The goals of this paper are two-fold: I wish to clarify the Aristotelian conception of the law of non-contradiction as a metaphysical rather than a semantic or logical principle, and to defend the truth of the principle in this sense. First I will explain what it in fact means that the law of non-contradiction is a metaphysical principle. The core idea is that the law of non-contradiction is a general principle derived from how things are in (...) the world. For example, there are certain constraints as to what kind of properties an object can have, and especially: some of these properties are mutually exclusive. Given this characterisation, I will advance to examine what kind of challenges the law of non-contradiction faces; the main opponent here is Graham Priest. I will consider these challenges and conclude that they do not threaten the truth of the law of non-contradiction understood as a metaphysical principle. (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)

Critical discussion of Alan Code's paper "Aristotle's Investigation of a Basic Logical Principle: Which Science Investigates the Principle of Non-Contradiction?".

The present article deals with the not very common opinion among medieval philosophers according to which the identity principle is the true first principle, undermining the primacy of the principle of non-contradiction. Following a refutation of this position in the logical work of the Franciscan Geraldus Odonis, we intend to investigate its target as well as other cases of the same dispute in 14th century authors: Antoine Andre, John of Buridan, John of Baconthorpe and Nicolas of Autrecourt. (...) We defend that Odonis presents a successful response to this position. (shrink)

In his Metaphysics Γ.4, Aristotle defends the principle of non-contradiction (PNC). The PNC says that all contradictions are false. So if some contradictions are true, then PNC is false. Even if PNC’s contrary is false, PNC’s contradictory might still be true. But it’s been noted in the literature for over a century that Aristotle seems to be exclusively interested in attacking PNC’s contrary (‘All contradictions are true’) rather than PNC’s contradictory (‘Some contradictions are true’). So his defense of PNC (...) seems to fail. This would be a surprising error from the inventor of formal logic. It is especially puzzling because we have plenty of evidence showing that Aristotle is keenly aware of the distinction between contraries and contradictories, and because Aristotle distinguishes between PNC’s contradictory and PNC’s contrary in Γ.4. I defend Aristotle against these charges: (1) I explain that one important reason for Aristotle’s focus on PNC’s contrary is that he took it to be deeply connected to views held by thinkers such as Anaxagoras, Cratylus, Democritus, Empedocles, Heraclitus, Parmenides, Protagoras, and Xenophanes; (2) Aristotle’s defense of PNC must be a particular kind of indirect defense rather than a direct demonstration; (3) I argue that if Aristotle’s defense of PNC is read as centering around his argument for what counts as a thing that is determinate, Aristotle demonstrates the reliance of coherent communication on non-contradiction. Read this way, he gives a fairly compelling case to reject not just PNC’s contrary but also PNC’s contradictory. (shrink)

It has long been recognized that negation in Aristotle’s term logic differs syntactically from negation in classical logic: modern external negation attaches to propositions fully formed, whereas Aristotelian internal negation forms propositions from sentential constituents. Still, modern external negation is used to render Aristotelian internal negation, as may be seen in formalizations of Aristotle’s semantic principles of non-contradiction and of excluded middle. These principles govern the distribution of truth values among pairs of contradictory propositions, and Aristotelian contradictories always (...) consist of an affirmation and a denial. So how should we formalize a false denial? In the literature, we find that a false denial is formalized by means of two negation signs attached to a one-place predicate. However, it can be shown that this rendering leads to an incorrect picture of Aristotle’s principles. In this paper, I propose a solution to this technical problem by devising a formal notation especially for Aristotelian propositions in which internal negation is differentiated from external negation. I will also analyze both principles, each of which has two logically equivalent forms, a positive and a negative one. The fact that Aristotle’s principles are distinct and complementary is reflected in my new formalizations. (shrink)

Papineau has suggested that the Principle of Non-Contradiction is a logical law that ?verificationists? are not entitled to claim as a prioritrue. The Principle, like that of Excluded Middle, is not sufficiently grounded in the ?miserly? epistemology of verificationism to be proven in ?verificationist logic?. We examine who might be challenged by this claim: who are the ?verificationists?? We defend our candidates against Papineau's criticisms and other attacks, but this leaves the verificationist open to a different criticism.

