"One of the most careful and intensive among the introductory texts that can be used with a wide range of students. It builds remarkably sophisticated technical skills, a good sense of the nature of a formal system, and a solid and extensive background for more advanced work in logic.... The emphasis throughout is on natural deduction derivations, and the text's deductive systems are its greatest strength. Lemmon's unusual procedure of presenting derivations before truth tables is very effective." --Sarah Stebbins, The (...) Journal of Symbolic Logic. (shrink)

"One of the most careful and intensive among the introductory texts that can be used with a wide range of students. It builds remarkably sophisticated technical skills, a good sense of the nature of a formal system, and a solid and extensive background for more advanced work in logic.... The emphasis throughout is on natural deduction derivations, and the text's deductive systems are its greatest strength. Lemmon's unusual procedure of presenting derivations before truth tables is very effective." --Sarah Stebbins, _The (...) Journal of Symbolic Logic_. (shrink)

This brief volume supplements Lemmon's classic introductory logic text with almost 200 new exercises, many of them solved, solutions to selected exercises in _Beginning Logic_ itself, a helpful commentary on Lemmon’s use of key technical terms, alternative formulations, and advice to students.

Yet another introductory text of symbolic logic and a very good one, at that. The four principal chapters of this book treat propositional logic and predicate calculus, devoting two chapters to each. Each of these subsections is roughly organized as follows: An initial chapter presents the basic notational devices, translation methods, and intuitive discussion of arguments and validity. The subsequent chapter gives exact formation and transformation rules, proofs and metalinguistic considerations of questions of consistency and completeness. Lemmon's notation is similar (...) to that of Suppes, and his system bears some similarity to the rules of inference in Fitch's 1951 text, although Lemmon's system is a "standard" one. Two appendices, dealing with normal forms and elementary set theory, and an annotated bibliography of relevant works complete this high-quality text. An obviously promising career was cut tragically short by the premature death of the author in 1966.—H. P. K. (shrink)

The paper begins with a more carefully stated version of ontologically neutral (ON) logic, originally introduced in (Hailperin, 1997). A non-infinitistic semantics which includes a definition of potential infinite validity follows. It is shown, without appeal to the actual infinite, that this notion provides a necessary and sufficient condition for provability in ON logic.

This article discusses two topics, both commonly referred to using the label “the beginning of Hegel's Logic”: (1) Hegel's justification for the claim that a science of logic must begin by considering the concept of “pure being”. (2) Hegel's account of the concepts “being”, “nothing”, and “becoming” in the first chapter of his Logic. Discussing recent work on both of these topics, two primary claims are defended: Regarding (1): the strongest interpretations of Hegel's case for beginning a science (...) of logic with the concept of pure being are those which take him to argue that this concept must be necessarily both “immediate” and “mediated” at the same time. Regarding (2): The widespread tendency to take Hegel's treatment of the concepts of being, nothing, and becoming as an example with which to illustrate his “dialectical method” should be replaced with an interpretation of that chapter which understands it to rather make possible the kind of dialectical transitions which make up the rest of Hegel's Science of Logic. (shrink)

Recent discussions concerning the beginning problem of Hegel’s Logic have reached the agreement that any promised interpretation of the beginning of the Logic must reject opposition between the immediacy and mediation and embrace their unity instead. It is how this unity is understood that divides interpreters. Either the mediation precedes the immediacy and justifies it first, or a somewhat one-sided immediacy occurs first and waits to be mediated later in a circular justification. However, both concepts are confronted with (...) their own difficulties. To avoid these difficulties, I propose that the pure immediacy or pure being is justified to be the Logic’s beginning in virtue of its alternativelessness. Only it can measure up to the rigorous requirement implied by the nature of the beginning. (shrink)

A philosophical movement, correctly called logical pragmatism, is growing up around the philosophy of W. V. O. Quine. Soviet scholars follow this development with clear and well-grounded understanding of the origins and tenets of the system. This essay continues the "dialogue" between contemporary Marxism-Leninism and logical pragmatism recommended by Soviet scholars.

In this section, Bibler discusses the minutia of logic, specifically the development and maturation of logic as well as the self-justification of logic. Bibler points out that, in the science of lo...

