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In addition to a somewhat detailed exposition of certain portions of what may be called the book-work offormallogic, 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 withformal 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)

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Proof and Disproof inFormalLogic is a lively and entertaining introduction toformallogic providing an excellent insight into how a simplelogic works.Formallogic 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 aformal 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 usinglogic as a tool: making and usingformal proofs and disproofs of particular logical claims. Thelogic 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 toformallogic with a short history oflogic and explanations of some technical words. Part II "Formal Syntactic Proof" show you how to do calculations in aformal 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)

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This book aids in the rehabilitation of the wrongfully deprecated work of William Parry, and is the only full-length investigation into Parry-type propositional logics. A central tenet of the monograph is that the sheer diversity of the contexts in which the mereological analogy emerges – its effervescence with respect to fields ranging from metaphysics to computer programming – provides compelling evidence that the study of logics of analytic implication can be instrumental in identifying connections between topics that would otherwise remain (...) hidden. More concretely, the book identifies and discusses a host of cases in which analytic implication can play an important role in revealing distinct problems to be facets of a larger, cross-disciplinary problem. It introduces an element of constancy and cohesion that has previously been absent in a regrettably fractured field, shoring up those who are sympathetic to the worth of mereological analogy. Moreover, it generates new interest in the field by illustrating a wide range of interesting features present in such logics – and highlighting these features to appeal to researchers in many fields. (shrink)

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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 oflogic, especiallyformallogic. Part One and Two of the work present a detailed explication of Husserl's and Heidegger's philosophy oflogic which are respectively characterized as an archeology oflogic based upon transcendental phenomenological criticism and a radical phenomenology oflogic (...) 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 oflogic in order to achieve a more penetrating understanding of the relationship between Husserl and Heidegger. Thus, in one sense, this work takes the domain oflogic 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)

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The first beginninglogic text to employ the tree method--a completeformal system of first-orderlogic that is remarkably easy to understand and use--this text allows students to take control of the nuts and bolts offormallogic 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 thelogic 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 offormal methods that markedlogic’s coming of age in the 20th century: Godel’s completeness and incompleteness theorems for first and second-orderlogic, and the Church-Turing theorem on the undecidability of first-orderlogic. 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)

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"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 transcendentallogic of the Kantian critique, the dialecticallogic of Hegel, and the mathematicallogic of the Principia Mathematica of Whitehead and Russell. The term "FormalLogic", following Kant is generally used to distinguishformal logical reasonings, precisely asformal, from the remaining universal truths (...) based on reason. (Cf. SCHOLZ, 1931). A text-book example of aformal-logical inference which from "Some men are philosophers" and "All philosophers are wise" concludes that "Some men are wise" is calledformal, 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 oflogic 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)

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This paper addresses the question of the relationship between the object offormallogic and the object of ontology. The history oflogic and philosophy shows a kinship and overlapping between the two sciences. The analyses were conducted on the basis of three approaches toformallogic, i.e. Aristotle’slogic Rus­sell’s and Whitehead’slogic, and Leśniewski’slogic. At the same time, it sought to grasp its material andformal 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 oflogic are interpreted by way of ontology, theirformal 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 byformallogic. (shrink)

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The aim of the text is not so much to stress the importance of Peirce'sformal contributions to the semantic view informallogic as to argue that Peirce's semantic trend is part and parcel of his semiotic treatment of a general theory of meaning, understanding, and interpretation, a theory of how signs function which enables him to classify different sorts of signs in a natural way.

