What is the Principle of (Non-)Contradiction, Precisely? The Struggle at the Dawn of Formal Logic

This is a companion article to the translation of ‘Zasada sprzeczności a logika symboliczna’, the appendix on symbolic logic of Jan Łukasiewicz's 1910 book O zasadzie sprzeczności u Arytotelesa (On the Principle of Contradiction in Aristotle). While the appendix closely follows Couturat's 1905 book L'algebra de la logique (The Algebra of Logic), footnotes show that Łukasiewicz was aware of the work of Peirce, Huntington and Russell (before Principia Mathematica). This appendix was influential in the development of the Polish school of logic, directly inspiring Stanisław Leśniewski and Leon Chwistek and more widely by serving as a text of the new symbolic logic. This appendix was an important source of the dominant algebraic logic in Poland, but also indicates that Łukasiewicz appreciated Russell's axiomatic approach to logic.

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?

Luego de culminado el juicio oral, en los delitos de concusión (colusión), peculado y corrupción de funcionarios (cohecho) frecuentemente no existe prueba directa. La «prueba indiciaria» debe valorarse utilizando el «principio de sana crítica», desarrollando los elementos de esta: a) la lógica, con sus principios de identidad, contradicción, razón suficiente, tercero excluido; b) las máximas de la experiencia (criterios normativos o reglas no jurídicas); c) conocimientos científicamente aceptados.El método de valoración de la prueba indiciaria implica utilizar la «inferencia deductiva», lo cual es adecuado. Este «método de valoración de la prueba indiciaria» debe aplicarse a la valoración de la «prueba directa», por ser más eficaz, para acercarnos a la verdad.

The task of the criminal procedure defined in terms of clarifying and resolving the main case of a criminal proceedings – criminal matter (causa criminalis) is achieved by the application of legally standardized basic procedural principles. The possibility given to the parties in a criminal proceedings to actively participate in it represents one of the basic features of the accusatory system of criminal proceedings. The principle of contradiction refers to the right of any procedural party to declare itself to the procedural actions of the opposing party before the competent court bases its decision on these procedural actions. The principle of contradiction is one of the general elements of the right to a fair trial. Therefore, the right of contradiction (contradicere) opposing the allegations of the other party is, together with the principle of immediacy, in a direct function of achieving the principle of a fair trial in our legal system. It is present during the whole criminal proceedings, but it mostly becomes prominent to expressing the evidence at the main trial. The Code of Criminal Procedure of Republic of Serbia has explicitly not envisaged the principle of contradiction. However, in addition of being involved in the main trial, as the main phase of the criminal proceedings, this principle extends through the entire criminal procedure, both the first instance and the second instance. Although not explicitly proclaimed in the European Convention for the Protection of Human Rights and Fundamental Freedoms, the principle of contradiction has developed in the practice of the European Court of Human Rights as a general element of the right to a fair trial.

THE ACT OF ANALYSIS IT WOULD SEEM FAIR to say that the prevailing view of analysis has been largely shaped by Kant's account of the analytic judgment. In the Critique of Pure Reason that judgment is presented as one in which" the predicate B belongs to the subject A, as somewhat which is contained (though covertly) in the conception A." It is a judgment " in which the connection of the predicate with the subject is cogitated through identity." Forming it, therefore, involves no " recourse to the testimony of experience," no need to "go outside the sphere of my own conceptions." For example, " I need not go beyond the conception of body to find extension connected with it." I have but to " analyze the conception "-that is, to "become conscious of the manifold properties which I think in that conception." The judgment upon any notion demands only that we " analyze it into its constituent conceptions," thereby bringing into the full light of consciousness those elements "which were already in the subject, although in a confused manner." The product of the analytic act is thus " a proposition that stands firm a priori;" the mind needs" only to extract the predicate from the conception, according to the principle of contradiction, and thereby at the same time become aware of the necessity of the judgment." 1 The core of the above is also found in the Prolegomena to Any Future Metaphysics: "Analytic judgments express nothing in the predicate but what has already been thought in the concept of the subject, though not so distinctly or with the same full consciousness." 2 1 Critique of Pure Reason, trans. Meicklejohn, pp. 7-8. (Willey Book Company, New York, 1900). •Prologue to Any Future Metaphysics, p. 14. (The Library of Liberal Arts, Bobbs-Merrill, New York, 1950). 45 46 JOHN D. BEACH Important problems attach to Kant's account of analysis and its product, problems rooted in the stated covert, confused or indistinct presence in the subject-concept of the elements that reflective thought brings to the light of day. Granting the product of analysis that initial status, we are led to ask what, if anything, is given overtly and distinctly to thought. Putting this somewhat differently, we must ask in terms of what, if anything , the subject of analysis is initially known or identified. This, in turn, prompts a question concerning the relationship between what is initially known and that which analysis yields. But these are matters more appropriately dealt with in another context. Let us proceed for now on the assumption that the product of analysis, the extracted predicate, is so given as an element of the subject-concept that the act of analysis unfolds exclusively in accordance with the principles of identity and contradiction. The prevailing view of analysis is based upon that very assumption. Its adherents stress that, as initially present to thought, the subject is seen to contain the predicate -i.e., the subject is known as a complex having the predicate as one of its discrete parts. Accordingly, they would note, the analytic proposition is simply the expression of the knowledge that the predicate, the extracted constituent, is such a part of the complex. And this is to say that there is a relation of direct and complete identity between the relevant part of the complex and the extracted predicate, and a relation of direct but partial identity between the extracted element and the initial complex. The favored examples of the latter, partial identity are the relationship between male and bachelor, or unmarried male, and that between sibling and brother, or male sibling. Proponents of this essentially conventional approach toward analysis also emphasize that the necessity involved is based on the laws of identity and contradiction. Contracted to the case in hand, these principles tell us, respectively, that a given cogitated element 0£ a subject-concept must be such an element and that it isimpossible for it not to be such an element. Thus male must be and cannot not be an element of unmarried male. THE ACT OF ANALYSIS 47 Rephrased, an unmarried male must be and cannot not be a male. Whence what is now the standard characterization...

