Mathematics textbooks teach logical reasoning by example, a practice started by Euclid; while logic textbooks treat logic as a subject in its own right without practical application to mathematics. Stuck in the middle are students seeking mathematical proficiency and educators seeking to provide it. To assist them, the article explains in practical detail how to teach logic-based skills such as: making mathematical reasoning fully explicit; moving from step to step in a mathematical proof in logically correct ways; (...) and checking to make sure inferences are logically correct. The methodology can easily be extended beyond the four examples analyzed. (shrink)

This paper is about teaching elementary logic to blind or visually impaired students. The targeted audience are teachers who all of sudden have a blind or visually impaired student in their introduction to logic class, find limited help from disability centers in their institution, and have no idea what to do. We provide simple techniques that allow direct communication between a teacher and a visually impaired student. We show how the use of what is known as Polish (...) notation simplifies communication, and pedagogically is a great notation for a Braille reader. (shrink)

We are much better equipped to let the facts reveal themselves to us instead of blinding ourselves to them or stubbornly trying to force them into preconceived molds. We no longer embarrass ourselves in front of our students, for example, by insisting that “Some Xs are Y” means the same as “Some X is Y”, and lamely adding “for purposes of logic” whenever there is pushback. Logic teaching in this century can exploit the new spirit of objectivity, (...) humility, clarity, observationalism, contextualism, and pluralism. Besides the new spirit there have been quiet developments in logic and its history and philosophy that could radically improve logic teaching. This lecture expands points which apply equally well in first, second, and third courses, i.e. in “critical thinking”, “deductive logic”, and “symbolic logic”. (shrink)

This paper considers the nature of the changes that took place in logic teaching at the University of Oxford from the beginning of the sixteenth century, when students attended university lectures on Aristotle’s texts as well as studying short works dealing with specifically medieval developments, to the beginning of the eighteenth century when teaching was centred in the colleges, the medieval developments had largely disappeared, and manuals summarizing Aristotelian logic were used. The paper also considers the (...) reasons for these changes, including changes in English society, and the effect of humanism and the more scholarly Aristotelianism that it produced. (shrink)

We are much better equipped to let the facts reveal themselves to us instead of blinding ourselves to them or stubbornly trying to force them into preconceived molds. We no longer embarrass ourselves in front of our students, for example, by insisting that “Some Xs are Y” means the same as “Some X is Y”, and lamely adding “for purposes of logic” whenever there is pushback. Logic teaching in this century can exploit the new spirit of objectivity, (...) humility, clarity, observationalism, contextualism, and pluralism. Besides the new spirit there have been quiet developments in logic and its history and philosophy that could radically improve logic teaching. One rather conspicuous example is that the process of refining logical terminology has been productive. Future logic students will no longer be burdened by obscure terminology and they will be able to read, think, talk, and write about logic in a more careful and more rewarding manner. Closely related is increased use and study of variable-enhanced natural language as in “Every proposition x that implies some proposition y that is false also implies some proposition z that is true”. Another welcome development is the culmination of the slow demise of logicism. No longer is the teacher blocked from using examples from arithmetic and algebra fearing that the students had been indoctrinated into thinking that every mathematical truth was a tautology and that every mathematical falsehood was a contradiction. A fifth welcome development is the separation of laws of logic from so-called logical truths, i.e., tautologies. Now we can teach the logical independence of the laws of excluded middle and non-contradiction without fear that students had been indoctrinated into thinking that every logical law was a tautology and that every falsehood of logic was a contradiction. This separation permits the logic teacher to apply logic in the clarification of laws of logic. This lecture expands the above points, which apply equally well in first, second, and third courses, i.e. in “critical thinking”, “deductive logic”, and “symbolic logic”. (shrink)

Peer Instruction is a simple and effective technique you can use to make lectures more interactive, more engaging, and more effective learning experiences. Although well known in science and mathematics, the technique appears to be little known in the humanities. In this paper, we explain how Peer Instruction can be applied in philosophy lectures. We report the results from our own experience of using Peer Instruction in undergraduate courses in philosophy, formal logic, and critical thinking. We have consistently found (...) it to be a highly effective method of improving the lecture experience for both students and the lecturer. (shrink)

PARC is an "appended numeral" system of natural deduction that I learned as an undergraduate and have taught for many years. Despite its considerable pedagogical strengths, PARC appears to have never been published. The system features explicit "tracking" of premises and assumptions throughout a derivation, the collapsing of indirect proofs into conditional proofs, and a very simple set of quantificational rules without the long list of exceptions that bedevil students learning existential instantiation and universal generalization. The system can be used (...) with any Copi-style set of inference rules, so it is quite adaptable to many mainstream symbolic logic textbooks. Consequently, PARC may be especially attractive to logic teachers who find Jaskowski/Gentzen-style introduction/elimination rules to be far less "natural" than Copi-style rules. The PARC system is also keyboard-friendly in comparison to the widely adopted Jaskowski-style graphical subproof system of natural deduction, viz., Fitch diagrams and Copi "bent arrow" diagrams. (shrink)

