In a recent paper Johan van Benthem reviews earlier work done by himself and colleagues on ‘natural logic’. His paper makes a number of challenging comments on the relationships between traditional logic, modern logic and natural logic. I respond to his challenge, by drawing what I think are the most significant lines dividing traditional logic from modern. The leading difference is in the way logic is expected to be used for checking arguments. For traditionals the checking is local, (...) i.e. separately for each inference step. Between inference steps, several kinds of paraphrasing are allowed. Today we formalise globally: we choose a symbolisation that works for the entire argument, and thus we eliminate intuitive steps and changes of viewpoint during the argument. Frege and Peano recast the logical rules so as to make this possible. I comment also on the traditional assumption that logical processing takes place at the top syntactic level, and I question Johan’s view that natural logic is ‘natural’. (shrink)

Since the time of Aristotle's students, interpreters have considered Prior Analytics to be a treatise about deductive reasoning, more generally, about methods of determining the validity and invalidity of premise-conclusion arguments. People studied Prior Analytics in order to learn more about deductive reasoning and to improve their own reasoning skills. These interpreters understood Aristotle to be focusing on two epistemic processes: first, the process of establishing knowledge that a conclusion follows necessarily from a set of premises (that is, on the (...) epistemic process of extracting information implicit in explicitly given information) and, second, the process of establishing knowledge that a conclusion does not follow. Despite the overwhelming tendency to interpret the syllogistic as formal epistemology, it was not until the early 1970s that it occurred to anyone to think that Aristotle may have developed a theory of deductive reasoning with a well worked-out system of deductions comparable in rigor and precision with systems such as propositional logic or equational logic familiar from mathematical logic. When modern logicians in the 1920s and 1930s first turned their attention to the problem of understanding Aristotle's contribution to logic in modern terms, they were guided both by the Frege-Russell conception of logic as formal ontology and at the same time by a desire to protect Aristotle from possible charges of psychologism. They thought they saw Aristotle applying the informal axiomatic method to formal ontology, not as making the first steps into formal epistemology. They did not notice Aristotle's description of deductive reasoning. Ironically, the formal axiomatic method (in which one explicitly presents not merely the substantive axioms but also the deductive processes used to derive theorems from the axioms) is incipient in Aristotle's presentation. Partly in opposition to the axiomatic, ontically-oriented approach to Aristotle's logic and partly as a result of attempting to increase the degree of fit between interpretation and text, logicians in the 1970s working independently came to remarkably similar conclusions to the effect that Aristotle indeed had produced the first system of formal deductions. They concluded that Aristotle had analyzed the process of deduction and that his achievement included a semantically complete system of natural deductions including both direct and indirect deductions. Where the interpretations of the 1920s and 1930s attribute to Aristotle a system of propositions organized deductively, the interpretations of the 1970s attribute to Aristotle a system of deductions, or extended deductive discourses, organized epistemically. The logicians of the 1920s and 1930s take Aristotle to be deducing laws of logic from axiomatic origins; the logicians of the 1970s take Aristotle to be describing the process of deduction and in particular to be describing deductions themselves, both those deductions that are proofs based on axiomatic premises and those deductions that, though deductively cogent, do not establish the truth of the conclusion but only that the conclusion is implied by the premise-set. Thus, two very different and opposed interpretations had emerged, interestingly both products of modern logicians equipped with the theoretical apparatus of mathematical logic. The issue at stake between these two interpretations is the historical question of Aristotle's place in the history of logic and of his orientation in philosophy of logic. This paper affirms Aristotle's place as the founder of logic taken as formal epistemology, including the study of deductive reasoning. A by-product of this study of Aristotle's accomplishments in logic is a clarification of a distinction implicit in discourses among logicians--that between logic as formal ontology and logic as formal epistemology. (shrink)

