noncommuting variables are irreducibly interdependent. The logic of such dependence relations is the author's independence-friendly (IF) logic, extended by adding to it sentence-initial contradictory negation ¬ over and above the dual (strong) negation ∼. Then in a Hilbert space ∼ turns out to express orthocomplementation. This can be extended to any logical space, which makes it possible to define the dimension of a logical space. The received Birkhoff and von Neumann "quantum (...) logic" can be interpreted by taking their "disjunction" to be ¬(∼A & ∼ B). Their logic can thus be mapped into a Boolean structure to which an additional operator ∼ has been added. (shrink)

Two of the most important outcomes of The Contradictory Christ include: identifying Christ as an unproblematically contradictory being as well as laying the foundations of an investigation of the logical consequences of the existence of Christ, qua contradictory, within a particular 'theory'. In light of the enormous reluevance of Beall’s The contradictory Christ for the study of inconsistency, my main concern here is to explore the effect of some methodological choices behind Beall’s proposal -this in order (...) to recognize in more detail the scope of Beall’s contribution. To do so, I will focus on three main questions: 1. What is required for the identification of a contradiction? 2. How can we recognize a true contradiction from either an apparent or a temporal contradiction? 3. If we identify a true contradiction within a theory, where can we actually go from there? (shrink)

The central thesis of JC Beall’s paraconsistent Christology is that Christ, being human and divine, is a contradictory being, and a rational Christology can accept it, since logic nowadays does not exclude the possibility of true contradictions. In this paper, I move from Beall’s theory and I present an alternative view. I quote seven statements of the so-called ‘Athanasian Creed’ which synthesizes the results of conciliar Christology. The aim of the Creed is to combat monophysitism by stressing the duplicity (...) and unity of Christ: two (incompatible) natures inseparably joined in only one person. I note that the two-in-one principle, so intended, may be seen as an ancestor of what has been called ‘conjunctive paraconsistency’, whereby there could be true contradictions but contradictories cannot be separately true. I specifically oppose this view to Beall’s idea of Christ’s human divinity (or divine humanity) as a glut, showing that in the conjunctive account, true contradictions do not require any overlapping or joint ascription of truth and falsity. (shrink)

We present epistemic multilateral logic, a general logical framework for reasoning involving epistemic modality. Standard bilateral systems use propositional formulae marked with signs for assertion and rejection. Epistemic multilateral logic extends standard bilateral systems with a sign for the speech act of weak assertion (Incurvati and Schlöder 2019) and an operator for epistemic modality. We prove that epistemic multilateral logic is sound and complete with respect to the modal logic S5 modulo an appropriate translation. The logical framework developed provides the (...) basis for a novel, proof-theoretic approach to the study of epistemic modality. To demonstrate the fruitfulness of the approach, we show how the framework allows us to reconcile classical logic with the contradictoriness of so-called Yalcin sentences and to distinguish between various inference patterns on the basis of the epistemic properties they preserve. (shrink)

It is an unusual property for a logic to prove a formula and its negation without ending up in triviality. Some systems have nonetheless been observed to satisfy this property: one group of such non-trivial negation inconsistent logics has its archetype in H. Wansing’s constructive connexive logic, whose negation-implication fragment already proves contradictions. N. Francez and Y. Weiss subsequently investigated relevant subsystems of this fragment, and Weiss in particular showed that they remain negation inconsistent. In this note, we take (...) a closer look at this phenomenon in the systems of Francez and Weiss, and point out two types of necessary conditions, one proof-theoretic and one relevant, which any contradictory formula must satisfy. As a consequence, we propose a nine-fold classification of provable contradictions for the logics. (shrink)