Fourteen philosophers share their experience teaching Peirce to undergraduates in a variety of settings and a variety of courses. The latter include introductory philosophy courses as well as upper-level courses in American philosophy, philosophy of religion, logic, philosophy of science, medieval philosophy, semiotics, metaphysics, etc., and even an upper-level course devoted entirely to Peirce. The project originates in a session devoted to teaching Peirce held at the 2007 annual meeting of the Society for the Advancement of American (...) Philosophy. The session, organized by James Campbell and Richard Hart, was co-sponsored by the American Association of Philosophy Teachers. (shrink)

Philosophy is the study of the most general and fundamental problems of human life. The main areas of study in philosophy includes metaphysics, epistemology, logic, ethics and aesthetics etc. there are other several branches of philosophy which characterize different branches of knowledge. Philosophy being a very abstract branch of study, has not much scope of using equipment on a large scale to supplement the normal lecture schedules. However, in some papers/areas there are comparatively better scope to make the lectures (...) more concrete and interesting through proper use of various teaching aids and modes. We include logic, philosophy of science, applied philosophy, applied ethics, social and political philosophy, philosophy of mind, philosophy of cognitive science and history of philosophy etc., we can use various modern aids. In this article my attempt is to draw out an outlines of aids and modes for effective philosophy teaching. (shrink)

This book treats ancient logic: the logic that originated in Greece by Aristotle and the Stoics, mainly in the hundred year period beginning about 350 BCE. Ancient logic was never completely ignored by modern logic from its Boolean origin in the middle 1800s: it was prominent in Boole’s writings and it was mentioned by Frege and by Hilbert. Nevertheless, the first century of mathematical logic did not take it seriously enough to study the ancient (...) class='Hi'>logic texts. A renaissance in ancient logic studies occurred in the early 1950s with the publication of the landmark Aristotle’s Syllogistic by Jan Łukasiewicz, Oxford UP 1951, 2nd ed. 1957. Despite its title, it treats the logic of the Stoics as well as that of Aristotle. Łukasiewicz was a distinguished mathematical logician. He had created many-valued logic and the parenthesis-free prefix notation known as Polish notation. He co-authored with Alfred Tarski’s an important paper on metatheory of propositional logic and he was one of Tarski’s the three main teachers at the University of Warsaw. Łukasiewicz’s stature was just short of that of the giants: Aristotle, Boole, Frege, Tarski and Gödel. No mathematical logician of his caliber had ever before quoted the actual teachings of ancient logicians. -/- Not only did Łukasiewicz inject fresh hypotheses, new concepts, and imaginative modern perspectives into the field, his enormous prestige and that of the Warsaw School of Logic reflected on the whole field of ancient logic studies. Suddenly, this previously somewhat dormant and obscure field became active and gained in respectability and importance in the eyes of logicians, mathematicians, linguists, analytic philosophers, and historians. Next to Aristotle himself and perhaps the Stoic logician Chrysippus, Łukasiewicz is the most prominent figure in ancient logic studies. A huge literature traces its origins to Łukasiewicz. -/- This Ancient Logic and Its Modern Interpretations, is based on the 1973 Buffalo Symposium on Modernist Interpretations of Ancient Logic, the first conference devoted entirely to critical assessment of the state of ancient logic studies. (shrink)

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)

This paper corrects a mistake I saw students make but I have yet to see in print. The mistake is thinking that logically equivalent propositions have the same counterexamples—always. Of course, it is often the case that logically equivalent propositions have the same counterexamples: “every number that is prime is odd” has the same counterexamples as “every number that is not odd is not prime”. The set of numbers satisfying “prime but not odd” is the same as the set of (...) numbers satisfying “not odd but not not-prime”. The mistake is thinking that every two logically-equivalent false universal propositions have the same counterexamples. Only false universal propositions have counterexamples. A counterexample for “every two logically-equivalent false universal propositions have the same counterexamples” is two logically-equivalent false universal propositions not having the same counterexamples. The following counterexample arose naturally in my sophomore deductive logic course in a discussion of inner and outer converses. “Every even number precedes every odd number” is counterexemplified only by even numbers, whereas its equivalent “Every odd number is preceded by every even number” is counterexemplified only by odd numbers. Please let me know if you see this mistake in print. Also let me know if you have seen these points discussed before. I learned them in my own course: talk about learning by teaching! (shrink)

Argument mapping is a way of diagramming the logical structure of an argument to explicitly and concisely represent reasoning. The use of argument mapping in critical thinking instruction has increased dramatically in recent decades. This paper overviews the innovation and provides a procedural approach for new teaches wanting to use argument mapping in the classroom. A brief history of argument mapping is provided at the end of this paper.