Filling the need for an accessible, carefully structured introductory text in symbolic logic, Modern Logic has many features designed to improve students' comprehension of the subject, including a proof system that is the same as the award-winning computer program MacLogic, and a special appendix that shows how to use MacLogic as a teaching aid. There are graded exercises at the end of each chapter--more than 900 in all--with selected answers at the end of the book. Unlike competing texts, (...) class='Hi'>Modern Logic gives equal weight to semantics and proof theory and explains their relationship, and develops in detail techniques for symbolizing natural language in first-order logic. After a general introduction featuring the notion of logical form, the book offers sections on classical sentential logic, monadic predicate logic, and full first-order logic with identity. A concluding section deals with extensions of and alternatives to classical logic, including modal logic, intuitionistic logic, and fuzzy logic. For students of philosophy, mathematics, computer science, or linguistics, Modern Logic provides a thorough understanding of basic concepts and a sound basis for more advanced work. (shrink)

While for a long time the study of medieval logic focused on editorial projects and reconstructions of central medieval doctrines such as the theories of signification, supposition, consequences, and obligations, nowadays the spectrum of analysis has broadened and is increasingly informed by modern logical research, whose perspective is then applied to medieval logic. Promoting this tendency, logicians and researchers concerned with semantics in the Gesellschaft für Philosophie des Mittelalters und der Renaissance (GPMR) founded a working group bringing together medieval (...) logic and modern applied logic. The present volume is a seminal document of these interests and activities. It analyzes theories in medieval logic which are useful for solving questions of recent logic and explains crucial parts of medieval logic, philosophy, and theology by applying techniques of present-day logic. (shrink)

As the title suggests, this is a book devoted not merely to logic; students will also examine the philosophical debates that led to the development of the field.

The final work of the esteemed philosopher William Demopoulos supplants logical empiricism's accounts of physical theories, which fail to satisfactorily engage modern physics. Arguing for a new appreciation of the tightly woven character of theory and evidence, Demopoulos offers novel insights into the distinctive nature of quantum reality.

Preface -- 1. Propositional logic : proofs from axioms and inference rules -- 2. First order logic : proofs with quantifiers -- 3. Set theory : proofs by detachment, contraposition, and contradiction -- 4. Mathematical induction : definitions and proofs by induction -- 5. Well-formed sets : proofs by transfinite induction with already well-ordered sets -- 6. The axiom of choice : proofs by transfinite induction -- 7. applications : Nobel-Prize winning applications of sets, functions, and relations -- 8. Solutions (...) to some odd-numbered exercises. (shrink)

This book treats ancient logic: the logic that originated in Greece by Aristotle and the Stoics, mainly in the hundred year period beginning about 350 BCE. Ancient logic was never completely ignored by modern logic from its Boolean origin in the middle 1800s: it was prominent in Boole’s writings and it was mentioned by Frege and by Hilbert. Nevertheless, the first century of mathematical logic did not take it seriously enough to study the ancient logic texts. A renaissance in (...) ancient logic studies occurred in the early 1950s with the publication of the landmark Aristotle’s Syllogistic by Jan Łukasiewicz, Oxford UP 1951, 2nd ed. 1957. Despite its title, it treats the logic of the Stoics as well as that of Aristotle. Łukasiewicz was a distinguished mathematical logician. He had created many-valued logic and the parenthesis-free prefix notation known as Polish notation. He co-authored with Alfred Tarski’s an important paper on metatheory of propositional logic and he was one of Tarski’s the three main teachers at the University of Warsaw. Łukasiewicz’s stature was just short of that of the giants: Aristotle, Boole, Frege, Tarski and Gödel. No mathematical logician of his caliber had ever before quoted the actual teachings of ancient logicians. -/- Not only did Łukasiewicz inject fresh hypotheses, new concepts, and imaginative modern perspectives into the field, his enormous prestige and that of the Warsaw School of Logic reflected on the whole field of ancient logic studies. Suddenly, this previously somewhat dormant and obscure field became active and gained in respectability and importance in the eyes of logicians, mathematicians, linguists, analytic philosophers, and historians. Next to Aristotle himself and perhaps the Stoic logician Chrysippus, Łukasiewicz is the most prominent figure in ancient logic studies. A huge literature traces its origins to Łukasiewicz. -/- This Ancient Logic and Its Modern Interpretations, is based on the 1973 Buffalo Symposium on Modernist Interpretations of Ancient Logic, the first conference devoted entirely to critical assessment of the state of ancient logic studies. (shrink)

