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)
Van Heijenoort’s account of the historical development of modern logic was composed in 1974 and first published in 1992 with an introduction by his former student. What follows is a new edition with a revised and expanded introduction and additional notes.
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)
Comparing the three-form reasoning of new Hetu-vidya with Western logic, scholars have put forward four perspectives. Combining their strengths and shortcomings, and the examples of Hetu-vidya reasoning, we can conclude that the three-form reasoning should have four forms: (1) the affirmative expression of formal implication; (2) the modus ponens of hypothetical reasoning concerning sufficient conditions after universal instantiation; (3) the negative expression of a formal implication; and (4) the modus tollens of hypothetical reasoning concerning sufficient conditions after universal instantiation.
I use van Heijenoort’s published writings and manuscript materials to provide a comprehensive overview of his conception of modern logic as a first-order functional calculus and of the historical developments which led to this conception of mathematical logic, its defining characteristics, and in particular to provide an integral account, from his most important publications as well as his unpublished notes and scattered shorter historico-philosophical articles, of how and why the mathematical logic, whose he traced to Frege and the culmination (...) of its formative period in the incompleteness results of Gödel, became modern logic, as distinct from the traditional logic of Aristotle, and why and how the logistic tradition that led from Frege through Russell, rather than the algebraic tradition that led from De Morgan and Boole through Peirce and Schröder, came, in his view, to define modern logic. (shrink)
This paper aims to outline an analysis and interpretation of the process that led to First-Order Logic and its consolidation as a core system of modern logic. We begin with an historical overview of landmarks along the road to modern logic, and proceed to a philosophical discussion casting doubt on the possibility of a purely rational justification of the actual delimitation of First-Order-Logic. On this basis, we advance the thesis that a certain historical tradition was essential to the (...) emergence of modern logic; this traditional context is analyzed as consisting in some guiding principles and, particularly, a set of exemplares (i.e., paradigmatic instances). Then, we proceed to interpret the historical course of development reviewed in section 1, which can broadly be described as a two-phased movement of expansion and then restriction of the scope of logical theory. We shall try to pinpoint ambivalencies in the process, and the main motives for subsequent changes. Among the latter, one may emphasize the spirit of modern axiomatics, the situation of foundational insecurity in the 1920s, the resulting desire to find systems well-behaved from a proof-theoretical point of view, and the metatheoretical results of the 1930s. Not surprisingly, the mathematical and, more specifically, the foundational context in which Firs-Order-Logic matured will be seen to have played a primary role in its shaping. (shrink)
Philip R. Shields shows that ethical and religious concerns inform even the most technical writings on logic and language, and that, for Wittgenstein, the need to establish clear limitations is both a logical and an ethical demand. Rather than merely saying specific things about theology and religion, major texts from the Tractatus to the Philosophical Investigations express their fundamentally religious nature by showing that there are powers which bear down upon and sustain us. Shields finds a religious view of the (...) world at the very heart of Wittgenstein's philosophy. "Shields argues that the appearance throughout Wittgenstein's writings of such concepts as ritual, limit, transgression, a change of will, pride, temptation, and judgment implies a relation between religion and the logical aspects of Wittgenstein's philosophy."-- Choice "Of the many recent books about Wittgenstein, Logic and Sin is one of the very few that are well worth having"--Fergus Kerr, Modern Theology "What Shields has uncovered in Wittgenstein's religious sensibility is something genuine and profound. . . . Shields has not just written an important book on Wittgenstein but an enlightening work that invites further reflection."--Eric O. Springsted, Cross Currents. (shrink)
The mnemonic arts and the idea of a universal language that would capture the essence of all things were originally associated with cryptology, mysticism, and other occult practices. And it is commonly held that these enigmatic efforts were abandoned with the development of formal logic in the seventeenth century and the beginning of the modern era. In his distinguished book, Logic and the Art of Memory Italian philosopher and historian Paolo Rossi argues that this view is belied by an (...) examination of the history of the idea of a universal language. Based on comprehensive analyses of original texts, Rossi traces the development of this idea from late medieval thinkers such as Ramon Lull through Bruno, Bacon, Descartes, and finally Leibniz in the seventeenth century. The search for a symbolic mode of communication that would be intelligible to everyone was not a mere vestige of magical thinking and occult sciences, but a fundamental component of Renaissance and Enlightenment thought. Seen from this perspective, modern science and combinatorial logic represent not a break from the past but rather its full maturity. Available for the first time in English, this book (originally titled Clavis Universalis ) remains one of the most important contributions to the history of ideas ever written. In addition to his eagerly anticipated translation, Steven Clucas offers a substantial introduction that places this book in the context of other recent works on this fascinating subject. A rich history and valuable sourcebook, Logic and the Art of Memory documents an essential chapter in the development of human reason. (shrink)