The Symbolic Logic Study Guide is designed to accompany the widely used symbolic logic textbook Language, Proof and Logic (LPL), by Jon Barwise and John Etchemendy (CSLI Publications 2003). The guide has two parts. The first part contains condensed, essential lecture notes, which streamline and systematize the first fourteen chapters of the book into seven teaching sections, and thus provide a clear, well-designed roadmap for the understanding of the text. The second part consists of twelve sample (...) quizzes and solutions. The Symbolic Logic Study Guide is essential for all instructors and students who use LPL in their symbolic logic classes. (shrink)

The informal logic movement began as an attempt to develop – and teach – an alternative logic which can account for the real life arguing that surrounds us in our daily lives – in newspapers and the popular media, political and social commentary, advertising, and interpersonal exchange. The movement was rooted in research and discussion in Canada and especially at the University of Windsor, and has become a branch of argumentation theory which intersects with related traditions and approaches (...) (notably formal logic, rhetoric and dialectics in the form of pragma-dialectics). In this volume, some of the best known contributors to the movement discuss their views and the reasoning and argument which is informal logic’s subject matter. Many themes and issues are explored in a way that will fuel the continued evolution of the field. Federico Puppo adds an insightful essay which considers the origins and development of informal logic and whether informal logicians are properly described as a “school” of thought. In considering that proposition, Puppo introduces readers to a diverse range of essays, some of them previously published, others written specifically for this volume. (shrink)

Information-theoretic approaches to formal logic analyse the "common intuitive" concept of propositional implication (or argumental validity) in terms of information content of propositions and sets of propositions: one given proposition implies a second if the former contains all of the information contained by the latter; an argument is valid if the conclusion contains no information beyond that of the premise-set. This paper locates information-theoretic approaches historically, philosophically and pragmatically. Advantages and disadvantages are identified by examining such approaches in themselves (...) and by contrasting them with standard transformation-theoretic approaches. Transformation-theoretic approaches analyse validity (and thus implication) in terms of transformations that map one argument onto another: a given argument is valid if no transformation carries it onto an argument with all true premises and false conclusion. Model-theoretic, set-theoretic, and substitution-theoretic approaches, which dominate current literature, can be construed as transformation-theoretic, as can the so-called possible-worlds approaches. Ontic and epistemic presuppositions of both types of approaches are considered. Attention is given to the question of whether our historically cumulative experience applying logic is better explained from a purely information-theoretic perspective or from a purely transformation-theoretic perspective or whether apparent conflicts between the two types of approaches need to be reconciled in order to forge a new type of approach that recognizes their basic complementarity. (shrink)

This paper introduces the special issue on the Concept of God of the Journal of Applied Logics (College Publications). The issue contains the following articles: Logic and the Concept of God, by Stanisław Krajewski and Ricardo Silvestre; Mathematical Models in Theology. A Buber-inspired Model of God and its Application to “Shema Israel”, by Stanisław Krajewski; Gödel’s God-like Essence, by Talia Leven; A Logical Solution to the Paradox of the Stone, by Héctor Hernández Ortiz and Victor Cantero; No New Solutions (...) to the Logical Problem of the Trinity, by Beau Branson; What Means ‘Tri-’ in ‘Trinity’ ? An Eastern Patristic Approach to the ‘Quasi-Ordinals’, by Basil Lourié; The Éminence Grise of Christology: Porphyry’s Logical Teaching as a Cornerstone of Argumentation in Christological Debates of the Fifth and Sixth Centruies, by Anna Zhyrkova; The Problem of Universals in Late Patristic Theology, by Dirk Krasmüller; Intuitionist Reasoning in the Tri-unitrian Theology of Nicolas of Cues, by Antonino Drago. (shrink)

Despite increasing evidence that the traditional lecture is inefficient for student learning, such methods remain the central paradigm for teaching logic. In this paper, I identify the deficits of the lecture model and outline the many benefits of flipping the logic classroom—namely that students can absorb information at their own pace, freeing classroom time for active learn- ing activities, and allowing the students to come prepared to actively engage in deeper levels of learning. I provide advice for (...) curricular change from the traditional model, and guidance for flipped classroom implementation. I also offer suggestions for the best use of newly available class time, and advice for keeping students accountable for learning the information prior to class. Last, I consider common challenges with the flipped classroom model. I acknowledge possible obstacles to flipping the undergraduate logic course and address these challenges with potential solutions. (shrink)

The importance of teaching critical thinking skills at the college level cannot be overemphasized. Teaching a subcategory of these skills—argument analysis—we believe is especially important for first-year students with their college careers, as well as their lives, ahead of them. The struggle, however, is how to effectively teach argument analysis skills that will serve students in a broad range of disciplines.

