This brief paperback is designed for symbolic/formal logic courses. It features the tree method proof system developed by Jeffrey. The new edition contains many more examples and exercises and is reorganized for greater accessibility.

This book was designed primarily as a textbook; though the author hopes that it will prove to be of interests to others beside logic students. Part I of this book covers the fundamentals of the subject the propositional calculus and the theory of quantification. Part II deals with the traditional formal logic and with the developments which have taken that as their starting-point. Part III deals with modal, three-valued, and extensional systems.

Not focusing on the history of classical logic, this book provides discussions and quotes central passages on its origins and development, namely from a philosophical perspective. Not being a book in mathematical logic, it takes formal logic from an essentially mathematical perspective. Biased towards a computational approach, with SAT and VAL as its backbone, this is an introduction to logic that covers essential aspects of the three branches of logic, to wit, philosophical, mathematical, and (...) computational. (shrink)

This work has been selected by scholars as being culturally important and is part of the knowledge base of civilization as we know it. This work is in the public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be (...) preserved, reproduced, and made generally available to the public. To ensure a quality reading experience, this work has been proofread and republished using a format that seamlessly blends the original graphical elements with text in an easy-to-read typeface. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

After critiquing the arguments against using formal logic to teach critical thinking, this paper argues that for theoretical, practical, and empirical reasons, instruction in the fundamentals of formal logic is essential for critical thinking, and so should be included in every class that purports to teach critical thinking.

This paper presents and illustrates a formal logic for the abduction of singular hypotheses. The logic has a semantics and a dynamic proof theory that is sound and complete with respect to the semantics. The logic presupposes that, with respect to a specific application, the set of explananda and the set of possible explanantia are disjoint . Where an explanandum can be explained by different explanantia, the logic allows only for the abduction of their disjunction.

The wider topic to which the content of this paper belongs is that of the relationship between formal logic and real argumentation. Of particular potential interest in this connection are held to be substantive arguments constructed by philosophers reputed equally as authorities in logical theory. A number of characteristics are tentatively indicated by the author as likely to be encountered in such arguments. The discussion centers afterwards, by way of specification, on a remarkable piece of argument quoted in (...) Cicero’s dialog On Divination and ascribed to Stoic thinkers. The Stoics’ formal theory of inference is summarily referred to in this context, with special emphasis on their basic deductive schemata (‘indemonstrables’), some of them recognizable as links in the overall structure of the quoted argument. The main lines of Cicero’s criticism of the Stoic argument are next commented upon, with emphasis on his implied view as to the requirements of a good argument. Towards the end of the paper, a few considerations are added on the changes in the prevailing style of argumentation conspicuous in the three famous Roman Stoics. (shrink)

_Formal Logic_ is an undergraduate text suitable for introductory, intermediate, and advanced courses in symbolic logic. The book’s nine chapters offer thorough coverage of truth-functional and quantificational logic, as well as the basics of more advanced topics such as set theory and modal logic. Complex ideas are explained in plain language that doesn’t presuppose any background in logic or mathematics, and derivation strategies are illustrated with numerous examples. Translations, tables, trees, natural deduction, and simple meta-proofs are (...) taught through over 400 exercises. A companion website offers supplemental practice software and tutorial videos. (shrink)

Classical logic yields counterintuitive results for numerous propositional argument forms. The usual alternatives (modal logic, relevance logic, etc.) generate counterintuitive results of their own. The counterintuitive results create problems—especially pedagogical problems—for informal logicians who wish to use formal logic to analyze ordinary argumentation. This paper presents a system, PL– (propositional logic minus the funny business), based on the idea that paradigmatic valid argument forms arise from justificatory or explanatory discourse. PL– avoids the pedagogical difficulties (...) without sacrificing insight into argument. (shrink)