This chapter contains sections titled: 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11.

The Law of Non-Contradiction - that no contradiction can be true - has been a seemingly unassailable dogma since the work of Aristotle, in Book G of the Metaphysics. It is an assumption challenged from a variety of angles in this collection of original papers. Twenty-three of the world's leading experts investigate the 'law', considering arguments for and against it and discussing methodological issues that arise whenever we question the legitimacy of logical principles. The result is a balanced inquiry into (...) a venerable principle of logic, one that raises questions at the very centre of logic itself. The aim of this volume is to present a comprehensive debate about the Law of Non-Contradiction, from discussions as to how the law is to be understood, to reasons for accepting or re-thinking the law, and to issues that raise challenges to the law, such as the Liar Paradox, and a 'dialetheic' resolution of that paradox. The editors contribute an introduction which surveys the issues and serves to frame the debate, and a useful bibliography offering a guide to further reading. This volume will be of interest to anyone working on philosophical logic, and to anyone who has ever wondered about the status of logical laws and about how one might proceed to mount arguments for or against them. (shrink)

The question of whether the Pyrrhonist adheres to certain logical principles, criteria of justification, and inference rules is of central importance for the study of Pyrrhonism. Its significance lies in that, whereas the Pyrrhonist describes his philosophical stance and argues against the Dogmatists by means of what may be considered a rational discourse, adherence to any such principles, criteria, and rules does not seem compatible with the radical character of his skepticism. Hence, if the Pyrrhonist does endorse them, one must (...) conclude that he is inconsistent in his outlook. Despite its import, the question under consideration has not received, in the vast literature on Pyrrhonism of the past three decades, all the attention it deserves. In the present paper, I do not propose to provide a full examination of the Pyrrhonist’s attitude towards rationality, but to focus on the question of whether he endorses the law of non-contradiction (LNC). However, I will also briefly tackle the question of the Pyrrhonist’s outlook on both the canons of rational justification at work in the so-called Five Modes of Agrippa and the logical rules of inference. In addition, given that the LNC is deemed a fundamental principle of rationality, determining the Pyrrhonist’s attitude towards it will allow us to understand his general attitude towards rationality. (shrink)

On the ground of Kant’s reformulation of the principle of con- tradiction, a non-classical logic KC and its extension KC+ are constructed. In KC and KC+, \neg(\phi \wedge \neg\phi),  \phi \rightarrow (\neg\phi \rightarrow \phi), and  \phi \vee \neg\phi are not valid due to specific changes in the meaning of connectives and quantifiers, although there is the explosion of derivable consequences from {\phi, ¬\phi} (the deduc- tion theorem lacking). KC and KC+ are interpreted as fragments of an (...) S5-based first-order modal logic M. The quantification in M is combined with a “subject abstraction” device, which excepts predicate letters from the scope of modal operators. Derivability is defined by an appropriate labelled tableau system rules. Informally, KC is mainly ontologically motivated (in contrast, for example, to Jaśkowski’s discussive logic), relativizing state of affairs with respect to conditions such as time. (shrink)

Łukasiewicz distinguishes three formulations of the principle of contradiction in Aristotle’s works: ontological, logical, and psychological. The first two formulations are equivalent though not synonymous, but neither of them is equivalent to the psychological one, which expresses not a principle but only an empirical law. Furthermore, the principle of contradiction is neither a simple and ultimate law nor is it necessary for conducting an inference, because the syllogism is independent of it. The further explanation of this concept (...) leads Łukasiewicz to formulate the idea of a non-Aristotelian logic, that is, a logic operating without the principle of contradiction. If the principle of contradiction shall be valid, it must be proved. A proof can be supplied only on the basis of a definition of object, as something that cannot have and not have the same property at the same time. However, this definition does not hold for all objects, i.e., for contradictory objects. In virtue of its ontological character the Aristotelian principle of contradiction is then different from that of symbolic logic. (shrink)

