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)

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)

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)

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)

The word "modem" in the title of this book refers primarily to post-medieval discussions, but it also hints at those medieval mo dal theories which were considered modem in contradistinction to ancient conceptions and which in different ways influenced philosophical discussions during the early modem period. The me dieval developments are investigated in the opening paper, 'The Foundations of Modality and Conceivability in Descartes and His Predecessors', by Lilli Alanen and Simo Knuuttila. Boethius's works from the early sixth century belonged (...) to the sources from which early medieval thinkers obtained their knowledge of ancient thought. They offered extensive discus sions of traditional modal conceptions the basic forms of which were: (1) the paradigm of possibility as a potency striving to realize itself; (2) the "statistical" interpretation of modal no tions where necessity means actuality in all relevant cases or omnitemporal actuality, possibility means actuality in some rel evant cases or sometimes, and impossibility means omnitemporal non-actuality; and (3) the "logical" definition of possibility as something which, being assumed, results in nothing contradic tory. Boethius accepted the Aristotelian view according to which total possibilities in the first sense must prove their met tle through actualization and possibilities in the third sense are assumed to be realized in our actual history. On these presump tions, all of the above-mentioned ancient paradigms imply the Principle of Plenitude according to which no genuine possibility remains unrealized. (shrink)

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

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)

Chapter One: Constructions. Entities, constructions, and functions When one travels from Los Angeles to New York, going, say, by way of St. Louis, Chicago, ...

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)

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 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...

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)

Review of Karel Lambert, Meinong and the Principle of Independence: Its Place in Meinong's Theory of Objects and Its Significance in Contemporary Philosophical Logic.

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)

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)

This edited volume presents a comprehensive history of modern logic from the Middle Ages through the end of the twentieth century.

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)

Geoffrey Lloyd engages in a wide-ranging exploration of what we can learn from the study of ancient civilizations that is relevant to fundamental problems, both intellectual and moral, that we still face today. These include, in philosophy of science, the question of the incommensurability of paradigms, the debate between realism and relativism or constructivism, and between correspondence and coherence conceptions of truth. How far is it possible to arrive at an understanding of alien systems of belief? Is it possible to (...) talk meaningfully of 'science' and of its various constituent disciplines, 'astronomy' 'geography' 'anatomy' and so on, in the ancient world? Are logic and its laws universal? Is there one ontology - a single world - to which all attempts at understanding must be considered to be directed? When we encounter apparently very different views of reality, how far can that be put down to a difference in conceptions of what needs explaining, or of what counts as an explanation, or to different preferred modes of reasoning or styles of inquiry? Do the notions of truth and belief represent reliable cross-cultural universals? In another area, what can ancient history teach us about today's social and political problems? Are the discourses of human nature and of human rights universally applicable? What political institutions do we need to help secure equity and justice within nation states and between them? Lloyd sets out to answer all these questions, and to argue that the study of the science and culture of ancient Greece and China provided a precious resource in order to advance a wealth of modern debates. (shrink)

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.

This paper deals with Ágnes Heller’s suggestion, in A Theory of Modernity (1999), to ascribe to science a central role in the ongoing development of modernity. As we shall argue, this is not merely a historical issue but, rather, a historical-philosophical one that entails the problem of defining modernity, science and technology and their mutual interconnections. As for modernity, according to Heller, it is a free developmental project without any foundations other than freedom itself. In particular, the evolution of science (...) and technology is one of its main developmental tendencies. Science, she argues, has become the dominant world explanation while technological thought is the corresponding dominant (but not unique) imaginary institution. Her attempt to isolate the essential cultural features of science-technology owes much to Habermas’s analyses of the 1960s on the technocratic developments of contemporary societies legitimized through an ideological employment of science. Like Habermas, Heller embraces an instrumental and problem-solving conception of technology and ‘normal’ science which appears questionable, however, in the light of conclusions on the intrinsic creativity of the science-technology interface deriving from historical epistemology and an externalist history of science. Considerations and examples derived from the recent agenda of historical epistemology could integrate Heller’s philosophical and cultural analysis of the role of science in modernity while challenging some of her assumptions. (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)

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)

This paper is about the putative theoretical virtue of strength, as it might be used in abductive arguments to the correct logic in the epistemology of logic. It argues for three theses. The first is that the well-defined property of logical strength is neither a virtue nor a vice, so that logically weaker theories are not—all other things being equal—worse or better theories than logically stronger ones. The second thesis is that logical strength does not entail the looser characteristic of (...) scientific strength, and the third is that many modern logics are on a par—or can be made to be on a par—with respect to scientific strength. (shrink)

Individualism: The Cultural Logic of Modernity is an edited collection of sixteen essays on the idea of the modern sovereign individual in the western cultural tradition. Reconsidering the eighteenth-century realist novel, twentieth-century modernism, and underappreciated topics on individualism and literature, this volume provocatively revises and enriches our understanding of individualism as the generative premise of modernity itself.

The use of the primitive notion of mereological fusion (also known as composition and sum) has been considered by various philosophers and logicians, including Aristotle, G. Leibniz, S. Leśniewski, K. Fine, J. Ketland, T. Schindler, and S. Kleishmid. The problem of finding an axiomatization of Classical Mereology with primitive fusion, instead of the primitive notion of being a part, is quite old and was formally considered by C. Lejewski. Lejewski somehow axiomatized classical mereology using primitive fusion (1962, and also later (...) in 1984). However, his mereology is formulated in a very rich nonclassical language of Leśniewski’s ontology with the essential use of quantification over function symbols, including function symbols “part of” and “sum of.” Lejewski’s approach is not expressible in modern mereologies, and we do not use it in any way. We prove that classical mereology is axiomatizable using primitive mereological fusion within the framework of two-sorted logic. The theory presented here is the first contemporary axiomatization of classical mereology with primitive mereological fusion. (shrink)

First published in 1956 The Language of Modern Physics gives a complete account of the concepts both of classical and quantum physics. It deals with themes like logic and semantics; basic ideas of physics and the methods scientists use for confirming their hypotheses.

History and Philosophy of Modern Mathematics was first published in 1988. Minnesota Archive Editions uses digital technology to make long-unavailable books once again accessible, and are published unaltered from the original University of Minnesota Press editions. The fourteen essays in this volume build on the pioneering effort of Garrett Birkhoff, professor of mathematics at Harvard University, who in 1974 organized a conference of mathematicians and historians of modern mathematics to examine how the two disciplines approach the history of (...) mathematics. In History and Philosophy of Modern Mathematics, William Aspray and Philip Kitcher bring together distinguished scholars from mathematics, history, and philosophy to assess the current state of the field. Their essays, which grow out of a 1985 conference at the University of Minnesota, develop the basic premise that mathematical thought needs to be studied from an interdisciplinary perspective. The opening essays study issues arising within logic and the foundations of mathematics, a traditional area of interest to historians and philosophers. The second section examines issues in the history of mathematics within the framework of established historical periods and questions. Next come case studies that illustrate the power of an interdisciplinary approach to the study of mathematics. The collection closes with a look at mathematics from a sociohistorical perspective, including the way institutions affect what constitutes mathematical knowledge. (shrink)

Whatever its merits and difficulties, the concept of logic embedded in much of the "new philosophy" of the early modern period was then understood to supplant contemporary views of formal logic. The notion of compiling a natural history of the understanding constituted the basis of this new concept of logic. The following paper attempts to trace this view of logic through some of the major and numerous minor texts of the period, centering on the development and influence of John (...) Locke's understanding of the analysis of the cognitive faculties as the discipline of logic. (shrink)