Concerned with the issue of the I-thou encounter and the question of how to overcome the problem of the confrontation that occurs in the worldly existence among individuals, the Japanese philosopher Ueda Shizuteru (1926-), a leading member of the Kyoto School, addressed this issue in his phenomenology of the self. Ueda develops his ideas as a hermeneutical practice in the reading of the well-known Zen classic parable Ten Ox-Herding pictures, given that Zen Buddhism is the main tradition upon which he (...) critically and creatively draws. In accordance with Zen, Ueda argues that the emptied self is a non-self self boundlessly open and compassionate. As a result, the selfless self is open to the other in such a way that a genuine encounter with the other can take place. In the encounter, the closed ego realizes that its self-identity includes also its self-negation, such that, for the selfless self the other’s concerns are one’s concerns. The main aim of this contribution is to look at Ueda’s understanding of the interpersonal relations in what he conceives as a twofold world, that is, the worldly existence and the infinite openness that imbues this existence. In the infinite openness, the fundamental difference between self and other disappears into the state where the individual becomes emptiness while maintaining their “otherness” as separate and distinct. It means that the nullified self still preserves its identity, the other is accepted as different, and at the same time both share this common selfless ground. Thus, a particular conception of identity arises in which we can trace its roots back to Nishida Kitarō’s notion of “self-contradictory identity” (矛盾的自己同一 mujunteki jikodōitsu) and Nishitani Keiji’s “non-duality of self and other” (自他不二 jitafuni). According to the first, each of us, as individual bodies, also contains absolute negation within. The latter refers to a mode of being where self and other are one and simultaneously each one remains what it is in itself, probably inspired in Mahāyāna Buddhist logic whereby two different things are neither one nor two and yet both one and two. Given that the identification of the subject with its context and the emphasis on the sameness can entail the risk of suppressing other’s alterity and subordinating the individual to the whole, these views have been regarded as ambivalent. Therefore, the criticism and implications of this standpoint of “contradictory identity” are also examined. However, the assumption here is that due to the ethical import of this standpoint, it could certainly be of interest if applied to social and intercultural relations. (shrink)

Although some scholars propagate cross-sector partnerships (CSPs) as a panacea for addressing the grand challenges of the 21st century, scholars also acknowledge that this type of collaboration faces significant barriers since the institutional logics of partners such as business, civil society, and government potentially have contradicting interests and future visions. This inductive longitudinal case study on integrating skilled migrants into the German labor market examines the institutional work by which CSP members, particularly government actors, deliberately rein in contradictory (...) logics to reproduce the dominant government logic, and in doing so, establish a high level of institutional coherence. In disclosing the processes by which the dominant logic maintains its hegemony, I contribute new insight into the institutional dynamics within government-sponsored CSPs. I identify various actions and I illuminate CSP tensions and dark sides. (shrink)

In this paper I argue that Kierkegaard endorses Hegel's theory of mediation, the view that relative opposites are mediated. However, I show that Kierkegaard denies Hegel's thesis that there are all and only relative opposites. I develop two of his arguments against this thesis. The first is existential. This argument comes from the dramatic interplay between A, the often disagreeable aesthete of Either/Or I, and Judge William, the dutiful ethicist of Either/Or II. Judge William convincingly argues that the possibility of (...) future aiming human projects is rendered obsolete if aesthetic and ethical forms of existence are mediated opposites. The second is philosophical. This second argument is offered by Johannes Climacus in Concluding Unscientific Postscript. I show that Climacus issues a reductio argument that I call "the argument from insufficient difference.". (shrink)

Claims concerning what is or is not possible abound in contemporary philosophy. The epistemology of such claims, however, remains largely unexplored. Anything imaginable is possible, we are told, with the proviso that imagination be governed by logic. Many who defend this methodology argue that logic frees us from recourse to some mysterious a priori faculty of intuition. Anything is possible so long as it does not contain a contradiction—and we don’t need intuition to tell us what is contradictory, just (...) logic. Logic is thus supposed to provide us with a non-mysterious, intuition-free, modal epistemology. The purpose of this paper is to argue that logic does not deliver on this promise. The argument proceeds along two fronts. First, I argue that logic is not sufficient for the task. Not only are there consistent but impossible propositions, there are even inconsistent propositions that are possible. Second, and more fundamentally, I question the epistemic independence of appeals to logical consistency. I will argue that in judging a proposition to be consistent we rely upon our intuitions concerning what is possible. In other words, our modal beliefs undergird our judgments about consistency and not vice-versa. (shrink)

The fundamental problem of Christology is the apparent contradiction of Christ as recorded at Chalcedon. Christ is human and Christ is divine. Being divine entails being immutable. Being human entails being mutable. Were Christ two different persons there’d be no apparent contradiction. But Chalcedon rules as much out. Were Christ only partly human or only partly divine there’d be no apparent contradiction. But Chalcedon rules as much out. Were the very meaning of ‘mutable’ and/or ‘immutable’ other than what they are, (...) there’d be no apparent contradiction. But the meaning is what it is, and changing the meaning of our terms to avoid the apparent contradiction of Christ is an apparent flight from reality. What, in the end, is the explanation of the apparent contradiction of Christ? Theologians and philosophers have long advanced many consistency-seeking answers, all of which increase the metaphysical or semantical complexity of the otherwise strikingly simple but radical core of Christianity’s GodMan. In this paper, I put the simplest explanation on the theological table: namely, Christ appears to be contradictory because Christ is contradictory. This explanation may sound complicated to the many who are steeped in the mainstream account of logic according to which logic precludes the possibility of true contradictions. But the mainstream account of logic can and should be rejected. Ridding theology of the dogma of mainstream logic illuminates the simple though striking explanation of the apparent contradiction of Christ — namely, that Christ is a contradictory being. Just as the simplest explanation to the apparent roundness of the earth has earned due acceptance, so too should the simplest explanation of the apparent contradiction of Christ. (shrink)