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Apart from the knowledge of specialists in the field of Islamic law, the perception of Islamic law, for the most part, is that of a law with no structure and, therefore, no logical consistency. This depiction is often theorized by using the term Kadijustiz. In this essay, I seek to undermine this view and propose that Islamic legal theorists developed a systematic process of employingformallogic in Islamic law. I argue that these arguments exist, were negotiated regarding (...) the nuances of their form and modes of applicability, and were utilized in concrete cases in books of Islamic legal theory. Byformallogic, I am referring to inferential forms of reasoning. The paper is divided into two main parts: first, I describe the notion of Kadijustiz and offer objections to this view. Second, I explain the primary arguments that fall under the category of qiyas. These include, broadly, deduction, induction, and legal analogy. My hope is that this essay contributes to the appreciation of Islamic Law as a rigorous legal system and serves as a reference for some of the modes of thinking and processes of legal reasoning when engaging with Islamic law. (shrink)

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Evidence is given that implication (and its special case, negation) carry the logical strength of a system offormallogic. This is done by proving normalization and cut elimination for a system based on combinatorylogic or λ-calculus with logical constants for and, or, all, and exists, but with none for either implication or negation. The proof is strictly finitary, showing that this system is very weak. The results can be extended to a "classical" version of the (...) system. They can also be extended to a system with a restricted set of rules for implication: the result is a system of intuitionistic higher-order BCKlogic with unrestricted comprehension and without restriction on the rules for disjunction elimination and existential elimination. The result does not extend to the classical version of the BCKlogic. (shrink)

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

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This volume presents 38 classic texts informal epistemology, and strengthens the ties between research into this area of philosophy and its neighbouring intellectual disciplines. The editors provide introductions to five subsections: Bayesian Epistemology, Belief Change, Decision Theory, Interactive Epistemology and EpistemicLogic. 'Formal epistemology' is a term coined in the late 1990s for a new constellation of interests in philosophy, the origins of which are found in earlier works of epistemologists, philosophers of science and logicians. It (...) addresses a growing agenda of problems concerning knowledge, belief, certainty, rationality, deliberation, decision, strategy, action and agent interaction - and it does so using methods fromlogic, probability, computability, decision and game theory. The volume also includes a thorough index and suggestions for further reading, and thus offers a complete teaching and research package for students as well as research scholars offormal epistemology, philosophy,logic, computer science, theoretical economics and cognitive psychology. (shrink)

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Originally published in 1966. This is a self-instructional course intended for first-year university students who have not had previous acquaintance withLogic. The book deals with "propositional"logic by the truth-table method, briefly introducing axiomatic procedures, and proceeds to the theory of the syllogism, thelogic of one-place predicates, and elementary parts of thelogic 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)

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In Other Logics: Alternatives toFormalLogic in the History of Thought and Contemporary Philosophy , edited by Admir Skodo, an array of historical and philosophical chapters decenter the idea offormallogic as the most accurate, timeless, and abstract description of all thought and reasoning.

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Rigorous yet intuitive and accessible, Introduction toFormalLogic provides a focused, "nuts-and-bolts" introduction toformal deductivelogic that covers syntax, semantics, translation, and natural deduction for propositional and predicate logics. For instructors who want to go beyond a basic introduction to explore the connection betweenformallogic techniques and philosophy, Oxford also publishes Introduction toFormalLogic with Philosophical Applications, an extended version of this text that incorporates two chapters of stand-alone (...) essays onlogic and its application in philosophy and beyond. (shrink)

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_A Pocket Guide toFormal Logic_ is a succinct primer meant especially for those without any prior background inlogic. Its brevity makes it well-suited to introductory courses in critical thinking or introductory philosophy with aformallogic component, and its friendly tone offers a welcoming introduction to this often-intimidating subject. The book provides a focused presentation of common methods used in statementlogic, including translations, truth tables, and proofs. Supplemental materials—including more detailed treatments of (...) select methods and concepts as well as additional sample questions and answers—are available on a companion website. (shrink)

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

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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 oflogic lies behind them? (4) Can there be alogic of the concept of God? I address them by presenting aformal-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-orderlogic (FOL) with identity, and then show that its simplest modal extension (SQML, or the simplest quantified modallogic) 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)

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Derrida onFormalLogic: An Interpretive Essay develops the dominant themes in Jacques Derrida's texts on the principles offormallogic, especially identity and contradiction, as these themes emerged from Derrida's discussion of Joyce's Ulysses. For students of Derrida and his place as a critic of western metaphysics, the work provides a clear account and critical evaluation of implications drawn from Derrida's conclusions concerning the strength offormallogic.