Łukasiewicz distinguishes three formulations of the principle of contradiction in Aristotle’s works: ontological, logical, and psychological. The first two formulations are equivalent though not synonymous, but neither of them is equivalent to the psychological one, which expresses not a principle but only an empirical law. Furthermore, the principle of contradiction is neither a simple and ultimate law nor is it necessary for conducting an inference, because the syllogism is independent of it. The further explanation of this concept (...) leads Łukasiewicz to formulate the idea of a non-Aristotelian logic, that is, a logic operating without the principle of contradiction. If the principle of contradiction shall be valid, it must be proved. A proof can be supplied only on the basis of a definition of object, as something that cannot have and not have the same property at the same time. However, this definition does not hold for all objects, i.e., for contradictory objects. In virtue of its ontological character the Aristotelian principle of contradiction is then different from that of symbolic logic. (shrink)

The Law of Non-Contradiction - that no contradiction can be true - has been a seemingly unassailable dogma since the work of Aristotle. It is an assumption challenged from a variety of angles in this collection of original papers. Twenty-three of the world's leading experts investigate the 'law', considering arguments for and against it and discuss methodological issues that arise. The result is a balanced inquiry into a venerable principle of logic, one that raises questions at the very (...) centre of logic itself. (shrink)

The origin of paraconsistent logic is closely related with the argument, ‘from the assertion of two mutually contradictory statements any other statement can be deduced’; this can be referred to as ex contradictione sequitur quodlibet (ECSQ). Despite its medieval origin, only by the 1930s did it become the main reason for the unfeasibility of having contradictions in a deductive system. The purpose of this article is to study what happened earlier: from Principia Mathematica to that time, when it became (...) well established. The two main historical claims that I am going to advance are the following: (1) the first explicit use of ECSQ as the main argument for supporting the necessity of excluding any contradiction from deductive systems is to be found in the first edition of the book Grundzüge der Theoretischen Logik (Hilbert, D. and Ackermann, W. 1928. Grundzüge der Theoretischen Logik. Berlin: Julius Springer Verlag); (2) Łukasiewicz's position regarding the logical constraints against contradictions varies considerably from his studies on the principle of (non-) contradiction in Aristotle, published in 1910 and what is stated in his `authorized lectured notes' on mathematical logic that appeared in 1929. The two texts are: 1) a paper in German (Łukasiewicz, J. 1910. `Über den Satz des Widerspruchs bei Aristotles'. Bulletin International de l'Académie des sciences de Cracovie, Classe d'Histoire et de Philosophie, pp. 15–38) [English translation: Łukasiewicz, J. 1971. ‘On the principle of contradiction in Aristotle’, Review of Metaphysics, XXIV, 485–509]; and 2) a book in Polish. Łukasiewicz, J. 1910. O zasadzie sprzecznosci u Aristotelesa Studium krytyczne, Warsaw: Panstwowe Wydawnictwo Naukowe [German translation: Łukasiewicz, J. 1993. Über den Satz des Widerspruchs bei Aristotles. Hildesheim: Georg Olms Verlag]. The lecture notes were then published as a book (Łukasiewicz, J. 1958. Elementy Logiki Matematycznej. Warszawa: Panstwowe Wydawnictwo Naukowe [PWN] and then translated into English (Łukasiewicz, J. 1963. Elements of Mathematical Logic. Oxford, New York: Pergamon Press/the Macmillan Company). The second half of this article will concentrate on Łukasiewicz's position on ECSQ. This will lead me to propose that to regard him as a forerunner of paraconsistent logic by virtue of those early writings is accurate only if his book published in Polish is considered but not if the analysis is restricted to the paper originally published in German (as has been the case for the principal reconstructions of the history of paraconsistent logic). Furthermore, I will stress that in the 1929 book he presented one formalization of ECSQ as an axiom for sentential calculus and, also, he used ECSQ to defend the necessity of consistency, apparently independently of Hilbert and Ackermann's book. At the end, I will suggest that the aim of twentieth century usage of ECSQ was to change from the centuries-long philosophical discussion about contradictions to a more ‘technical’ one. But with paraconsistent logic viewed as a technical solution to this restriction, then, the philosophical problem revives but having now at one's disposal an improved understanding of it. Finally, Łukasiewicz's two different positions about ECSQ open an interesting question about the history of paraconsistent logic: do we have to attempt a consistent reconstruction of it, or are we prepared to admit inconsistencies within it? (shrink)