The logic of academic writing is the argumentative strategy on which our papers, our sections, and our paragraphs are based. It is a strategy, as it is a plan that connects different steps and has a specific goal, namely convincing the audience of an original and important idea. And it is argumentative, for two reasons. First, we can defend our idea and we can convince our audience only through arguments, which only in very few disciplines are formal deductions. In (...) most cases, the arguments that we use are based on premises accepted by a community and the conclusions are drawn from principles that in the ancient dialectics were called “maxims,” principles shared by everyone. Second, a paper is a dialogue between the author and his or her readers. An idea can be considered as interesting and worth reading only when it addresses a topic that is perceived as important by the readers and tackles a problem that is open and needs to be solved. Our arguments are acceptable when they start from the premises of our community of readers, avoiding repeating what is obvious for them or taking for granted what is obscure or unknown to them. In this book, we present the argumentative approach to academic writing that we used in classroom. What characterizes it and makes it unique is the perspective that is adopted. We do not start from preexisting ideas that only need to be presented in a way that is suitable to an academic public. We intend to show that writing academically is a consequence of thinking academically, or rather “strategically.” We want to explain how the linguistic and presentational devices are the result of a much deeper plan underlying them, and how mastering the logic of a paper leads to understanding and even developing academic styles. The logic of academic writing is not aimed at teaching how to use language and write texts academically, but at enabling readers to create their own style based on their own argumentative strategies. (shrink)

A new method of mental development of children (and adults) is considered, the basis of which is the logical operation of comparison. The method was created taking into account the Aristotelian understanding of philosophy as "the science of first causes and beginnings." It can act as a fundamentally new pedagogical approach to school education, since comparative concepts make it possible to demarcate between reason, two forms of intelligence and wisdom. Therefore, the new thinking proposed in this article, which consists in (...) the ability to identify the different and distinguish the identical (same) within the framework of a cumulative series of comparative concepts, is a tool for integrating knowledge, which, in my opinion, should be introduced into the sphere of traditional school education. In this connection, around each of the comparative concepts, the main subjects will be reorganized and lined up in a certain order. As a result, behind different physical or social phenomena, students - schoolchildren or students will learn to see the same content, so the problem with memorizing large amounts of educational material disappears. (shrink)

Should the state teach ethics? There is widespread disagreement on whether (and how) secular states should be in the business of promoting a particular moral viewpoint. This article attempts to schematize, and evaluate, these stances. It does so by posing three, simple questions: (1) Should the state explicitly promote certain ethical values over others? (2) Should the state have ultimate justifications for the values it promotes? (3) Should the state compel its citizens to accept these ultimate justifications? Logically, each question (...) in this series is a prerequisite for considering those questions further down the list. The result is that responses can be categorized into one of four possible permutations or ‘camps.’ These are: (1) The Libertarian (“No” to all three questions) (2) The Pluralist (“Yes” to question 1; “No” to questions 2 and 3) (3) The Rationalist Republican (“Yes” to questions 1 and 2; “No” to question 3) (4) The Rigorous Republican (“Yes” to all three questions) It will be shown that just one of these positions, the ‘rationalist republican,’ stands out from all the rest. For only the rationalist republican can account for a normative politics while also safeguarding the individual’s freedom of conscience. (shrink)

This paper is in two parts. Part I outlines three traditional approaches to the teaching of critical thinking: the normative, cognitive psychology, and educational approaches. Each of these approaches is discussed in relation to the influences of various methods of critical thinking instruction. The paper contrasts these approaches with what I call the “visualisation” approach. This approach is explained with reference to computer-aided argument mapping (CAAM) which uses dedicated computer software to represent inferences between premise and conclusions. The paper (...) presents a detailed account of the CAAM methodology, and theoretical justification for its use, illustrating this with the argument mapping software Rationale™. A number of Rationale™ design conventions and logical principles are outlined including the principle of abstraction, the MECE principle, and the “Holding Hands” and “Rabbit Rule” heuristics. Part II of this paper outlines the growing empirical evidence for the effectiveness of CAAM as a method of teaching critical thinking. (shrink)

The truth table method, natural deduction, and the truth tree method, the three validity proving methods standardly taught in an introductory logic course, are too clumsy for the battlefield of real-life. The “short truth table” test is handy at times, but it stumbles at many other times. In this paper, we set up a general method that can beat all the methods mentioned above in a contest of speed. Furthermore, the procedure can be step-by-step paraphrased in a natural language, (...) so that, unlike the other methods, a real-life logical problem can be analyzed and explained in a real-life language too. (shrink)

