This paper discusses the history of the confusion and controversies over whether the definition of consequence presented in the 11-page 1936 Tarski consequence-definition paper is based on a monistic fixed-universe framework?like Begriffsschrift and Principia Mathematica. Monistic fixed-universe frameworks, common in pre-WWII logic, keep the range of the individual variables fixed as the class of all individuals. The contrary alternative is that the definition is predicated on a pluralistic multiple-universe framework?like the 1931 Gödel incompleteness paper. A pluralistic multiple-universe framework recognizes multiple (...) universes of discourse serving as different ranges of the individual variables in different interpretations?as in post-WWII model theory. In the early 1960s, many logicians?mistakenly, as we show?held the ?contrary alternative? that Tarski 1936 had already adopted a Gödel-type, pluralistic, multiple-universe framework. We explain that Tarski had not yet shifted out of the monistic, Frege?Russell, fixed-universe paradigm. We further argue that between his Principia-influenced pre-WWII Warsaw period and his model-theoretic post-WWII Berkeley period, Tarski's philosophy underwent many other radical changes. (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)

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)

We compare Jan ?ukasiewicz's and Karl Popper's views on induction. The English translation of the two ?ukasiewicz's papers is included in the Appendix.

The study of information based on the approach of Shannon was detached from problems of meaning. Also, it did not allow analysis of the structural characteristics of information, nor describe the way structures carry information. An outline of a different theory of information, including its semantics, was earlier proposed by the author. This theory was using closure spaces to model information. In the present paper, structures (called syllogistics) underlying syllogistic reasoning as well as ethnoscientific classifications are identified together with the (...) conditions for the lattice of closed subsets describing information to allow its existence. The structures can be used for logical analysis outside of language at the more general level of information, which in turn can be applied to the description of semantic relations in the context of information. (shrink)

Leśniewski’s metalogic is often considered to be difficult to understand because it differs greatly from its standard formulation. In this paper I try to explain the reasons of these idiosyncrasies. I claim that they have mainly two sources. First of all there is Leśniewski’s conviction that a formal system should be conceived as a set of concrete marks that can always physically and syntactically be expanded by the addition of new theses. Secondly there is Leśniewski’s conviction that definitions should neither (...) be formulas belonging to the metalanguage, nor deduction rules, but formulas belonging to the object-language and expressed with the help of the biconditional functor. The realisation of the first point is linked to the second one in so far as the metalinguistic rule for the writing out of definitions has to be formulated in a way that makes it possible to build the formal system in agreement with Leśniewski’s conception. While explaining these points I give an overview of the main peculiarities of Leśniewski’s metalogic. (shrink)

En la historia de la sociedad moderna –basada en la discriminación económica y en ideas competitivas de poder– la teoría de la evolución darwiniana, por medio de la selección natural de las especies, llegó como una justificación de aptitudes biológicas para su estructura socioeconómica. Pareció que el papel del individuo era contribuir a perpetuar la especie; todo lo que se tenía que hacer era dejar que los fenómenos naturales siguieran su curso. La ciencia, en especial la biología, aparecía justificando la (...) premisa “cualquier cosa por el bien de la humanidad”. (shrink)

This article features long-sought proofs with intriguing properties (such as the absence of double negation and the avoidance of lemmas that appeared to be indispensable), and it features the automated methods for finding them. The theorems of concern are taken from various areas of logic that include two-valued sentential (or propositional) calculus and infinite-valued sentential calculus. Many of the proofs (in effect) answer questions that had remained open for decades, questions focusing on axiomatic proofs. The approaches we take are of (...) added interest in that all rely heavily on the use of a single program that offers logical reasoning, William McCune's automated reasoning program OTTER. The nature of the successes and approaches suggests that this program offers researchers a valuable automated assistant. This article has three main components. First, in view of the interdisciplinary nature of the audience, we discuss the means for using the program in question (OTTER), which flags, parameters, and lists have which effects, and how the proofs it finds are easily read. Second, because of the variety of proofs that we have found and their significance, we discuss them in a manner that permits comparison with the literature. Among those proofs, we offer a proof shorter than that given by Meredith and Prior in their treatment of ukasiewicz's shortest single axiom for the implicational fragment of two-valued sentential calculus, and we offer a proof for the ukasiewicz 23-letter single axiom for the full calculus. Third, with the intent of producing a fruitful dialogue, we pose questions concerning the properties of proofs and, even more pressing, invite questions similar to those this article answers. (shrink)

The rich body of physical theories defines the foundation of our understanding of the world. Its mathematical formulation is based on classical Aristotelian logic. In the philosophy of science the ambiguities, paradoxes, and the possibility of subjective interpretations of facts have challenged binary logic, leading, among other developments, to Gotthard Günther’s theory of polycontexturality. Günther’s theory explains how observers with subjective perception can become aware of their own subjectivity and provides means to describe contradicting or even paradox observations in a (...) logically sound formalism. Here we summarize the formalism behind Günther’s theory and apply it to two well-known examples from physics where different observers operate in distinct and only locally valid logical systems. Using polycontextural logic we show how the emerging awareness of these limitations of logical systems entails the design of mathematical transformations, which then become an integral part of the theory. In our view, this approach offers a novel perspective on the structure of physical theories and, at the same time, emphasizes the relevance of the theory of polycontexturality in modern sciences. (shrink)

With material on his early philosophical views, his contributions to set theory and his work on nominalism and higher-order quantification, this book offers a uniquely expansive critical commentary on one of analytical philosophy’s great ...

In the article we consider the relationship of traditional provisions of basic logical concepts and confront them with new and modern approaches to the same concepts. Logic is characterized in different ways when it is associated with syllogistics (referential – semantical model of logic) or with symbolic logic (inferential – syntactical model of logic). This is not only a difference in the logical calculation of (1) concepts, (2) statements, and (3) predicates, but this difference also appears in the treatment of (...) the calculative abilities of logical forms, the ontological-referential status of conceptual content and the inferential-categorical status of logical forms. The basic markers or basic ideas that separate ontologically oriented logic from categorically oriented logic are the (1) concept of truth, the (2) concept of meaning, the (3) concept of identity, and the (4) concept of predication. Here, this differences are explicitly demonstrated by the introduction of differential terminology. From this differential methodology follows a new set of characterizations of logic. (shrink)

Although the theory of the assertoric syllogism was Aristotle's great invention, one which dominated logical theory for the succeeding two millenia, accounts of the syllogism evolved and changed over that time. Indeed, in the twentieth century, doctrines were attributed to Aristotle which lost sight of what Aristotle intended. One of these mistaken doctrines was the very form of the syllogism: that a syllogism consists of three propositions containing three terms arranged in four figures. Yet another was that a syllogism is (...) a conditional proposition deduced from a set of axioms. There is even unclarity about what the basis of syllogistic validity consists in. Returning to Aristotle's text, and reading it in the light of commentary from late antiquity and the middle ages, we find a coherent and precise theory which shows all these claims to be based on a misunderstanding and misreading. (shrink)