In terms of ontology, the so-called dialectical contradiction is precisely the unity of opposites of the objective world itself, in other words, the objective dialectics. In terms of subjective thinking, dialectical contradiction is the reflection of the objective world, the unity of opposites inherent in thinking and understanding, in other words, the subjective dialectics. Said Engels: "The so-called objective dialectics dominates the entire nature; the so-called subjective dialectics, that is, dialectical thinking, is nothing but the reflection of contradictory (...) motions prevalent everywhere in nature …" . Therefore there is a fundamental difference between dialectical contradiction and logical contradiction. (shrink)

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)

According to Kant’s Universal Law Formula, maxims that cannot be conceived as universal laws denote duties of perfect obligation. In the recent literature, two versions of the Contradiction in Conception test have received the most attention. When acting on a maxim would violate a perfect duty, according to the Logical Contradiction Interpretation (LCI), universalizing the maxim would make it literally impossible to perform the action as described in the original maxim. According to the Practical Contradiction Interpretation (...) (PCI), the locus of the contradiction is as follows: the agent acts on the maxim in order to achieve some purpose, but were the agent’s maxim universalized, the purpose would be unattainable. Having examined the most widely accepted versions of both interpretations, I argue that i) PCI cannot generate contradictions for the maxims of any actions that violate perfect duties beyond those generated by LCI, ii) despite claims of its proponents, PCI cannot solve the vexing problem of relevant descriptions that Kant’s account faces, and iii) other arguments in favor of PCI are at best inconclusive. I therefore conclude that since LCI better coheres with the text, sympathetic interpreters should focus their efforts on LCI, not PCI. (shrink)

The solution of this problem is extremely important to the process of the development of logic. Its solution is of great theoretical significance in upholding the methodology of dialectical materialism; criticizing the sophistry of metaphysics, skepticism, and relativism; developing formal logic and symbolic logic; forming the scientific system of dialectical logic in the strict sense; and conducting research to strengthen all the concrete sciences. It also has enormous significance for the criticism of "leftist" and right-wing opportunism and revisionism, and especially (...) for criticizing the counterrevolutionary revisionist line of the "gang of four.". (shrink)

A common response to those who question the Law of Non-Contradiction is that it is impossible to debate such a fundamental law of logic. The reasons for this response vary, but what seems to underlie them is the thought that there is a minimal set of logical resources without which rational debate is impossible. This chapter argues that this response is misguided. First, it defends non-apriorism in logic: the view that logic is in the same epistemic boat as (...) other scientific theories. It then offers an account of logical theory change in terms of this epistemology. The LNC is discussed in terms of this account of logical theory change, and it is shown that rational debate over this law can, and does, proceed. Finally, arguments for and against the LNC are discussed, and how and where non-a priori considerations arise in these arguments are illustrated. (shrink)

There is a principle in things, about which we cannot be deceived, but must always, on the contrary, recognize the truth – viz. that the same thing cannot at one and the same time be and not be": with these words of the Metaphysics, Aristotle introduced the Law of Non-Contradiction, which was to become the most authoritative principle in the history of Western thought. However, things have recently changed, and nowadays various philosophers, called dialetheists, claim that this Law does (...) not hold unrestrictedly – that in peculiar circumstances the same thing may at the same time be and not be, and contradictions may obtain in the world. This book opens with an examination of the famous logical paradoxes that appear to speak on behalf of contradictions (e.g., the Liar paradox, the set-theoretic paradoxes such as Cantor’s and Russell’s), and of the reasons for the failure of the standard attempts to solve them. It provides, then, an introduction to paraconsistent logics – non-classical logics in which the admission of contradictions does not lead to logical chaos –, and their astonishing applications, going from inconsistent data base management to contradictory arithmetics capable of circumventing Gödel’s celebrated Incompleteness Theorem. The final part of the book discusses the philosophical motivations and difficulties of dialetheism, and shows how to extract from Aristotle’s ancient words a possible reply to the dialetheic challenge. How to Sell a Contradiction will appeal to anyone interested in non-classical logics, analytic metaphysics, and philosophy of mathematics, and especially to those who consider challenging our most entrenched beliefs the main duty of philosophical inquiry. (shrink)