The first beginning logic text to employ the tree method--a complete formal system of first-order logic that is remarkably easy to understand and use--this text allows students to take control of the nuts and bolts of formal logic quickly, and to move on to more complex and abstract problems. The tree method is elaborated in manageable steps over five chapters, in each of which its adequacy is reviewed; soundness and completeness proofs are extended at each (...) step, and the decidability proof is extended at the step from truth functions to the logic of nonoverlapping quantifiers with a single variable, after which undecidability is demonstrated by example. The first three chapters are bilingual, with arguments presented twice, in logical notation and in English. The last three chapters consider the discoveries defining the scope and limits of formal methods that marked logic’s coming of age in the 20th century: Godel’s completeness and incompleteness theorems for first and second-order logic, and the Church-Turing theorem on the undecidability of first-order logic. This new edition provides additional problems, solutions to selected problems, and two new Supplements: Truth-Functional Equivalence reinstates material on that topic from the second edition that was omitted in the third, and Variant Methods, in which John Burgess provides a proof regarding the possibility of modifying the tree method so that it will always find a finite model when there is one, and another, which shows that a different modification—once contemplated by Jeffrey--can result in a dramatic speed--up of certain proofs. (shrink)

In this paper I try to answer four basic questions: (1) How the concept of God is to be represented? (2) Are there any logical principles governing it? (3) If so, what kind of logic lies behind them? (4) Can there be a logic of the concept of God? I address them by presenting a formal-logical account to the concept of God. I take it as a methodological desideratum that this should be done within the simplest existing (...) logical formalism. I start with first-order logic (FOL) with identity, and then show that its simplest modal extension (SQML, or the simplest quantified modal logic) is enough for us to formalize a minimally satisfactory theory of the concept of God. I focus exclusively on the monotheistic concept of God. (shrink)

Formal logic provides us with a powerful set of techniques for criticizing some arguments and showing others to be valid. These techniques are relevant to all of us with an interest in being skilful and accurate reasoners. In this highly accessible book, Peter Smith presents a guide to the fundamental aims and basic elements of formal logic. He introduces the reader to the languages of propositional and predicate logic, and then develops formal systems for (...) evaluating arguments translated into these languages, concentrating on the easily comprehensible 'tree' method. His discussion is richly illustrated with worked examples and exercises. A distinctive feature is that, alongside the formal work, there is illuminating philosophical commentary. This book will make an ideal text for a first logic course, and will provide a firm basis for further work in formal and philosophical logic. (shrink)

Many texts on logic are written with a mathematical emphasis, and focus primarily on the development of a formal apparatus and associated techniques. In other, more philosophical texts, the topic is often presented as an indulgent collection of musings on issues for which technical solutions have long since been devised. What has been missing until now is an attempt to unite the motives underlying both approaches. Paul Hoyningen-Huene’s Formal Logic seeks to find a balance between the (...) necessity of formal considerations and the importance of full reflection and explanation about the seemingly arbitrary steps that occasionally confound even the most serious student of logic. Alex Levine’s artful translation conveys both the content and style of the German edition. Filled with examples, exercises, and a straightforward look at some of the most common problems in teaching the subject, this work is eminently suitable for the classroom. (shrink)

"Logic", one of the central words in Western intellectual history, compre hends in its meaning such diverse things as the Aristotelian syllogistic, the scholastic art of disputation, the transcendental logic of the Kantian critique, the dialectical logic of Hegel, and the mathematical logic of the Principia Mathematica of Whitehead and Russell. The term "Formal Logic", following Kant is generally used to distinguish formal logical reasonings, precisely as formal, from the remaining universal truths (...) based on reason. (Cf. SCHOLZ, 1931). A text-book example of a formal-logical inference which from "Some men are philosophers" and "All philosophers are wise" concludes that "Some men are wise" is called formal, because the validity of this inference depends only on the form ofthe given sentences -in particular it does not depend on the truth or falsity of these sentences. (On the dependence of logic on natural language, English, for example, compare Section 1 and 8). The form of a sentence like "Some men are philosophers", is that which remains preserved when the given predicates, here "men" and "philosophers" are replaced by arbitrary ones. The form itself can thus be represented by replacing the given predicates by variables. Variables are signs devoid of meaning, which may serve merely to indicate the place where meaningful constants (here the predicates) are to be inserted. As variables we shall use - as did Aristotle - letters, say P, Q and R, as variables for predicates. (shrink)

Central topics in medieval logic are here treated in a way that is congenial to the modern reader, without compromising historical reliability. The achievements of medieval logic are made available to a wider philosophical public then the medievalists themselves. The three genres of logica moderna arising in a later Middle Ages are covered: obligations, insolubles and consequences - the first time these have been treated in such a unified way. The articles on obligations look at the role of (...) logical consistence in medieval disputation techniques. Those on insolubles concentrate on medieval solutions to the Liar Paradox. There is also a systematic account of how medieval authors described the logical content of an inference, and how they thought that the validity of an inference could be guaranteed. (shrink)