Proof and Disproof in Formal Logic is a lively and entertaining introduction to formal logic providing an excellent insight into how a simple logic works. Formal logic allows you to check a logical claim without considering what the claim means. This highly abstracted idea is an essential and practical part of computer science. The idea of a formal system-a collection of rules and axioms, which define a universe of logical proofs-is what gives us programming languages and modern-day programming. This (...) book concentrates on using logic as a tool: making and using formal proofs and disproofs of particular logical claims. The logic it uses-natural deduction-is very small and very simple; working with it helps you see how large mathematical universes can be built on small foundations. The book is divided into four parts: Part I "Basics" gives an introduction to formal logic with a short history of logic and explanations of some technical words. Part II "Formal Syntactic Proof" show you how to do calculations in a formal system where you are guided by shapes and never need to think about meaning. Your experiments are aided by Jape, which can operate as both inquisitor and oracle. Part III "Formal Semantic Disproof" shows you how to construct mathematical counterexamples to shoe that proof is impossible. Jape can check the counterexamples you build. Part IV " Program Specification and Proof" describes how to apply your logical understanding to a real computer science problem, the accurate description and verification of programs. Jape helps, as far as arithmetic allows. Aimed at undergraduates and graduates in computer science, logic, mathematics and philosophy, the text includes reference to and exercises based on the computer software package Jape, an interactive teaching and research tool designed and hosted by the author that is freely available on the web. (shrink)
This is the first comprehensive study of the early modern logic of ideas. It is also a profound contribution to our understanding of the interaction between Aristotelianism and new philosophy and between rationalism and empiricism.
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 article investigates the nature of Aristotelian syllogistics and shows that the categorical syllogism is fundamentally about showing the connection, in the premises of the syllogism, between the major and minor terms as stated in the conclusion. It discusses how this is important for the use of the syllogism in scientific demonstration. The article then examines modern deductive logic with an eye to they way in which it contrasts with Aristotelian syllogistics. It shows howmodern logic is about making necessary (...) connections between each proposition by means of external or second order rules. In the syllogism, on the other hand, the necessity between the premises as a whole unit and the conclusion is based on the internal middle term. The article concludes with a discussion of Günther Patzig’s claim that Aristotelian syllogisms are best thought of as tautological propositions. If this were the case, then the differences asserted to exist between syllogistic and modern logic would not hold. However, it is shown that Patzig’s assimilation of syllogistics to modern deductive logic is illegitimate. (shrink)
Philotheus boehner's "medieval logic" gives the impression that medieval supposition theory and modern quantification theory agree on their interpretation of particular propositions but differ on their interpretation of universal propositions. Matthews shows that this impression is mistaken: they differ on both particular and universal propositions, And the basic reason is that the medievals quantify over terms while modern logicians quantify over variables. (staff).
Hilbert’s unpublished 1917 lectures on logic, analyzed here, are the beginning of modern metalogic. In them he proved the consistency and Post-completeness (maximal consistency) of propositional logic -results traditionally credited to Bernays (1918) and Post (1921). These lectures contain the first formal treatment of first-order logic and form the core of Hilbert’s famous 1928 book with Ackermann. What Bernays, influenced by those lectures, did in 1918 was to change the emphasis from the consistency and Post-completeness of a logic to (...) its soundness and completeness: a sentence is provable if and only if valid. By 1917, strongly influenced by PM, Hilbert accepted the theory of types and logicism -a surprising shift. But by 1922 he abandoned the axiom of reducibility and then drew back from logicism, returning to his 1905 approach of trying to prove the consistency of number theory syntactically. (shrink)
Many historians and philosophers of logic have claimed that during the modern classical era there was a long period of stagnation or even of decline in the field of logic. The aim of this paper is to convince the audience that this standard evaluation of the development of modern logic during the period from Leibniz to Frege is misdirected and needs to be corrected. Even though it is true that the now usual way of understanding logic merely as (...) the doctrine of syntax and semantics of explicit languages would not have appealed even to most 19th century logicians, it is still not the case that there is nothing worth discussing with regard to the development of logic during the modern classical period. The algebraic period culminated with Schroder's contribution and neither Herbartian formal logic nor Trendelenburg's critical epistemology aroused much interest among the 20th century mathematical logicians and analytic philosophers. Nevertheless, the development of symbolic logic can only be understood properly by relating its emergence to the immediately preceding philosophically-oriented discussion about the reform of logic. (shrink)
The author considers the question of whether modern mathematical logic is an adequate framework for the explication and formalization of the kind of reasoning which occurs in everyday life in society, as well as with regard to the kind of more refined reasoning that is represented by the social scientist. (staff).