In Philosophical Logic, the Liar Paradox has been used to motivate the introduction of both truth value gaps and truth value gluts. Moreover, in the light of “revenge Liar” arguments, also higher-order combinations of generalized truth values have been suggested to account for so-called hyper-contradictions. In the present paper, Graham Priest's treatment of generalized truth values is scrutinized and compared with another strategy of generalizing the set of classical truth values and defining an entailment relation on the resulting sets of (...) higher-order values. This method is based on the concept of a multilattice. If the method is applied to the set of truth values of Belnap's “useful four-valued logic”, one obtains a trilattice, and, more generally, structures here called Belnap-trilattices. As in Priest's case, it is shown that the generalized truth values motivated by hyper-contradictions have no effect on the logic. Whereas Priest's construction in terms of designated truth values always results in his Logic of Paradox, the present construction in terms of truth and falsity orderings always results in First Degree Entailment. However, it is observed that applying the multilattice-approach to Priest's initial set of truth values leads to an interesting algebraic structure of a “bi-and-a-half” lattice which determines seven-valued logics different from Priest's Logic of Paradox. (shrink)

This paper grapples with two objections against Max Weber’s methodology that arise because Weber borrows some ideas from Heinrich Rickert’s neo-Kantian philosophical system. The first objection (“the contradiction argument”) is raised by Julius J. Schaaf who disagrees with Weber’s claim that historical objects are constituted through retrospectively and hypothetically applied selections of value relations and that we can understand these objects. Weber’s idea that the relating ideal type constructions are also non-arbitrary—i.e., not merely subjective—and can be rectified, Schaaf maintains, (...) contradicts his own characterization of historical objects and reality. The second objection (“the incom¬patibility argument”) was made by Gerhard Wagner and Heinz Zipprian. It focuses on the conception of objective possibility. According to this, counter¬factual chains of historical events are contrived to figure out the relevant set of causes by which a certain event can be said to necessarily occur. The critics claim that it is only based on the ontological assumption of a preexisting real object that this thought-experiment can operate. That is, the pre-emptive assumption of realism is indis¬pensable for explaining the historical individual. To refute these two objections, I clarify some foundational notions and ideas of Weber’s methodology. (shrink)

The apparently paradoxical nature of self-deception has attracted a great deal of controversy in recent years. Focussing on those aspects of the phenomenon which involve the holding of "contradictory" beliefs, it is our intention to argue that this presents no "paradox" if a non-classical, "paraconsistent", doxastic logic is adopted. (On such logics, see, for example, N. C. A. da Costa, 'On the theory of inconsistent formal systems', Notre Dame J Formal Logic 11(1974), 497-510, and A. I. Arruda, 'A survey of (...) paraconsistent logic', in A. I. Arruda, N. C. A da Costa and R Chuaqui, _Mathematical Logic in Latin America, North-Holland, 1984, pp. 1-41.). (shrink)

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

The papers in this volume present some of the most recent results of the work about contradictions in philosophical logic and metaphysics; examine the history of contradiction in crucial phases of philosophical thought; consider the relevance of contradictions for political and philosophical actuality. From this consideration a common question emerges: the question of the irreducibility, reality and productive force of (some) contradictions.

Almost all contemporary philosophers on the subject agree that emotions play an indispensable role in the justification (as opposed to the mere causation) of other mental states and actions. However, how this role is to be understood is still an open question. At the core of the debate is the phenomenon of conflict without contradiction: why is it that an emotion need not be revised in the light of better judgment and knowledge? Conflict without contradiction has been explained (...) either by difference in content between emotion and judgment, or by a difference in the respective attitude towards content. I argue that conflict without contradiction is due to both differences, where difference in content is prior to difference in attitude. (shrink)

A dialectical contradiction can be appropriately described within the framework of classical formal logic. It is in harmony with the law of noncontradiction. According to our definition, two theories make up a dialectical contradiction if each of them is consistent and their union is inconsistent. It can happen that each of these two theories has an intended model. Plenty of examples are to be found in the history of science.

Epistemic contradictions are now a well-known and often discussed phenomenon among those who study epistemic modals. These contradictions are expressed by sentences like ‘It is raining and it might not be raining’ whose oddness to the common ear demands an explanation. However, it has turned out to be a rather controversial enterprise to provide such an explanation in a sufficiently precise and general manner. According to pragmatic explanations, epistemic contradictions are semantically consistent but pragmatically defective. According to semantic explanations, one (...) should regard epistemic contradictions as plain semantic inconsistencies. Endorsing such a semantic solution is, however, tantamount to rejecting classical logic, which predicts the consistency of epistemic contradictions. After arguing that all existing solutions to this problem face as yet unmet challenges, I will present a new solution that adequately overcomes these challenges. I will propose to view epistemic contradictions as semantically consistent but epistemically defective sentences. The main thesis emerging will be that we need neither abandon the inference rules of classical logic nor the classical truth-conditional approach to semantics to deal adequately with epistemic contradictions. (shrink)