Logical reflection in early modern philosophy (EMP) is marked by the instability of the period, although it is more lasting (the Port-Royal Logic was nevertheless used as a handbook in philosophy courses until the end of the nineteenth century). It started in the sixteenth century and ended in the nineteenth century, a period of 300 years during which there were deep transformations in the conceptions of authority and scientific method. For the history of twentieth-century philosophy, it was the period (...) of “classical logic,” which lasted from the Renaissance to the linguistic turn conducted by Gottlob Frege. The period was used to be thought of as centuries of little or no original contribution to logic, in which conceptions of logic were tainted by rhetoric, epistemology, and psychologism in the worst sense (Kneale and Kneale 1962; Michael 1997). From the last decades of the twentieth century, however, scholars began to regard this period more accurately with respect to reflection and changes in logic and semantics. It has recently become a promising field for historical and conceptual research; today we can say that the legacy of early modern logical reformism has a philosophical, logical, and semantic value in itself. (shrink)

While there are good reasons to think that Hegel would not engage with modern scepticism in the Science of Logic, this article argues that he nevertheless does so in a way that informs the text's conception of logic as the latter pertains to metaphysics. Hegel engages with modern scepticism's general concerns that philosophy should begin without unexamined presuppositions and should come to attain not only knowledge of truth, but corresponding second-order knowledge: knowledge of knowing truth. These concerns inform (...) two needs that Hegel formulates for first philosophy, which logic—by unifying with metaphysics, which is traditionally synonymous with first philosophy—is to satisfy. However, logic, for the Logic, is unified with metaphysics as a science of absolute knowing, the form of thinking involved in traditional metaphysics. As such, logic, for the Logic, is neither anti-metaphysical nor reducible to metaphysics, but is rather a science of metaphysical thinking, which, for Hegel, includes metaphysics. The article emphasizes how Hegel's construal of logic as a science of absolute knowing avoids running into the ‘swimming problem’ that Hegel raises against, broadly, epistemological forms of first philosophy. (shrink)

This paper intends to explain key differences between Aristotle’s understanding of the relationships between nous, epistêmê, and the art of syllogistic reasoning(both analytic and dialectical) and the corresponding modern conceptions of intuition, knowledge, and reason. By uncovering paradoxa that Aristotle’s understanding of syllogistic reasoning presents in relation to modern philosophical conceptions of logic and science, I highlight problems of a shift in modern philosophy—a shift that occurs most dramatically in the seventeenth century—toward a project of construction, a (...) pervasive desire for rational certainty, and a general insistence on the reducibility of the sciences. The major motivation of this analysis is my intention to show that modern attempts to reduce science/epistêmê to a single science/method of inquiry occlude dialectical and ethico-political dimensions of “reason” and, hence, also impoverish philosophy’s critical capacities. (shrink)

This paper examines the theory of sustainable development presented by Jeffrey Sachs in The Age of Sustainable Development. While Sustainable Development ostensibly seeks to harmonise the conflict between ecological sustainability and human development, the paper argues this is impossible because of the conceptual frame it employs. Rather than allowing for a re-conceptualisation of the human-nature relation, Sustainable Development is simply the latest and possibly last attempt to advance the core idea of western modernity — the notion of self-determination. Drawing upon (...) Hegel's account of historical development it is argued that Sustainable Development and the notion of planetary boundaries cannot break out of a dualism of nature and self-determining agents. (shrink)