A history of the logic of action is outlined, beginning with St Anselm. Five modern authors are discussed in some detail: von Wright, Fitch, Kanger, Chellas and Pratt.

The study of the interdependence of grammar and logic at the beginning of the twelfth century is a difficult subject and progress here has been slow. With the recent publication of the Notae Dunelmenses, however, we are now able to see rather more clearly how closely the two disciplines were bound to one another. The following article draws upon this newly published material and on unpublished material from contemporary commentaries on Aristotle’s Categories to investigate how the grammarians’ account of (...) number was reconciled with that given by Aristotle. It considers in particular the problem of the meaning of numerical terms such as ‘pair’ and of collective names such as ‘people’ and how attempting to solve it shaped thinking about the metaphysics of number. (shrink)

According to the leading commentators and the author himself, Edmund Husserl's Formal and transcendental Logic is the most important work on phenomenological logic ever written. Nonetheless, it has, in general, gained far less attention than theLogical Investigations and the Ideas on a Pure Phenomenology and Phenomenological Philosophy. In particular, the argument of § 1 of the Logic, namely, that it is fruitful to start with the meanings of the expression “logos” in order to develop a genuinely transcendental logic, has received (...) virtually no consideration. This paper takes a step towards filling this empty space by analyzing and criticizing the argument of § 1 as a problem to which (a) solution(s) must be found: First, it offers an introduction to the problem per se, which is one of the relationship between speech and reason; second, it tries to bring the given senses of “logos” to a higher grade of conceptual clarity and distinctness than that in the text; third, it attempts to decide whether and how far these senses of the word can be documented according to principles of Classical philology; fourth, it endeavors to determine exactly the relationship between the meanings of “logos” in § 1 and the senses of “logic” in §§ 1–107; finally, it strives to show that, with respect to the account of the relationship between speech and reason provided by Husserl in the Logic, there is, at best, a conflict and, at worst, a contradiction between the strategy outlined in § 1 and the tactics adopted in § 2ff. Throughout, the paper reads Husserl's “descriptions” as ‘arguments’ for his positions, thereby avoiding any of the obscurity sometimes infecting work in “Continental philosophy”. (shrink)

This article is concerned with some of the criteria which Hegel believes apply to a scientific treatment of logic. I briefly address criteria which I take Hegel to inherit from German rationalism before focusing on two fairly idiosyncratic criteria: the requirement that a science of logic exhibit a circular structure and that it begin with the concept of pure being. I offer an explanation of these criteria which understands them as motivated by anti-sceptical concerns, before arguing that Hegel’s mature treatment (...) of the latter criterion is problematically ambiguous. (shrink)

This paper argues firstly that the argument of Hegel's Phenomenology of Spirit is necessary for the justification of the beginning of his logical project, and secondly that Hegel's attempt to secure the beginning of his Science of Logic by relying upon the argument of the Phenomenology fails. I argue firstly that the position taken up at the beginning of Hegel's Logic is constructed in such a fashion that it relies upon the argument of the Phenomenology to justify (...) it. I then offer some support for the view of the relationship between the two texts defended by Maker in order to see how the two might be thought to be compatible. Finally, in the longest part of the paper, I offer a number of reasons for thinking that attempts to render the two compatible in this fashion fail. I therefore conclude that, as it stands, the beginning of Hegel's Logic is not secured against objection in the way that Hegel wants it to be. (shrink)

In three exemplary essays, author Peter Bornedal promotes Deconstruction as a cogent analytical method whose distinctive critical object is foundational knowledge. In this, he wants to restore Deconstruction as a rational discourse, while continuing to emphasize it as a critique of metaphysics.

In this paper we investigate those extensions of the bimodal provability logic \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\vec CSM}_{0}$\end{document} (alias \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\vec PRL}_{1}$\end{document} or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\vec F}^{-})$\end{document} which are subframe logics, i.e. whose general frames are closed under a certain type of substructures. Most bimodal provability logics are in this class. The main result states that (...) all finitely axiomatizable subframe logics containing \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\vec CSM}_{0}$\end{document} are decidable. We note that, as a rule, interesting systems in this class do not have the finite model property and are not even complete with respect to Kripke semantics. (shrink)

Classical logic, beginning with the work of Aristotle, has developed into a powerful and rigorous mathematical theory with many applications in mathematics and ...