The discussion on the principle of non-contradiction (1946–1957) between Marxist and non-Marxist philosophers was one of the major philosophical discussions in Polish philosophy of this period. In my text, I carefully reconstruct this discussion and outline its relation to Soviet debates on the subject. I show that the change in Schaff’s position happened in the early 1950s under the combined influence of the Lvov–Warsaw School and the changes in the official Soviet position regarding formal logic. I discuss the aftermath (...) following Schaff’s change in attitude towards the analytic tradition for the development of Polish philosophy, as well as the critique of this change by Jarosław Ładosz. In my reconstruction of the latter, I focus on the problem of the historical development of science. I refer to Ilyenkov’s critique of Schaff, opposing synchronic (“positivist”) and diachronic (“dialectical”) concepts of knowledge. As I argue, these opposing concepts of science can be seen as a genuine issue at stake in the Polish discussion as well, especially in the polemic between Schaff and Ładosz. (shrink)

We introduce the simpler and shorter proof of Hajek’s theorem that the mathematical induction on ω implies a contradiction in the set theory with the comprehension principle within Łukasiewicz predicate logic Ł ${\forall}$ (Hajek Arch Math Logic 44(6):763–782, 2005) by extending the proof in (Yatabe Arch Math Logic, accepted) so as to be effective in any linearly ordered MV-algebra.

The book Paraconsistent Logic: Consistency, Contradiction and Negation by Walter Carnielli and Marcelo Coniglio is the most thorough study of Logics of Formal Inconsistency...

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)

Chapter VI discusses a few assumptions which underlie the proposed reconstruction of Hegel's procedures. It is shown that certain equivalents of such assumptions are either explicitly accepted by Hegel, or they are consequences of theses he subscribed to. Finally, it is suggested that some of these assumptions envisage a conception of language and philosophy which has an interesting parallel in Wittgenstein's later work. Such a conception sets philosophy sharply apart from the sciences, and deemphasizes the formation of contradictions. The general (...) relationship between vagueness and contradiction is briefly explored to show that some familiar contradiction-generating procedures can be seen to be based on vagueness of the relevant theoretical expressions. ;Chapter V deals with the Aufhebung. After an examination of Hegel's own theory of sublation, and an analysis of some textual examples, it is shown that some Aufhebung-procedures can be modelled by certain patterns of argument concerning simple algebraic structures. A few Hegelian concepts, such as "opposition," "unity," "passing over," "reflection," are partially explicated in the process. ;Chapter IV plays a pivotal role in the dissertation. It explains the generation of dialectical contradictions as stemming from syntactic and intensional indeterminacy of the theoretical terms. It is shown that Hegel's conceptual terms have no well-determined sense, so that their senses can be identified in different and possibly inconsistent ways. It is also shown that they can be made to play different syntactic roles, for no rigid determination of their syntactic function is assumed. This, again, may create contradictions. Both kinds of indeterminacy are related to Hegel's borrowing of his conceptual terms from a "natural" philosophical koine. ;Chapter III is an analysis of Hegel's language in the Logic. It tries to determine the grammatical and semantical status of Hegel's conceptual terms and to explicate the Hegelian use of the copula as occurring in certain typical sentential forms. ;The presence of the so-called "dialectical contradictions" is the most evident logico-linguistic peculiarity of Hegel's text. After a discussion of Hegel's theory of contradiction, and of some of its interpretations, it is argued that any satisfactory account of the dialectical method must explain the formation of contradictions on the basis of other logico-linguistic features of Hegel's text. ;Chapter II deals with a few such explanations which have been offered by modern interpreters of Hegel. Use of contemporary analytic concepts in the logical and linguistic analysis of Hegel's text is justified with reference to the Carnapian categories of "explication" and "rational reconstruction." ;The literature on Hegel's dialectic has made important contributions to an understanding of its nature both as a metaphysical theory of reality and as a philosophical attitude towards the human world. However, many recent interpreters agree that no satisfactory progress has been made in understanding dialectic as a method. This dissertation identifies the dialectical method as the specific form of Hegel's philosophical discourse, and tries to bring out its basic logico-linguistic features. The Science of Logic is chosen as the field of inquiry, because of its crucial position among Hegel's mature works. (shrink)