In this paper, we propose a framework for fostering argumentative skills in a systematic way in Philosophy and Ethics classes. We start with a review of curricula and teaching materials from the German-speaking world to show that there is an urgent need for standards for the teaching and learning of argumentation. Against this backdrop, we present a framework for such standards that is intended to tackle these difficulties. The spiral-curricular model of argumentative competences we sketch helps teachers introduce (...) the relevant concepts and skills to students early on in their school career. The focus is on secondary schools, but the proposal can also be of use for learning and teaching in universities, especially in introductory classes. (shrink)

A Fortiori Logic: Innovations, History and Assessments is a wide-ranging and in-depth study of a fortiori reasoning, comprising a great many new theoretical insights into such argument, a history of its use and discussion from antiquity to the present day, and critical analyses of the main attempts at its elucidation. Its purpose is nothing less than to lay the foundations for a new branch of logic and greatly develop it; and thus to once and for all dispel the (...) many fallacious ideas circulating regarding the nature of a fortiori reasoning. -/- The work is divided into three parts. The first part, Formalities, presents the author’s largely original theory of a fortiori argument, in all its forms and varieties. Its four (or eight) principal moods are analyzed in great detail and formally validated, and secondary moods are derived from them. A crescendo argument is distinguished from purely a fortiori argument, and similarly analyzed and validated. These argument forms are clearly distinguished from the pro rata and analogical forms of argument. Moreover, we examine the wide range of a fortiori argument; the possibilities of quantifying it; the formal interrelationships of its various moods; and their relationships to syllogistic and analogical reasoning. Although a fortiori argument is shown to be deductive, inductive forms of it are acknowledged and explained. Although a fortiori argument is essentially ontical in character, more specifically logical-epistemic and ethical-legal variants of it are acknowledged. -/- The second part of the work, Ancient and Medieval History, looks into use and discussion of a fortiori argument in Greece and Rome, in the Talmud, among post-Talmudic rabbis, and in Christian, Moslem, Chinese and Indian sources. Aristotle’s approach to a fortiori argument is described and evaluated. There is a thorough analysis of the Mishnaic qal vachomer argument, and a reassessment of the dayo principle relating to it, as well as of the Gemara’s later take on these topics. The valuable contribution, much later, by Moshe Chaim Luzzatto is duly acknowledged. Lists are drawn up of the use of a fortiori argument in the Jewish Bible, the Mishna, the works of Plato and Aristotle, the Christian Bible and the Koran; and the specific moods used are identified. Moreover, there is a pilot study of the use of a fortiori argument in the Gemara, with reference to Rodkinson’s partial edition of the Babylonian Talmud, setting detailed methodological guidelines for a fuller study. There is also a novel, detailed study of logic in general in the Torah. -/- The third part of the present work, Modern and Contemporary Authors, describes and evaluates the work of numerous (some thirty) recent contributors to a fortiori logic, as well as the articles on the subject in certain lexicons. Here, we discover that whereas a few authors in the last century or so made some significant contributions to the field, most of them shot woefully off-target in various ways. The work of each author, whether famous or unknown, is examined in detail in a dedicated chapter, or at least in a section; and his ideas on the subject are carefully weighed. The variety of theories that have been proposed is impressive, and stands witness to the complexity and elusiveness of the subject, and to the crying need for the present critical and integrative study. But whatever the intrinsic value of each work, it must be realized that even errors and lacunae are interesting because they teach us how not to proceed. -/- This book also contains, in a final appendix, some valuable contributions to general logic, including new analyses of symbolization and axiomatization, existential import, the tetralemma, the Liar paradox and the Russell paradox. (shrink)

Part I of this paper outlined the three standard approaches to the teaching of critical thinking: the normative (or philosophical), cognitive psychology, and educational taxonomy approaches. The paper contrasted these with the visualisation approach; in particular, computer-aided argument mapping (CAAM), and presented a detailed account of the CAAM methodology and a theoretical justification for its use. This part develops further support for CAAM. A case is made that CAAM improves critical thinking because it minimises the cognitive burden of prose (...) and the demands that arguments in prose typically place on memory. CAAM also has greater usability, complements the imperfect human cognitive system, and adopts a logic of semi-formality which is both natural and intuitive. The paper claims that CAAM is an important advance given that traditional stand-alone critical thinking courses do not teach critical thinking as well as they as they are assumed to do. It is also important given that tertiary education fails to deliver improvements in critical thinking gains for too many students. The paper outlines results from a number of empirical studies that demonstrate that CAAM yields robust gains in critical thinking as measured by independent tests. Students themselves also believe CAAM to be beneficial as noted in coded responses to surveys. I conclude the paper by comparing the traditional approaches to the teaching of critical thinking to the visualisation approach. I argue that CAAM should taken seriously in the context of contemporary educational practices. (shrink)