This chapter discusses three fundamental principles of Leibniz's philosophy: the Principle of Contradiction, the Principle of Sufficient Reason, and the Principle of Identity of Indiscernibles. It evaluates various formulations of these principles, their axiomatic character, and some attempts to demonstrate them. In particular, the chapter discusses in detail the derivation of the Principle of Sufficient Reason in Primary Truths, and argues that Leibniz does not use the Principle of Contradiction in that derivation. It also discusses an attempt, in the correspondence with Clarke, to prove the Principle of Sufficient Reason empirically. Finally, the article examines the argument for the Principle of Identity of Indiscernibles in the Discourse on Metaphysics.

In a world full of arguments and opinions, the ability to understand the logic of thinking is a strong foundation for creating persuasive arguments. Logical thinking involves using meaningful rules and principles to form reasonable arguments based on available information. The basic principles of logic, such as the identity principle, the contradiction principle, and the refutation principle, are the basis for consistent and correct reasoning. Understanding and avoiding logical fallacies is also important for developing critical thinking and strong reasoning. In argumentation, logic plays an important role in creating strong and convincing arguments. By using strong evidence, assessing relevance, and thinking critically, you can avoid reasoning errors and create stronger arguments. In argumentation and debate, arguments that are based on logic and strong evidence are generally accepted and taken more seriously by opponents and listeners. Therefore, it is important to develop a good understanding of logical thinking and use it effectively to create strong and convincing arguments. Understanding the basic principles of logic will help you identify weaknesses in arguments, evaluate evidence critically, and create more effective arguments. Thus, logical thinking becomes an important basis for creating strong and convincing arguments.

The Conceptions of Self-Evidence in the Finnis Reconstruction of Natural Law

The abstract status of Kant's account of his ‘general logic’ is explained in comparison with Gödel's general definition of a formal logical system and reflections on ‘abstract’ (‘absolute’) concepts. Thereafter, an informal reconstruction of Kant's general logic is given from the aspect of the principles of contradiction, of sufficient reason, and of excluded middle. It is shown that Kant's composition of logic consists in a gradual strengthening of logical principles, starting from a weak principle of contradiction that tolerates a sort of contradictions in predication, and then proceeding to the (constructive) principle of sufficient reason, and to a classical-like logic, which includes the principle of excluded middle. A first-order formalisation is applied to this reconstruction, which reveals implicit modalities in Kant's account of logic, and confirms the implementability of Kant's logic into a sound and complete formal system.