INTRODUCTION TO LOGIC is a combined text and workbook for students beginning their study of logic. The workbook style allows students to proceed at their own pace, checking their progress in the end-of-chapter exercises. The text covers propositional logic and predicate logic with identity, the focus being on arguments. The methods of proof are truth-tables and truth-trees in the style of Jeffrey. This text is suitable for students of philosophy, computer science, mathematics and science in general.

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)

We apply and extend the theory and methods of algorithmic correspondence theory for modal logics, developed over the past 20 years, to the language LR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}_R$$\end{document} of relevance logics with respect to their standard Routley–Meyer relational semantics. We develop the non-deterministic algorithmic procedure PEARL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {PEARL}$$\end{document} for computing first-order equivalents of formulae of the language LR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} (...) \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}_R$$\end{document}, in terms of that semantics. PEARL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {PEARL}$$\end{document} is an adaptation of the previously developed algorithmic procedures SQEMA and ALBA. We then identify a large syntactically defined class of inductive formulae in LR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}_R$$\end{document}, analogous to previously defined such classes in the classical, distributive and non-distributive modal logic settings, and show that PEARL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {PEARL}$$\end{document} succeeds for every inductive formula and correctly computes a first-order definable condition which is equivalent to it with respect to frame validity. We also provide a detailed comparison with two earlier works, each extending the class of Sahlqvist formulae to relevance logics, and show that both are subsumed by simple subclasses of inductive formulae. (shrink)

Logical pluralism is the view that there is more than one logic. Logical normativism is the view that logic is normative. These positions have often been assumed to go hand-in-hand, but we show that one can be a logical pluralist without being a logical normativist. We begin by arguing directly against logical normativism. Then we reformulate one popular version of pluralism—due to Beall and Restall—to avoid a normativist commitment. We give three non-normativist pluralist views, the most promising of which depends (...) not on logic’s normativity but on epistemic goals. (shrink)

Epistemic logic has grown from its philosophical beginnings to find diverse applications in computer science, and as a means of reasoning about the knowledge and belief of agents. This book provides a broad introduction to the subject, along with many exercises and their solutions. The authors begin by presenting the necessary apparatus from mathematics and logic, including Kripke semantics and the well-known modal logics K, T, S4 and S5. Then they turn to applications in the context of distributed systems (...) and artificial intelligence. These include the notions of common knowledge, distributed knowledge, explicit and implicit belief, the interplays between knowledge and time, and knowledge and action, as well as a graded (or numerical) variant of the epistemic operators. The authors also discuss extensively the problem of logical omniscience. They cover Halpern & Moses' theory of honest formulas, and they make a digression into the realm of nonmonotonic reasoning and preferential entailment. They discuss Moore's autoepistemic logic, together with Levesque's related logic of "all I know". Furthermore, they show how one can base default and counterfactual reasoning on epistemic logic. Graduate students in philosophy or in computer science, especially those with an interest in AI, will find this book useful. (shrink)

Logic for Philosophy is an introduction to logic for students of contemporary philosophy. It is suitable both for advanced undergraduates and for beginning graduate students in philosophy. It covers (i) basic approaches to logic, including proof theory and especially model theory, (ii) extensions of standard logic that are important in philosophy, and (iii) some elementary philosophy of logic. It emphasizes breadth rather than depth. For example, it discusses modal logic and counterfactuals, but does not prove the central metalogical results (...) for predicate logic (completeness, undecidability, etc.) Its goal is to introduce students to the logic they need to know in order to read contemporary philosophical work. It is very user-friendly for students without an extensive background in mathematics. In short, this book gives you the understanding of logic that you need to do philosophy. (shrink)

In this paper I address the relationship between Hegel and Heidegger from the perspective of the dialectic between end and beginning. After the introduction, in the second part of the paper I analyze Hegel’s position in the history of being as the beginning of the end of metaphysics. Afterwards I address Heidegger’s interpretation of the beginning of Hegel’s Logic, showing how the essential beginning is the end of the completed system. In the fourth part of the (...) text, I discuss how Heidegger wants to position himself in relation to Hegel, mainly examining their different interpretation of negativity and initiality. I will argue that the key word for understanding their different positions is “event”. Whereas Hegel’s system leaves no room for any beginning as event, but every beginning is reabsorbed by the finished and concluded system, for Heidegger thought must time and time again make itself initial and evential. (shrink)