The dearth of resources inherent in the study of Logic effected and affected by lack of reading materials, the financial constraints characteristic of the plight of students, the utter difficulty in teaching the subject shorthanded by write-then-explain method, made possible this undertaking. This text is a compilation of scholarly works by noted logicians that have made their way through publication. This work pales in comparison to their works and no deliberate efforts were made to water-down portions of their (...) book that were not included herein. I beg one favor of my readers that this endeavor should not be interpreted as mimicry, instead as compilation with the purpose of putting together brilliant works for enlightenment of the young. Mere mention of the names of the authors that I have used in this project is not even Commensurate to the efforts they have laid. Notwithstanding this fact, it is out of deference and admiration that pages for bibliography are allotted. It is fervently prayed for that this work will ease the difficulty of comprehension on the part of students and likewise facilitate instruction because of its rudimentary and comprehensive substance. (shrink)

This book serves as a concise introduction to some main topics in modern formal logic for undergraduates who already have some familiarity with formal languages. There are chapters on sentential and quantificational logic, modal logic, elementary set theory, a brief introduction to the incompleteness theorem, and a modern development of traditional Aristotelian Logic.

The article is devoted to the historical and philosophical problems of the study of the text of the "Gridaya Sutra" ("Sherab Nyingpo") within the tradition of Tibetan Buddhism. As a prolegomena, an overview of the field of translation was chosen - for a better understanding of both the logic of the formation of the Buddhist tradition of the Prajna-paramita sutras in India (which is revealed during translations from Sanskrit into Western languages), and the principles of their textual transmission to (...) Tibet (through the clarification of the principles of Tibetan translation from Sanskrit in the Lotsa tradition). The purpose of the study is to identify the mechanism that made it possible to make a transition in the Buddhist religious and philosophical space from the intellectual understanding of the ideas of emptiness (explained in the Magayana sutras) to their direct practice, the meaning of which is the detection of emptiness in one's own stream of consciousness (proposed by the tantric teachings of Vajrayana).. And since the "Gridaya-sutra" is the quintessence of the entire tradition of Prajna-paramita texts and contains a unique Buddhist mantra of "consolation in suffering", it, as a representative of the teachings of the sutras, was chosen for comparative analysis with the tradition of Mantrayana texts, i.e. Tantrayana. The representative of the tantric tradition was chosen to be the teaching text of the highest class of tantras - the Anuttarayogatantra called Kalachakra, which, due to its systematic development and strong semantic connections with classical Buddhist sciences, is considered the most consistent and structured tantra. As a result of research in Kalachakra, a specific complication of the meanings of the "Gridaya-sutra" was revealed, which opened the way for the individual from contemplating the emptiness of dharmas to collapsing the space-time continuum of the inner experience of conceptuality into a transcendent singularity for the sake of understanding true knowledge (jnana). In addition, the key mechanism of such a transition, which made it possible for the third turning of the Dharma wheel to become a continuation of the practice of the second turning, and not just another separate method, turned out to be the mechanism of the mantra, which is transformed from a concise morphological formula in the "Gridaya-sutra" into a phonetic formula in the Kalachakra. Therefore, the "Gridaya-sutra" takes the place of a trigger in the paramitayana, which makes it possible to organically connect the "higher" arsenal of Buddhist tantra to the process of enlightenment. (shrink)

Logicians disagree about how validity—the very heart of logic—should be understood. Many different formal systems have been born due to this disagreement. This thesis examines how teachers explain the subject matter of logic to students in introductory logic textbooks, and demonstrates the different explanations teachers use. These differences help explain why logicians have different intuitions about validity.

Philosophers such as Nietzsche, Heidegger, Derrida, Deleuze and Gendlin pronounce that difference must be understood as ontologically prior to identity. They teach that identity is a surface effect of difference, that to understand the basis of logico-mathematical idealities we must uncover their genesis in the fecundity of differentiation. In this paper, I contrast Heidegger’s analyses of the present to hand logico-mathematical object, which he discuses over the course of his career in terms of the ‘as’ structure, temporalization and enframing , (...) with the approaches of Gendlin and Deleuze, supplementing this discussion with Husserl’s investigations of mathematical idealities. Deleuze and Gendlin distinguish between the representational power a logical pattern has in itself, apart from its virtual generative source, to exactly repeat itself, and the way this self-same pattern is generated and changed by the larger situational texture within which it is embedded. In so doing, they misconstrue the empty, meaningless temporalization of logical calculation as the explicitly preserving carrying-through of already instituted implicit sense. For Heidegger, by contrast, logical inference is less a supplement to or development of implicit experience than a narrowing of its scope , a deficient mode of handiness. Experiencing something as present to hand extension modifies the relevant usefulness of ‘as’ structured comportment by stripping away what is meaningful in our relation with beings, and in the process stripping away its intelligibility. Thus, contrary to the assertions of Deleuze and Gendlin, extensive repetition does not carry through intelligible, relevant meaning, it dissolves understanding into the nihilism of empty calculation. (shrink)