This paper examines the theory of sustainable development presented by Jeffrey Sachs in The Age of Sustainable Development. While Sustainable Development ostensibly seeks to harmonise the conflict between ecological sustainability and human development, the paper argues this is impossible because of the conceptual frame it employs. Rather than allowing for a re-conceptualisation of the human-nature relation, Sustainable Development is simply the latest and possibly last attempt to advance the core idea of western modernity — the notion of self-determination. Drawing upon (...) Hegel's account of historical development it is argued that Sustainable Development and the notion of planetary boundaries cannot break out of a dualism of nature and self-determining agents. (shrink)

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

This book is written for those who wish to learn some basic principles of formal logic but more importantly learn some easy methods to unpick arguments and assess their value for truth and validity. -/- The first section explains the ideas behind traditional logic which was formed well over two thousand years ago by the ancient Greeks. Terms such as ‘categorical syllogism’, ‘premise’, ‘deduction’ and ‘validity’ may appear at first sight to be inscrutable but will easily be understood with examples (...) bringing the subjects to life. Traditionally, Venn diagrams have been employed to test arguments. These are very useful but their application is limited and they are not open to quantification. The mid-section of this book introduces a methodology that makes the analysis of arguments accessible with the use of a new form of diagram, modified from those of the mathematician Leonhard Euler. These new diagrammatic methods will be employed to demonstrate an addition to the basic form of syllogism. This includes a refined definition of the terms ‘most’ and ‘some’ within propositions. This may seem a little obscure at the moment but one will readily apprehend these new methods and principles of a more modern logic. (shrink)

In face of persistent and notable efforts taking place in many parts of the world today to make modern logic a tool of legal thought, lawyers are inclined to ask: "What is the real significance of modern logic for us?" A sum mary answer to this question is: "Modern logic provides up-to-date principles and methods for tracing and display ing self-consistent thought, which is indispensable for ef ficient and proper performance of legal tasks." This answer may not (...) satisfy the questioner's curiosity about modern logic in the service of law. If he wants to know more, he is invited to join in an investigation of the various ways in which modern logic can prove useful in legal thought. The present book proposes to engage in such an investi gation, and in doing so it also endeavors to meet various challenges to legal logic. If the value of this logic is totally denied, a reaction to this denial is to ask what its reasonable alternative could possibly be. So far there appears to be no sensible reply to this question. The merit of many specific criticisms of legal logic is indisputable. (shrink)

Logic has attained in our century a development incomparably greater than in any past age of its long history, and this has led to such an enrichment and proliferation of its aspects, that the problem of some kind of unified recom prehension of this discipline seems nowadays unavoidable. This splitting into several subdomains is the natural consequence of the fact that Logic has intended to adopt in our century the status of a science. This always implies that the general optics, (...) under which a certain set of problems used to be con sidered, breaks into a lot of specialized sectors of inquiry, each of them being characterized by the introduction of specific viewpoints and of technical tools of its own. The first impression, that often accompanies the creation of one of such specialized branches in a diSCipline, is that one has succeeded in isolating the 'scientific core' of it, by restricting the somehow vague and redundant generality of its original 'philosophical' configuration. But, after a while, it appears that some of the discarded aspects are indeed important and a new specialized domain of investigation is created to explore them. By follOwing this procedure, one finally finds himself confronted with such a variety of independent fields of research, that one wonders whether the fact of labelling them under a common denomination be nothing but the contingent effect of a pure historical tradition. (shrink)

This chapter contains sections titled: The Key Ingredients of Logic Non‐Deductive Reasoning Forms Plausible Reasoning Induction and Abduction Confirmatory Induction Concluding Remarks.

First published in 1943, and revised for this 1952 edition, this book was intended for use by students of philosophy and as such traditional and modern developments in logic have been combined in a unified treatment. The author envisaged this volume as filling a gap for a simple, introductory text on formal logic, written from a modern point of view, unencumbered by traditional doctrine. This title provides a thorough introduction and grounding in the philosophy of logic, and was (...) later revised after the author’s death to correct a number of logical errors — making this edition the most complete version of the work. (shrink)

This study concentrates on exploring the ‘logic’ of modern universities in China-a working concept that promotes a more in-depth discourse on the implicit illustrations of the ‘logic’ of universiti...