In connection with the discussion on the problem of logic, both within our country and abroad , it has already turned from the general, comparatively abstract, and difficult-to-resolve problem of the "relationship between formal logic and dialectics" to the more concrete problem of the functional scope of formal logic. Some people have even utilized special articles to inquire about a certain law in formal logic: the functional scope of the law of identity or the law of contradiction. The purpose (...) of these philosophers or logicians is to try to define the functional scope of a certain law in formal logic as a starting point, and then to proceed to study the functional scope of formal logic in the entire thinking process of mankind. If the functional scope of formal logic in mankind's entire thinking process is clearly and precisely defined, the solution of this complicated problem of the "relationship between formal logic and dialectics" would certainly not be very difficult. The basic purpose of Comrade Chu-ko Yin-t'ung's article on the problem of "whether or not formal logic's law of contradiction can be contravened in the dialectical thinking process" of men, which appeared in issue No. 82 of the "Philosophy" supplement of Kuang-ming Daily on April 24, is like this. It should be considered as a rather practical method. But I do not quite agree with his basic points of argument and with his analysis of certain examples of dialectics. I intend to present my superficial views here and to study them with Comrade Chu-ko. (shrink)

This volume examines the entire logical and philosophical production of Nikolai A. Vasil’ev, studying his life and activities as a historian and man of letters. Readers will gain a comprehensive understanding of this influential Russian logician, philosopher, psychologist, and poet. The author frames Vasil’ev’s work within its historical and cultural context. He takes into consideration both the situation of logic in Russia and the state of logic in Western Europe, from the end of the 19th century to the beginning (...) of the 20th. Following this, the book considers the attempts to develop non-Aristotelian logics or ideas that present affinities with imaginary logic. It then looks at the contribution of traditional logic in elaborating non-classical ideas. This logic allows the author to deal with incomplete objects just as imaginary logic does with contradictory ones. Both logics are objects of interesting analysis by modern researchers. (shrink)

This article challenges the prevailing interpretations of Hegel's account of the concept "contradiction" in the Science of Logic by arguing that it is concerned with the principle of Hegel's method rather than with the classical law of non-contradiction. I first consider Hegel's Doctrine of Essence in view of Kant's discussion of the concepts of reflection in the first Critique. On this basis, I examine Hegel's account of the logical principles based on the concepts "identity," "opposition," and " (...) class='Hi'>contradiction." Finally, I point out how the principle Hegel derives from the concept of contradiction actually informs his own method. (shrink)

Western philosophy has mainly developed in accordance with the three laws of identity, noncontradiction and excluded middle, also known as “laws of thought”. Since Zen Buddhism often violates these apparently indisputable logical principles, a superficial reading may induce the idea that Zen Buddhism is a completely irrational, illogical doctrine. In this essay, I argue that Zen Buddhism is not absurd or illogical. Conversely, it relies on a different logic, which is perfectly consonant with the Buddhist view of the world.

Despite the tendency to be otherwise, some non-classical logics are known to validate formulas that are invalid in classical logic. A subclass of such systems even possesses pairs of a formula and its negation as theorems, without becoming trivial. How should these provable contradictions be understood? The present paper aims to shed light on aspects of this phenomenon by taking as samples the constructive connexive logic C, which is obtained by a simple modification of a system of constructible falsity, namely (...) N4, as well as its non-constructive extension C3. For these systems, various observations concerning provable contradictions are made, using mainly a proof-theoretic approach. The topics covered in this paper include: how new contradictions are found from parts of provable contradictions; how to characterise provable contradictions in C3 that are constructive; how contradictions can be seen from the relative viewpoint of strong implication; and as an appendix an attempt at generating provable contradictions in C3. (shrink)

Classical logic, of first or higher order, is extended with sentential operators and quantifiers, interpreted substitutionally over unrestricted substitution class. Operators mark a single layered, consistent metalanguage. Self-reference, arising from substitutional quantification over sentences, allows to express paradoxes which, unlike contradictions, do not lead to explosion. Semantics of the resulting language, using semi-kernels of digraphs, is non-explosive yet two-valued and has classical semantics as a special case for clasically consistent theories. A complete reasoning is obtained by extending LK with two (...) rules for sentential quantifiers. Adding (cut) yields a complete system for the explosive semantics. (shrink)

From the Editors [of Voprosy filosofii]: Recently the prominent Soviet philosopher, specialist in problems of dialectics, theory of knowledge, the history of philosophy, and esthetics Igor Sergeevich Narskii marked his sixtieth birthday. The editorial board and staff of Voprosy filosofii wish to extend their congratulations to him and wish him good health, happiness in his personal life, and continued success in his work.

In this review I briefly analyse the main elements of each chapter of the book centred in the general areas of logic, epistemology, philosophy and history of science. Most of them are developed around a fine-grained investigation on the principle of non-contradiction and the concept of consistency, inquired mainly into the broad area of paraconsistent logics. The book itself is the result of a work that was initiated on the Studia Logica conference "Trends in Logic XVI: Consistency, Contradiction, (...) Paraconsistency and Reasoning - 40 years of CLE", held at the State University of Campinas (Unicamp), Brazil, between September 12-15, 2016. (shrink)