This paper will present two contributions to teaching introductory logic. The first contribution is an alternative tree proof method that differs from the traditional one-sided tree method. The second contribution combines this tree system with an index system to produce a user-friendly tree method for sentential modal logic.

This article proposes that the prime ideals of the university - those of truth, knowledge, justice, and emancipation - are also those that currently produce unjust practices "outside" and "within". Using the work of Jacques Derrida and Paul Virilio, the article argues that the central problem of the university today consists not so much of a neo-liberalisation, but of the speeding-up of these ideals through their enmeshment with techniques of calculation, vision, and prediction. The current university therefore suffers from what (...) it with Derrida identifies as an "auto-immune disease," in which the acceleration of its foundational aspirations have led to a near-total subjugation of all and everything to an oppressive quest for transparency. However, the article proposes via Virilio that this totalising transparency paradoxically also produces more blindness, accidents, and unknowability. It hopes to illustrate this with some examples in the teaching scene as well by working through some of its own conceptual tensions. The other logic of the university today, the article finally proposes, consists of a "dark" or stealth functionality, opening up the promise of a radically different future and unanticipated resistance despite itself. (shrink)

A scientific perspective is a particular way of regarding something, using a scientific method. Scientific methodology is defined as a mode of research in which a problem is identified, relevant data gathered, a hypothesis formulated, and then empirically tested. Viewed from such a perspective, Guru Nanak's life was a continuous process of scientific experimentation and statement. Guru Nanak's life and writings are abundant in several such examples wherein his scientific approach to resolving several real-life situations with logic and rationality (...) is evident. (shrink)

he general theme of this paper is the issue of formalization in philosophy; in a more specific way, it deals with the issue of formalization of arguments in analytic philosophy of religion. One argument in particular – Anselm’s Proslogion II ontological argument – and one specific attempt to formalize it – Robert Adams’ formalization found in his paper “The Logical Structure of Anselm’s Arguments”, published in The Philosophical Review in 1971 – are taken as study cases. The purpose of the (...) paper is to critically analyze Adams’ formalization with the intent to shed some light on the following questions: What are the virtues of formally analyzing arguments and the contributions, if any, of such an enterprise to the debate on Anselm’s argument? Which lessons can Adam’s work teach us about the dangers and limitations of formalization? Do these virtues and dangers teach us something about analysis of arguments in general? (shrink)

The paper depicts the blueprint of an electronic wise indicating system for demonstrating learning French to understudies to overcome the inconveniences they go up against. The fundamental idea of this structure is a proficient introduction into learning French. The system shows the purpose of learning French and coordinates thusly made issues for the understudies to clarify. The system is logically balanced at run time to the understudy’s individual progress. The system gives unequivocal help to adaptable presentation to learners.

This short work presents a popular fringe theory as a source of case studies for use in teaching informal logic in an introductory course. It puts forward ancient astronaut theory as the candidate source, together with a characterization of why it fits the bill. The televised material associated with that theory is well suited to being used as case studies given that they are easy to follow, contain a surprising number of arguments and fallacies, and keep students reliably (...) engaged. The paper includes an overview of the forms of argumentation and fallacies that these cases may be used to teach, along with a sketch of how to best implement them in the classroom. (shrink)

Abstract: The paper depicts the blueprint of an electronic wise indicating system for demonstrating learning French to understudies to overcome the inconveniences they go up against. The fundamental idea of this structure is a proficient introduction into learning French. The system shows the purpose of learning French and coordinates thusly made issues for the understudies to clarify. The system is logically balanced at run time to the understudy’s individual progress. The system gives unequivocal help to adaptable presentation to learners.

- uneducated in the field authors who defend a consensus they are being TOLD when they enter offices of Ed-Sci, teaching and writing works on learning-theory - but never checked the facts, PART I and PART II.