The goal of this paper is to provide an account of the role played by logic in the context of what Husserl names the “crisis of European sciences.” Presupposing the analyses offered in the Krisis, I look at Formale und Transzendentale Logik to demonstrate that the crisis of logic stems from the deviation of its original meaning as a “theory of science” and from its restriction to a mere “theoretical technique.” Through a comparison between Aristotelian syllogistic and modern logic, (...) I show why the modern discovery of the purely formal dimension of knowledge which makes possible such a mathematical technization is a positive achievement that hinders at the same time the disclosure of the truly philosophical nature of logic. The correct appraisal of this ambiguous phenomenon will explain why the rise of modern logic represents a decisive challenge for the success of Husserl’s late phenomenological project. (shrink)

Being mainly concerned with the origins and development of formal logic, current “histories of logic” often devote scarce, if any, space to logic in the early modern period. In standard narratives, emphasis is put, on one side, on Aristotle’s Organon and on the Stoics’ logic of propositions, and on the other side, on the development of mathematical logic from Boole and Frege on. The picture often emerging from such reconstructions represents early modern philosophers—net of their criticisms of Aristotelian (...) syllogism— as largely estranged from the discipline. It is true that, the medieval period’s concern with semantic and inference issues has been accounted... (shrink)

There is a growing understanding that modern slavery is a phenomenon ‘hidden in plain sight’ in the home countries of multinational firms. Yet, business scholarship on modern slavery has so far focussed on product supply chains. To address this, we direct attention to the various institutional pressures on the UK construction industry, and managers of firms within it, around modern slavery risk for on-site labour. Based on a unique data set of 30 in-depth interviews with construction firm (...) managers and directors, we identify two institutional logics as being integral to explaining how these companies have responded to the Modern Slavery Act: a market logic and a state logic. While the institutional logics literature largely assumes that institutional complexity will lead to a conciliation of multiple logics, we find both complementarity and continued conflict in the logics in our study. Though we identify conciliation between aspects of the market logic and the state logic, conflict remains as engagement with actions which could potentially address modern slavery is limited by the trade-offs between the two logics. (shrink)

Scientists, more than ever before, are interested in the logic of science, which has been stimulated by various interrelated circumstances including the development of the postulation method in mathematics, the unprecedented expansion of modern technology, and recent advances in the biological, psychological, and anthropological sciences that have created a demand for more effective theory and system construction. In this book, an attempt is made to free the logic of science from the historical epistemologies and ontologies. The elimination of spiritistic (...) entities enables the logician to study the evolution of scientific constructs from the actual behavior of persons with objects and events. The naturalistic study of scientific work and its systemic products is based upon Interbehavioral Psychology, which demands that the scientist be studied in his various contacts with objects and processes, both in the field and laboratory. (shrink)

Not until the early 1920’s was it possible to distinguish Aristotelian or traditional logic from Aristotle’s own ancient logic. We can now recognize many aspects of his logical investigations that are themselves modern, in the sense that modern logicians are making discoveries that Aristotle had already made or had anticipated. Here we gather five salient features of Aristotle’s logical investigations that reveal a striking philosophical modernity: 1) Aristotle took logic to be that part of epistemology used to establish (...) knowledge of logical consequence; 2) Prior Analytics is a metalogical treatise on the syllogistic deduction system; 3) Aristotle recognized the epistemic efficacy of certain elemental argument patterns, and he explicitly formulated them as rules of natural deduction in corresponding sentences; 4) Prior Analytics is a proof-theoretic treatise in which Aristotle describes a natural deduction system and demonstrates certain of the logical relationships among syllogistic deduction rules ; and finally, 5) Aristotle worked with a notion of substitution sufficient for distinguishing logical syntax and semantics. In this connection he also distinguished validity from deducibility sufficiently well to note the completeness of his logic. (shrink)