We discuss in this paper the question of the scope of the principle of tolerance about languages promoted in Carnap's The Logical Syntax of Language and the nature of the analogy between it and the rudimentary conventionalism purportedly exhibited in the work of Poincaré and Hilbert. We take it more or less for granted that Poincaré and Hilbert do argue for conventionalism. We begin by sketching Coffa's historical account, which suggests that tolerance be interpreted as a conventionalism that allows us (...) complete freedom to select whatever language we wish—an interpretation that generalizes the conventionalism promoted by Poincaré and Hilbert which allows us complete freedom to select whatever axiom system we wish for geometry. We argue that such an interpretation saddles Carnap with a theory of meaning that has unhappy consequences, a theory we believe he did not hold. We suggest that the principle of linguistic tolerance in fact has a more limited scope; but within that scope the analogy between tolerance and geometric conventionalism is quite tight. (shrink)

Beginnings can be empirically described, philosophically debated, fictionally recounted or theatrically staged – each kind of discourse approaches beginnings via an examination of representation as an impossible return to source. The work of French philosopher Philippe Lacoue-Labarthe articulates the problem of beginnings by considering them as a form of subjective collapse, loss of integrity and aggravation of emotion resulting from the paradoxical logic of representation. While Lacoue-Labarthe’s position has been largely developed in his philosophical writings, this study focuses more specifically (...) on his literary production, namely his collection of short stories, L’ “Allégorie”, and his staged debate with Jean-Luc Nancy in Scène: suivi de Dialogue sur le dialogue, in which the two writers present opposing views on the theatre. The analysis of the above texts centres on the two key “locations” for beginnings, namely narrative voice and the theatrical stage. (shrink)

This junior/senior level text is devoted to a study of first-order logic and its role in the foundations of mathematics: What is a proof? How can a proof be justified? To what extent can a proof be made a purely mechanical procedure? How much faith can we have in a proof that is so complex that no one can follow it through in a lifetime? The first substantial answers to these questions have only been obtained in this century. The most (...) striking results are contained in Goedel's work: First, it is possible to give a simple set of rules that suffice to carry out all mathematical proofs; but, second, these rules are necessarily incomplete - it is impossible, for example, to prove all true statements of arithmetic. The book begins with an introduction to first-order logic, Goedel's theorem, and model theory. A second part covers extensions of first-order logic and limitations of the formal methods. The book covers several advanced topics, not commonly treated in introductory texts, such as Trachtenbrot's undecidability theorem. Fraissé's elementary equivalence, and Lindstroem's theorem on the maximality of first-order logic. (shrink)

In this paper we prove the completeness of three logical systems I LI, IL2 and IL3. IL1 deals solely with identities {a = b), and its deductions are the direct deductions constructed with the three traditional rules: (T) from a = b and b = c infer a = c, (S) from a = b infer b = a and (A) infer a = a(from anything). IL2 deals solely with identities and inidentities {a ± b) and its deductions include both (...) the direct and the indirect deductions constructed with the three traditional rules. IL3 is a hybrid of IL1 and IL2: its deductions are all direct as in IL1 but it deals with identities and inidentities as in IL2. IL1 and IL2 have a high degree of naturalness. Although the hybrid system IL3 was constructed as an artifact useful in the mathematical study of IL1 and IL2, it nevertheless has some intrinsically interesting aspects. The main motivation for describing and studying such simple systems is pedagogical. In teaching beginning logic one would like to present a system of logic which has the following properties. First, it exemplifies the main ideas of logic: implication, deduction, non-implication, counterargument(or countermodel), logical truth, self-contradiction, consistency,satisfiability, etc. Second, it exemplifies the usual general metaprinciples of logic: contraposition and transitivity of implication, cut laws, completeness,soundness, etc. Third, it is simple enough to be thoroughly grasped by beginners. Fourth, it is obvious enough so that its rules do not appear to be arbitrary or purely conventional. Fifth, it does not invite confusions which must be unlearned later. Sixth, it involves a minimum of presuppositions which are no longer accepted in mainstream contemporary logic. (shrink)