In this paper I inquire into the role of general logic in Fichte’s early formulations of his first principle. This inquiry contains three main parts. First, I summarize the role of general logic in Kant’s theoretical philosophy, as well as Gottlob Schulze’s critical claims regarding their relation in Reinhold’s Elementarphilosophie. Second, I examine the first three sections of Fichte’s private notes on the Elementarphilosophie, called the Eigene Meditationen, and closely follow his early attempts to provide a basic principle that is systematically prior both to Reinhold’s principle of consciousness, as well as the logical principle of contradiction. I examine Fichte’s struggles with relating the principles of a foundational transcendental philosophy to those of general logic, in order to emphasize his own doubts in systematically motivating the use of logical rules in the exhibition of his first principle. In the third section, I examine the manner in which these principles are introduced in the 1794 Wissenschaftslehre via the logical principles of identity and contradiction, and argue that Fichte’s procedure is problematic given the programmatic constraints on general logic put forward in the meditations on the philosophy of the elements. I conclude by briefly relating Fichte’s doubts in such a procedure, as well as an alternative procedure he already proposed in his private notes, to the method that he would later adopt in the Wissenschaftslehre nova methodo.

On a principle of contradiction in normative logic and jurisprudence

Since the time of Aristotle’s thinking of logic to being a tool for orderly think, has maintained its importance until the present times. So soon, studies of the non-classical logical calls (Abe in 4th International Workshop on Soft Computing Applications. IEEE, pp. 11–18, 2010 [1]) have become a powerful tool as an aid in the making of decisions. The Paraconsistent logic calls attention to the clarity of containing provisions contrary to some of the basic principles of Aristotelian logic, such as the principle of contradiction. In this article, the use of technology allows proposing a structured organization in the process for use of Paraconsistent artificial neural networks. This process aims to be a facilitator in supporting the construction of the decision support (Abe in 4th International Workshop on Soft Computing Applications. IEEE, pp. 11–18, 2010 [1]) with the announcements for project recount in the function point analysis technique.

The meeting “On the Questions of Dialectical Contradictions in the Light of Contemporary Science and Practice”, organized by the Institute of Philosophy of the Academy of Sciences of the USSR and held in Moscow in April of 1958, is noteworthy from several points of view even for those who do not normally occupy themselves with questions of Marxism-Leninism. First, to judge from the reports received, this meeting was remarkably objective. Several of the addresses gave evidence of a definite impartiality, unusual in the Soviet context. In the second place, this is the first time that Soviet philosophers have systematically discussed the “dialectical” contradiction, which is the most significant aspect of the Hegelian heritage of their philosophy. Although Hegel’s name appears very seldom, it is evident that the participants form two groups: those who defend the “dialectical” contradiction, “dialectical” logic, in a word, the legacy of Hegel as transmitted by Engels and Lenin; and those who defend formal logic, try to interpret “real” contradictions as polar oppositions and for whom “Hegelian” is almost an insult. Without a doubt, the “strategic” position of the first group is much more favourable since its members can almost invariably count on the support of the “classics”, who prized contradiction and “dialectical” logic very highly. The partisans of the second group find themselves in a particularly uncomfortable situation since they must either bring their defence of reason and logic into agreement with the statements of the “classics” or contradict these same “classics”. That this last possibility actually was realized is the third and most noteworthy fact about the meeting.KeywordsContradictory NatureReidel Publishing CompanySoviet PhilosopherUncomfortable SituationInfinitesimal MovementThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

With the rise of the discourse on dialectical materialism in the late 1920s, ideas related to the Marxist notion of dialectical logic started to circulate in the Chinese intellectual world. Not long after the first public discussions on dialectical materialism started to emerge in the early 1930s, the discussants on both sides started to address the question of the Marxist notion of logic and its relationship with Western formal logic. Consequently, over the 1930s, a series of separate public debates ensued, in which dialectical logic contended against the “conventional” forms of logic, such as traditional Aristotelian and modern formal logic. This paper outlines the major landmarks within the public as well as internal Marxist debates on logic in the 1930s. The discussion starts with a general overview of the intellectual background of the debates, and proceeds by analysing the principal developments in them, starting with Ye Qing’s and Zhang Dongsun’s polemic about “dynamic logic” from 1933, and concluding with the internal Marxist discussions on the sublation of formal logic in the last years of the decade.

The clash between Salafi and Sufi schools of thought has shaped academic discourse and religious practice among Muslims worldwide. Therefore, this study examines the methodology of matn criticism developed by Ibn al-Qayyim from the Salafi tradition and Mullā al-Qārī from the Sufi tradition. The research employs a qualitative design, with data analyzed inductively, emphasizing critical editions of texts alongside textual and comparative methods. The findings reveal that both scholars share three out of four fundamental principles in matn criticism: contradiction with the Qurʾān, contradiction with authentic hadith, and the attribution of expressions unbefitting of the Prophet. However, they differ on the principle of contradiction with reason. This study brings positive implications by proposing an integrative framework that combines the Salafi tradition’s textual approach with the Sufi tradition’s contextual sensitivity, strengthening the preservation of hadith authenticity against unverified narrations.