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Excerpt from Hegel's Doctrine ofFormalLogic: Being a Translation of the First, Section of the SubjectiveLogic It has been my great good fortune to have freely at my disposal during the preparation of this work the wide knowledge and wise judgement of my friend Dr. James Creed Meredith. I am indeed deeply in his debt for his valuable assistance, ever ready to my call but I can console myself by reflecting that the reader is still (...) more indebted to me for the unstinted use I have made of it. 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)

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Weakening classicallogic is one of the most popular ways of dealing with semantic paradoxes. Their advocates often claim that such weakening does not affect non-semantic reasoning. Recently, however, Halbach and Horsten have shown that this is actually not the case for Kripke’s fixed-point theory based on the Strong Kleene evaluation scheme. Feferman’s axiomatization $\textsf{KF}$ in classicallogic is much stronger than its paracomplete counterpart $\textsf{PKF}$, not only in terms of semantic but also in arithmetical content. This paper (...) compares the proof-theoretic strength of an axiomatization of Kripke’s construction based on the paraconsistent evaluation scheme of $\textsf{LP}$, formulated in classicallogic with that of an axiomatization directly formulated in $\textsf{LP}$, extended with a consistency operator. The ultimate goal is to find out whether paraconsistent solutions to the paradoxes that employ consistency operators fare better in this respect than paracomplete ones. (shrink)

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Many people understand the expression “formallogic” as meaning modern mathematicallogic by opposition to traditionallogic before the revolution that happened in the second part of the 19th century with Boole, Frege and others. But in fact this expression was created by Kant. Some people like to quote a excerpt of the preface of the second edition of the Critic of pure reason, where Kant says thatformallogic is a finished and closed (...) science: “logic … has not been able to advance a single step, and hence is to all appearances closed and complete”. Retrospectively, this remark by Kant seems pretty ridiculous. One may wonder how such a wise man could have been so wrong. On the other hand it is quite ironic that the expression created by this philosopher has turned to be used to name the newlogic that he was not able to prophesy. Of course “formallogic” is not the only expression used to denote the newlogic but it is quite popular and widely spread, maybe because it means several things at the same time. (shrink)

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This volume brings together some of Dr. Kuczynski's most important work on mathematicallogic. The crushing power of Kuczynski's intellect is on full display in these paper, in which he introduces the neophyte to the basic principles of set theory andlogic while at the very same time articulating new and important theorems of his own.

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The wider topic to which the content of this paper belongs is that of the relationship betweenformallogic 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)

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What does it mean to say thatlogic isformal? The short answer is: it means (or can mean) several different things. In this paper, I argue that there are (at least) eight main variations of the notion of theformal that are relevant for current discussions in philosophy andlogic, and that they are structured in two main clusters, namely theformal as pertaining to forms, and theformal as pertaining to rules. To (...) the first cluster belong theformal as schematic; theformal as indifference to particulars; theformal as topic-neutrality; theformal as abstraction from intentional content; theformal as de-semantification. To the second cluster belong theformal as computable; theformal as pertaining to regulative rules; theformal as pertaining to constitutive rules. I analyze each of these eight variations, providing their historical background and raising related philosophical questions. The significance of this work of ?conceptual archeology? is that it may enhance clarity in debates where the notion of theformal plays a prominent role (such as debates where it is expected to play a demarcating role), but where it is oftentimes used equivocally and/or imprecisely. (shrink)

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Central topics in medievallogic are here treated in a way that is congenial to the modern reader, without compromising historical reliability. The achievements of medievallogic 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)

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