It is sometimes said that there are two, competing versions of W. V. O. Quine’s unrelenting empiricism, perhaps divided according to temporal periods of his career. According to one, logic is exempt from, or lies outside the scope of, the attack on the analytic-synthetic distinction. This logic-friendly Quine holds that logical truths and, presumably, logical inferences are analytic in the traditional sense. Logical truths are knowable a priori, and, importantly, they are incorrigible, and so immune from revision. The (...) other, radical reading of Quine does not exempt logic from the attack on analyticity and a priority. Logical truths and inferences are themselves part of the web of belief, and the same global methodology applies to logic as to any other part of the web, such as theoretical chemistry or ordinary beliefs about ordinary objects. Everything, including logic, is up for grabs in our struggle for holistic confirmation. The purpose of this paper is to examine the law of non-contradiction, and the concomitant principle of ex falso quodlibet, from the perspective of the principles advocated by the radical Quine. I show that he has no compelling reason to accept either of these. To put it bluntly, neither the law of non-contradiction nor the rule of ex falso quodlibet is empirically confirmed, and these principles fare poorly on the various criteria for theory acceptance on the methodology of the radical Quine. So the radical Quine is led rather quickly and rather directly into something in the neighborhood of Graham Priest’s dialetheism. (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)

The topic of this paper is Aristotle’s ‘proof by refutation’ of the Principle of Non-Contradiction (Metaphysics Γ 4, 1006a11–1007a20). I consider a worry which has often been raised in connection with this proof. The worry is that, faced with an opponent who is prepared to tolerate contradictions, the argument is dialectically powerless: it is incapable of getting them to abandon their position. In reply, I argue that the proof needs to be seen in its proper context, that is, as (...) part of Aristotle’s response to an opponent who demands a demonstration of the principle. This allows us to clarify the purpose of the argument and answer the worry about its effectiveness. (shrink)

In "Metaphysics" Gamma 3 Aristotle declares that the philosopher investigates things that are qua things that are and that he therefore should be able to state the firmest principles of everything. The firmest principle of all is identified as the principle of non-contradiction (PNC). The main focus of Gamma 3 is Aristotle's proof for this identification. This paper begins with remarks about Aristotle's notion of the firmness of a principle and then offers an analysis of the firmness (...) proof for PNC. It focuses on some key assumptions of the proof and on the range and force of the proof. Aristotle closes Gamma 3 with the claim that PNC is ultimate in the sense that all other principles somehow rest on it. This, rather controversial, claim is given a defensible reading and shown to be central to the chapter's effort to establish PNC as the firmest principle of all. As such it completes the firmness proof and is not simply an appended remark. (shrink)