The logic of paradox, LP, is a first-order, three-valued logic that has been advocated by Graham Priest as an appropriate way to represent the possibility of acceptable contradictory statements. Second-order LP is that logic augmented with quantification over predicates. As with classical second-order logic, there are different ways to give the semantic interpretation of sentences of the logic. The different ways give rise to different logical advantages and disadvantages, and we canvass several of these, concluding that it will be (...) extremely difficult to appeal to second-order LP for the purposes that its proponents advocate, until some deep, intricate, and hitherto unarticulated metaphysical advances are made. (shrink)

I argue that the medieval form of dialectical disputation known as obligationes can be viewed as a logical game of consistency maintenance. The game has two participants, Opponent and Respondent. Opponent puts forward a proposition P; Respondent must concede, deny or doubt, on the basis of inferential relations between P and previously accepted or denied propositions, or, in case there is none, on the basis of the common set of beliefs. Respondent loses the game if he concedes a contradictory (...) set of propositions. Opponent loses the game if Respondent is able to maintain consistency during the stipulated period of time. The obligational rules are here formalised by means of familiar notational devices, and the application of some game-theoretical concepts, such as (winning) strategy, moves, motivation, allows for an analysis of some crucial properties of the game. In particular, the primacy of inferential (syntactic) relations over semantic aspects and the dynamic character of obligations are outlined. (shrink)

In this paper, I reflect on the idea, hinted at by Kant in a footnote to §16 of the B- Deduction that is not often discussed (KrV B 134n.), that transcendental logic is the ground of logic as a whole. This has important repercussions for the way we should see the role of transcendental logic with respect to the question of truth as well as the nature and scope of transcendental logic in relation to cognition, and in relation to general (...) or formal logic as such. To illustrate one of the ways in which transcendental logic is fundamental to our way of thinking, I address an issue that is brought up by Kant’’s counterfactual claim at B 132 that if a representation is not accompanied by an ‘‘I’’ thought, it is ‘‘either impossible’’ or at least ‘‘nothing for me’’, suggesting to some recent commentators that by the former Kant means the impossibility of thinking contradictory thoughts. Unlike these commentators, I do not think Kant is saying here that we cannot think contradictory thoughts. To believe he is betrays a misunderstanding of the metaphysical nature of transcendental logic, or so I shall argue. It is because transcen- dental logic is a metaphysical logic that Kant can claim that transcendental logic grounds even the whole of logic, including the possibility of thinking contradictory thoughts. (shrink)

Presumably, the exponent of logical normativity believes it to be the case that rational agents ought to reason logically. If the converse holds, and the exponent of logical normativity believes either that it is false that rational agents ought to reason logically or that the claim that rational agents ought to reason logically is not truth–functional, then any attempt to formulate sound arguments in support of their position will be either question–begging or self–contradictory. To argue in favour of the (...) normative status of logic involves the assumption that the issue is substantive. I draw a distinction between deontic and axiological normativity as applied to logical theory and demonstrate that because deontic sentences are not truth–functional, logic cannot be normative in the deontic sense. I conclude that the only sense in which logical theory may be thought of as normative is axiologically. (shrink)

In a forthcoming paper, Walter Carnielli and Abilio Rodrigues propose a Basic Logic of Evidence whose natural deduction rules are thought of as preserving evidence instead of truth. BLE turns out to be equivalent to Nelson’s paraconsistent logic N4, resulting from adding strong negation to Intuitionistic logic without Intuitionistic negation. The Carnielli/Rodrigues understanding of evidence is informal. Here we provide a formal alternative, using justification logic. First we introduce a modal logic, KX4, in which \ can be read as asserting (...) there is implicit evidence for X, where we understand evidence to permit contradictions. We show BLE embeds into KX4 in the same way that Intuitionistic logic embeds into S4. Then we formulate a new justification logic, JX4, in which the implicit evidence motivating KX4 is made explicit. KX4 embeds into JX4 via a realization theorem. Thus BLE has both implicit and explicit possibly contradictory evidence interpretations in a formal sense. (shrink)