This clear, accessible account of Hegelian logic makes a case for its enormous seductiveness, its surprising presence in the collective consciousness, and the dangers associated therewith. Offering comprehensive coverage of Hegel's important works, Bencivenga avoids getting bogged down in short-lived scholarly debates to provide a work of permanent significance and usefulness.
Kant's views on logic and logical theory play an important role in his critical writings, especially the Critique of Pure Reason. However, since he published only one short essay on the subject, we must turn to the texts derived from his logic lectures to understand his views. The present volume includes three previously untranslated transcripts of Kant's logic lectures: the Blumberg Logic from the 1770s; the Vienna Logic (supplemented by the recently discovered Hechsel Logic) from the early 1780s; and the (...) Dohna-Wundlacken Logic from the early 1790s. Also included is a new translation of the Jasche Logic, compiled at Kant's request and published in 1800 but which also appears to stem in part from a transcript of his lectures. Together these texts provide a rich source of evidence for Kant's evolving views on logic, on the relations between logic and other disciplines, and on a variety of topics (e.g. analysis and synthesis) central to Kant's mature philosophy. They also provide a portrait of Kant as lecturer, a role in which he was both popular and influential. This volume contains substantial editorial apparatus: a general introduction, linguistic and factual notes, glossaries of key terms (both German/English and English/German) and concordances relating Kant's lectures to Georg Frederich Meier's Excerpts from the Doctrine of Reason, the book on which Kant lectured throughout his life and in which he left extensive notes. (shrink)
John Dewey is celebrated for his work in the philosophy of education and acknowledged as a leading proponent of American pragmatism. His philosophy of logic, on the other hand, is largely unheard of. In Dewey's New Logic, Burke analyzes portions of the debate between Dewey and Bertrand Russell that followed the 1938 publication of Dewey's Logic: The Theory of Inquiry. Burke shows how Russell failed to understand Dewey, and how Dewey's philosophy of logic is centrally relevant to contemporary developments in (...) philosophy and cognitive science. Burke demonstrates that Russell misunderstood crucial aspects of Dewey's theory and contends that logic today, having progressed well beyond Russell's early views, is approaching Dewey's broader perspective. -/- "[This] book should be of substantial interest not only to Dewey scholars and other historians of twentieth-century philosophy, but also to devotees of situation theory, formal semantics, philosophy of mind, cognitive science, and Artificial Intelligence."--Georges Dicker, Transactions of the C.S. Peirce Society "No scholar, thus far, has offered such a sophisticated and detailed version of central themes and contentions in Dewey's Logic . This is a pathbreaking study."--John J. McDermott, editor of The Philosophy of John Dewey. (shrink)
Introduction -- Entering the gallery : Hegel's overall project and the project of the logic -- The skepticism of Hume and Kant -- Reason overgrasps reality -- Essential, necessary universals -- Reason drives itself : semantics and syntax -- Hegel's argument -- Hegel's overall project -- The conceptual and semantic project of the logic -- The syntactic project of the logic -- Introduction -- The doctrine of quality -- The doctrine of quantity -- The doctrine of measure -- Wrap up (...) being : comments on syntax -- Introduction -- Essence as the ground of existence -- The doctrine of appearance -- The doctrine of actuality -- Wrap up essence : comments on syntax -- Introduction -- The doctrine of the object -- The doctrine of the idea -- Wrap up concept : comments on syntax -- Epilogue: Hegel's materialism, optimism, and faith. (shrink)
This is the second edition of an important introduction to Leibniz's philosophy of logic and language first published in 1972. It takes issue with several traditional interpretations of Leibniz (by Russell amongst others) while revealing how Leibniz's thought is related to issues of great interest in current logical theory. For this new edition, the author has added new chapters on infinitesimals and conditionals as well as taking account of reviews of the first edition.