The Principle of Non-Contradiction is found and christened as “the firmest principle” by Aristotle. Since then, almost no objection has been raised against this principle and its firmness, until Łukasiewicz. By reflecting on the possibility of the emergence of new systems of logic with reference to the developments in non-Eucledian geometry, Łukasiewicz stated that the reconsideration of the principles of logic may lead to new systems. With the guidance of this idea, in his revolutionary work (...) “On the Principle of Contradiction in Aristotle” Łukasiewicz approaches Aristotle’s account on the Principle of Non-Contradiction and analyzes it thoroughly. Starting with the identification and the comparison of three distinct definitions of the Principle of Non-Contradiction in Metaphysics, the Polish logician criticizes Aristotle’s arguments regarding the unprovability and the firmness of the principle. Even though Aristotle defended that the principle cannot be proved by direct proof but can be proved by indirect ones, i.e. elenchus, Łukasiewicz realizes that Aristotle’s aforementioned distinction is actually built on sand, since the Stagirate had stated in Prior Analytics that these two types of proof can be used interchangeably. This statement means that there is no err in using direct proofs instead of elenchtic ones and thus it appears that if the principle can be proved by elenchus, it can be proved by direct proof as well. In other words, Aristotle runs into a contradiction by trying to prove the principle that cannot be proved. Having found that there lies a contradiction in the very foundations of the Principle of Non-Contradiction regarding its unprovability, he continues with scrutinizing Aristotle’s elenchtic proofs in Metaphysics. With the discovery of fallacies appealed by Aristotle in those proofs, Łukasiewicz is ready for raising his objection to the firmness of the Principle of Non-Contradiction. Starting with stating how contradictions were defined before and by Aristotle, how the Stagirate formulated the Principle of Non-Contradiction, our study aims to examine Łukasiewicz’s objection and summarize his arguments against the Principle of Non-Contradiction. (shrink)

We present two approaches to investigate the validity of connexive principles and related formulas and properties within coherence-based probability logic. Connexive logic emerged from the intuition that conditionals of the form if not-A, thenA, should not hold, since the conditional’s antecedent not-A contradicts its consequent A. Our approaches cover this intuition by observing that the only coherent probability assessment on the conditional event $${A| \overline{A}}$$ A | A ¯ is $${p(A| \overline{A})=0}$$ p ( A | A ¯ ) (...) = 0. In the first approach we investigate connexive principles within coherence-based probabilistic default reasoning, by interpreting defaults and negated defaults in terms of suitable probabilistic constraints on conditional events. In the second approach we study connexivity within the coherence framework of compound conditionals, by interpreting connexive principles in terms of suitable conditional random quantities. After developing notions of validity in each approach, we analyze the following connexive principles: Aristotle’s theses, Aristotle’s Second Thesis, Abelard’s First Principle, and Boethius’ theses. We also deepen and generalize some principles and investigate further properties related to connexive logic (like non-symmetry). Both approaches satisfy minimal requirements for a connexive logic. Finally, we compare both approaches conceptually. (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 "contradiction." Finally, (...) I point out how the principle Hegel derives from the concept of contradiction actually informs his own method. (shrink)

What sorts of things qualify as first principles of reasoning and what kind of justification for them can be offered? We think of principles like that of non-contradiction as first principles of reasoning. Laurence Bloom argues that Plato’s Republic, the first text that affords us a complete statement of the Principle of Non-contradiction, offers us a powerful, complex and detailed argument for taking form—specifically that of the good—as the first principle of both knowing and being.

Volume 47, Issue 3, November 2018, Page 461-466.

The discussion of the Principle of Non-Contradiction (PNC) in Aristotle’s Metaphysics Γ is usually taken to include three ‘versions’ of the principle: an ontological, psychological, and logical one. In this article I develop an interpretation of Metaphysics Γ3 and a parallel text, De interpretatione 14, in order to show that these texts are concerned with two related but different principles: a version of the Principle of Identity, and a corollary to this, which concerns the ability to accept (...) two ‘opposite’ items at the same time. I argue that these principles must be considered separately in order to properly understand Aristotle’s remarks about PNC in Metaphysics Γ, and in order to defend his approach in these remarks against certain objections raised against the discussion in Metaphysics Γ by Jan Łukasiewicz in his "Über den Satz des Widerspruchs bei Aristoteles". The main result of my interpretation is to distinguish in the discussion of PNC in Metaphysics Γ three principles: one concerning linguistic items (words and statements), one concerning thoughts (δόξαι), and one concerning objects or states of affairs. (shrink)