This work brings together three themes whose relationship has gone unexplored in the recent literature of philosophy: the transcendental phenomenology of Edmund Husserl, the phenomenological ontology of Martin Heidegger and the discipline of logic, especially formal logic. Part One and Two of the work present a detailed explication of Husserl's and Heidegger's philosophy of logic which are respectively characterized as an archeology of logic based upon transcendental phenomenological criticism and a radical phenomenology of logic (...) based upon phenomenological criticism which issues from a concern with the question of the meaning of Being. Part Three, then, offers a comparative analysis combined with a critical appraisal of these two philosophies of logic in order to achieve a more penetrating understanding of the relationship between Husserl and Heidegger. Thus, in one sense, this work takes the domain of logic and the varying approaches to it by Husserl and Heidegger as a clue to the similarities and dissimilarities within the thinking of these two phenomenologists, especially with regard to the understanding each has of the goal and method of phenomenology, and the results proper to phenomenological reflection. (shrink)

“Logic” nominally belongs to the classical trivium, the common ground, the crossroads of traditional liberal education, through which all educated persons would travel. But what sort of “logic” should or could fill that role today? Many teachers of logic today feel pulled in what seem two different directions: towards the more apparently practical utility of the emerging “informal logic” agenda; and towards the more apparently rigorous canon of formal logic, be it deductive or inductive. (...) These alternatives are neither mutually exclusive nor exhaustive of the possibilities of what might constitute an organon for the liberally educated person today. I will review some salient approaches to teaching logic, and some salient issues attending those approaches, focusing on issues regarding the utility of formal deductive logic in modern symbolic guise. (shrink)

In addition to a somewhat detailed exposition of certain portions of what may be called the book-work of formal logic, the following pages contain a number of problems worked out in detail and unsolved problems, by means of which the student may test his command over logical processes. In the expository portions of Parts I, II, and III, dealing respectively with terms, propositions, and syllogisms, the traditional lines are in the main followed, though with certain modifications; e.g., in (...) the systematisation of immediate inferences, and in several points of detail in connexion with the syllogism. For purposes of illustration Euler's diagrams are employed to a greater extent than is usual in English manuals. In Part IV, which contains a generalisation of logical processes in their application to complex inferences, a somewhat new departure is taken. So far as I am aware this part constitutes the first systematic attempt that has been made to deal with formal reasonings of the most complicated character without the aid of mathematical or other symbols of operation, and without abandoning the ordinary non-equational or predicative form of proposition. This attempt has on the whole met with greater success than I had anticipated; and I believe that the methods formulated will be found to be both as easy and as effective as the symbolical methods of Boole and his followers. The book concludes with a general and sure method of solution of what Professor Jevons called the inverse problem, and which he himself seemed to regard as soluble only by a series of guesses. (shrink)

Perfect for students with no background in logic or philosophy, Simple Formal Logic provides a full system of logic adequate to handle everyday and philosophical reasoning. By keeping out artificial techniques that aren’t natural to our everyday thinking process, Simple Formal Logic trains students to think through formal logical arguments for themselves, ingraining in them the habits of sound reasoning. Simple Formal Logic features: a companion website with abundant exercise worksheets, study (...) supplements (including flashcards for symbolizations and for deduction rules), and instructor’s manual two levels of exercises for beginning and more advanced students a glossary of terms, abbreviations and symbols. This book arose out of a popular course that the author has taught to all types of undergraduate students at Loyola University Chicago. He teaches formal logic without the artificial methods–methods that often seek to solve farfetched logical problems without any connection to everyday and philosophical argumentation. The result is a book that teaches easy and more intuitive ways of grappling with formal logic–and is intended as a rigorous yet easy-to-follow first course in logical thinking for philosophy majors and non-philosophy majors alike. (shrink)