One of the most challenging and fascinating issue in mathematical and theoretical physics concerns the possibility of identifying the logic underlying the so-called quantum universe, i.e., Quantum Mechanics and Quantum Gravity. Besides the sheer difficulty of the problem, inherent in the actual formulation of Quantum Mechanics—and especially of Quantum Gravity—to be used for such a task, a crucial aspect lies in the identification of the appropriate axiomatic logical proposition calculus to be associated to such theories. In this paper the (...) issue of the validity of the conventional principle of non-contradiction is called into question and is investigated in the context of non-relativistic Quantum Mechanics. In the same framework a modified form of the principle, denoted as 3-way PNC is shown to apply, which relates the axioms of quantum logic with the physical requirements placed by the Heisenberg Indeterminacy Principle. (shrink)

This article is an inquiry into how the relationship between the principle of non-contradiction and the limits of thought has been understood by thinkers as diverse as Hegel, Heidegger, Levinas, and Graham Priest. While Heidegger and Levinas focus on the question of temporality and Priest takes a formal approach, all these philosophers effectively maintain that the principle of non-contradiction imposes a restriction on thought that disables it from adequately accounting for its own limits and thus what lies beyond (...) those limits, the implication being that the violation of the principle is necessary for such an accounting to take place. However, the ultimate argument here is that, contrary to Priest’s interpretation, Hegel’s philosophy can be convincingly read as supporting the idea that the mind’s ability to go beyond any particular limit of thought can actually be said to involve an adherence to a normative demand to locate and dispel the contradictions that emerge through the very setting of determinative limits. This is a non-formal consistency that evinces a “logic” that is unknowingly followed by the Heideggerian and Levinasian phenomenological philosophies of transcendence. (shrink)

The argumentative structure of this article can be summarized as follows: 1) Spinoza was repeatedly accused of eleaticism; 2) there is a rupture with the eleaticism when one admits the ‘multivocity’ of the logical operators “is” and “is not”; as can be seen, for example, 2.1) in the discussion introducing the Great Genera in Plato’s Sophist, and 2.2) in a certain use made by Aristotle of the category’s doctrine in order to soften the Parmenidean version of the Principle of (...) Non-Contradiction. 3) Spinoza admits the ‘multivocity’ of “is” and “is not”. Hence, Spinoza should not be placed among the Eleatics but inside the brotherhood of the suspects of the so-called parricide against Parmenides. This reasoning, which almost sounds as a truism, will, however, suggest two interesting further considerations: one on how an important tradition of interpretation of Spinoza has misplaced the problem of the relationship of Spinoza’s ontology with the logical principles; and another on the type of logic that is implied by the theory of modes of perception or kinds of knowledge. (shrink)

The concept of truth arguably plays a central role in many areas of philosophical theorizing. Yet, what seems to be one of the most fundamental principles governing that concept, i.e. the equivalence between P and , is inconsistent in full classical logic, as shown by the semantic paradoxes. I propose a new solution to those paradoxes, based on a principled revision of classical logic. Technically, the key idea consists in the rejection of the unrestricted validity of the structural (...) principle of contraction. I first motivate philosophically this idea with the metaphysical picture of the states-of-affairs expressed by paradoxical sentences as being distinctively . I then proceed to demonstrate that the theory of truth resulting from this metaphysical picture is, in many philosophically interesting respects, surprisingly stronger than most other theories of truth endorsing the equivalence between P and (for example, the theory vindicates the validity of the traditional laws of excluded middle and of non-contradiction, and also vindicates the traditional constraint of truth preservation on logical consequence). I conclude by proving a cutelimination theorem that shows the consistency of the theory. (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)