This paper addresses the question of the relationship between the object of formal logic and the object of ontology. The history of logic and philosophy shows a kinship and overlapping between the two sciences. The analyses were conducted on the basis of three approaches to formal logic, i.e. Aristotle’s logic Rus­sell’s and Whitehead’s logic, and Leśniewski’s logic. At the same time, it sought to grasp its material and formal object. Now with (...) regard to ontology mainly Aristotelian and Leibnizean understanding of ontology was taken into account as an instance of philosophical ontology and set theory ontology as an example of ontology constructed with a view to play the role of a semantic background for the formulas of logical calculi. The considerations under study allow us to state that even if material objects in the two sciences overlap, especially in the case when the formulas of logic are interpreted by way of ontology, their formal objects seem to be different. Logic does not focus on the question of the cate­gorisation of reality; the essences belonging to respective ontological categories are only “truth-making” factors for the statements formulated by formal logic. (shrink)

Excerpt from A History of Formal Logic In this edition of the most considerable history Of formal logic yet published, the Opportunity has Of course been taken to make some adjustments seen to be necessary in the original, with the author's full concurrence. Only in 36, however, has the numeration of cited passages been altered owing to the introduction of new matter. Those changes are as follows. About the Publisher Forgotten Books publishes hundreds of thousands of (...) rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works. (shrink)

... and a reading knowledge of formal logical symbolism is essential too. (Philosophers often use bits of logical symbolism to clarify their arguments.) Because the artificial and simply formal languages of logic give us highly illuminating objects of comparison when we come thinking about how natural languages work. (Relevant to topics in ‘philosophical logic’ and the philosophy of language.) But mainly because it us the point of entry into the study of one of the major intellectual (...) achievements by philosophers of the 20th – i.e. the development of mathematical logic. (Not least in Cambridge: Russell and Whitehead, Ramsey, Turing.). (shrink)

This paper presents a way in which formal logic can be understood and reformulated in terms of argumentation that can help us unify formal and informal reasoning. Classical deductive reasoning will be expressed entirely in terms of notions and concepts from argumentation so that formal logical entailment is equivalently captured via the arguments that win between those supporting concluding formulae and arguments supporting contradictory formulae. This allows us to go beyond Classical Logic and smoothly connect (...) it with human reasoning, thus providing a uniform argumentation-based view of both informal and formal logic. (shrink)

Buddhist Formal Logic. A study of Dignaga's Hetucakra and K'uei-chi's Great Commentary on the Nyayapravesa. R.S.Y. Chi. Royal Asiatic Society, London 1969; revised edition, Motilal Banarsidass, Delhi 1984. lxxxii + 222 pp. Rs. 100.

Rigorous yet engaging and accessible, Introduction to Formal Logic with Philosophical Applications is composed of two parts. The first part provides a focused, "nuts-and-bolts" introduction to formal deductive logic that covers syntax, semantics, translation, and natural deduction forpropositional and predicate logics. The second part presents student-friendly essays on logic and its applications in philosophy and beyond, with writing prompts and suggestions for further reading.

This work is primarily an interpretation of Indian Logic preserved in China.

This book was designed primarily as a textbook; though the author hopes that it will prove to be of interests to others beside logic students. Part I of this book covers the fundamentals of the subject the propositional calculus and the theory of quantification. Part II deals with the traditional formal logic and with the developments which have taken that as their starting-point. Part III deals with modal, three-valued, and extensional systems.

Originally published in 1966. This is a self-instructional course intended for first-year university students who have not had previous acquaintance with Logic. The book deals with "propositional" logic by the truth-table method, briefly introducing axiomatic procedures, and proceeds to the theory of the syllogism, the logic of one-place predicates, and elementary parts of the logic of many-place predicates. Revision material is provided covering the main parts of the course. The course represents from eight to twenty hours (...) work. depending on the student's speed of work and on whether optional chapters are taken. (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)

“Formal logic”, an expression created by Kant to characterize Aristotelian logic, has also been used as a name for modern logic, originated by Boole and Frege, which in many aspects differs radically from traditional logic. We shed light on this paradox by distinguishing in this paper five different meanings of the expression “formal logic”: (1) Formal reasoning according to the Aristotelian dichotomy of form and content, (2) Formal logic as a (...) formal science by opposition to an empirical science, (3) Formal systems in the sense of Hilbert, Curry and the formalist school, (4) Symbolic logic, a science using symbols, such as Venn diagrams, (5) Mathematical logic, a mathematical approach to reasoning. We argue that these five meanings are independent and that the meaning (5) is the one which better characterized modern logic, which should therefore not be called “formal logic”. (shrink)