Aristotle discovered the law of contradiction more than 2000 years ago. Since then, this law has been regarded as one of the basic principles of logic. Aristotle considered this principle to be 'the most indisputable of all beliefs,' but nearly half a century ago, it began to be criticized. The voice of criticism came from a philosophical logic - paraconsistent logic. This study analyses in depth the specific properties of the positive logic plus approach, non-adjunctive (...) approach, and relevant approach of paraconsistent logic and presents three definitions of the law of contradiction. It also shows that there are two types of the law of contradiction in C-logics and that the law of contradiction with classical negation is valid in them. Furthermore, this study proves that the law of contradiction with classical negation is also valid in a non-adjunctive approach and explains the reason this law cannot be deduced in some relevant logic systems. Based on these, we can clarify what type of the law of contradiction is questioned by paraconsistent logics and thoroughly reveal the exact meaning of 'the law of contradiction is invalid' in paraconsistent logics. (shrink)

Five major stances on the problems of the possibility and fruitfulness of a debate on the principle of non-contradiction (PNC) are described: Detractors, fierce supporters, demonstrators, methodologists and calm supporters. We show what calm supporters have to say on the other parties wondering about the possibility and fruitfulness of a debate on PNC. The main claim is that one can find all the elements of calm supporters already in Aristotle’s works. In addition, we argue that the Aristotelian refutative strategy, (...) originally used for dealing with detractors of PNC in Metaphysics, has wider implications for the possibility and fruitfulness of an up-to-date debate on PNC, at least in exhibiting some serious difficulties for the other parties. (shrink)

In Book Γ of the Metaphysics Aristotle states and attempts to prove what he calls the basic principle of the science of being as being: the law of non‐contradiction. In this paper I defend an interpretation of his proof, inspired by Elizabeth Anscombe's 1961 essay in ‘Three Philosophers’, though some of its features were remarked on by Lukasiewicz in 1910, according to which Aristotle is proving this principle only for substance predicates, and that it is to be (...) understood as the basic principle of metaphysics rather than of logic. It is the principle that every substance has an essential nature, and if it is of the essential nature of x to be F then it cannot be of the essential nature of x to be not‐F since then x's being F and x's being not‐F would be the same. He puts it by saying that in such a case F and not‐F would ‘signify the same’ for x. Since they do not, x cannot be both. (shrink)

In 'Metaphysics' book 4 Aristotle offers several arguments in defense of the principle of noncontradiction (PNC). In this paper I want to focus on the stretch of argument from 1006a11 to 1006b34 which Aristotle calls a proof by refutation (elenktikos apodeixai), (1006a11). Contrary to Elizabeth Anscombe and others, I will argue that in this section of the defense Aristotle can defend a version of the principle that extends to nonessential predication, predication of properties, aggregates, and transcategorials.

The aim of this paper is to present and analyze arguments provided for the Psychological Principle of Non-Contradiction which states that one cannot have, or cannot be described as having, contradictory beliefs. By differentiating two possible interpretations of PNC, descriptive and normative, and examining arguments provided for each of them separately I point out the flaws in reasoning in these arguments and difficulties with aligning PNC with the empirical data provided by research done in cognitive and clinical psychology. I (...) claim that PNC cannot be derived from any metaphysical stance regarding the mental phenomena and that having contradictory beliefs should be regarded as possible. Furthermore, I argue that interpreting a subject as having contradictory beliefs, and therefore abandoning PNC, can be more effective in explaining the phenomena of contradictory beliefs and irrational behaviour than solutions consistent with the PNC. (shrink)

Can contradictions be meaningful? How can one assert 'P soku not-P' or 'P and yet not-P' without sacrificing intelligibility? Expanding on previous attempts, mainly by Dilworth and Heisig, to demystify the soku connective, a formal system is presented here for the logic of soku. Through a formal distinction between internal and external negation, grammatical features of the soku connective are shown to be logically irrelevant, and the principle of non-contradiction is preserved. Disparities with traditional logic are noted, (...) with a focus on negation rather than 'soku'. The formal examination of the logic of soku is intended to present the logic in a way acceptable to more analytically minded philosophers and thereby enhance East-West and Japanese-Anglo-American interaction and criticism. (shrink)