The argument diagramming method developed by Monroe C. Beardsley in his (1950) book Practical Logic, which has since become the gold standard for diagramming arguments in informal logic, makes it possible to map the relation between premises and conclusions of a chain of reasoning in relatively complex ways. The method has since been adapted and developed in a number of directions by many contemporary informal logicians and argumentation theorists. It has proved useful in practical applications and especially pedagogically (...) in teaching basic logic and critical reasoning skills at all levels of scientific education. I propose in this essay to build on Beardsley diagramming techniques to refine and supplement their structural tools for visualizing logical relationships in a number of categories not originally accommodated by Beardsley diagramming, including circular reasoning, reductio ad absurdum arguments, and efforts to dispute and contradict arguments, with applications and analysis. (shrink)

From the beginning of the 16th century to the end of the 18th century, there were not less than ten philosophers who focused extensively on Venn’s ostensible analytical diagrams, as noted by modern historians of logic (Venn, Gardner, Baron, Coumet et al.). But what was the reason for early modern philosophers to use logic or analytical diagrams? Among modern historians of logic one can find two theses which are closely connected to each other: M. Gardner states that (...) since the Middle Ages certain logic diagrams were used just in order to teach “dull-witted students”. Therefore, logic diagrams were just a means to an end. According to P. Bernhard, the appreciation of logic diagrams had not started prior to the 1960s, therefore the fact that logic diagrams become an end the point of research arose very late. The paper will focus on the question whether logic resp. analytical diagrams were just means in the history of (early) modern logic or not. In contrast to Gardner, I will argue that logic diagrams were not only used as a tool for “dull-witted students”, but rather as a tool used by didactic reformers in early modern logic. In predating Bernhard’s thesis, I will argue that in the 1820s logic diagrams had already become a value in themselves in Arthur Schopenhauer’s lectures on logic, especially in proof theory. (shrink)

The first beginnings of modern logic in Croatia are recognizable as early as in the middle of the 19th century in Vatroslav Bertić. At the turn of the 20th century, Albin Nagy, who was teaching in Italy, made contributions to algebraic logic and to the philosophy of logic. At that time, a distinctive author Mate Meršić stood out, also working on algebraic logic. In the Croatian academic philosophy, until the publication of Gajo Petrović's textbook (1964) (...) and the contributions by Heda Festini, a critical or indifferent attitude toward modern logic prevailed. The first comprehensive modern logic textbook in Croatian was written by mathematician Vladimir Devidé (1964), known, for example, for his axiomatizations of natural numbers and classical propositional logic. In the Croatian philosophy of the 1980s and 1990s, influential logicians are Goran Švob and mathematician Zvonimir Šikić. At the turn of the 21st century, logical investigations are being intensified and extended to various branches of the contemporary logic. (shrink)

How "red", "green" were defined? Through analyzing how two children with congenitally inverted color sensations corresponding to red flags and green grass accept their grand mothers’ teaching about colors, the paper get opposite conclusions against logical empiricism. The “red” and “green” and other names of properties of objects were defined by objective physical properties (or together with behavior, such as in defining “beauty”), instead our sensations. So language directly points to things in themselves passing through sensations and presentative world. (...) It is not that things in themselves are unknowable, but that sensations and presentative world cannot be exactly described by daily language. (shrink)

The paper discusses the place and the status of fallacies in Arnauld and Nicole’s Port-Royal Logic, which seems to be the first book to introduce a radical change from the traditional Aristotelian account of fallacies. The most striking innovation is not in the definition of a fallacy but in the publication of a new list of fallacies, dropping some Aristotelian ones and adding more than ten new ones. The first part of the paper deals with the context of the (...) book’s publication. We then show the influence of Cartesian and Augustinian/Pascalian philosophy on the whole book, especially their common critical views about logic, dialectic and their traditional academic teaching. The third part of the paper discusses the two chapters on fallacies. It focuses on their place in the book and their relation with its general orientation, before turning to their content, closely connected with some major concerns of the time. (shrink)

Language is what separates us from other animals, and is why we are able to solve complex logical problems, and, as far as we can tell, experience consciousness. Wittgenstein is the philosopher of language par excellence. He asks: what is the role of language in philosophy, and in the wider sphere of thinking in everyday life? This teaching paper gives a summary of his ideas.

In Winter 2017, the first author piloted a course in formal logic in which we aimed to (a) improve student engagement and mastery of the content, and (b) reduce maths anxiety and its negative effects on student outcomes, by adopting student oriented teaching including peer instruction and classroom flipping techniques. The course implemented a partially flipped approach, and incorporated group-work and peer learning elements, while retaining some of the traditional lecture format. By doing this, a wide variety of (...) student learning preferences could be provided for. (shrink)

*NEWEST VERSION OF THIS RESOURCE ONLINE @ Philosop-her dotcom This textbook has developed over the last few years of teaching introductory symbolic logic and critical thinking courses. It has been truly a pleasure to have benefited from such great students and colleagues over the years. As we have become increasingly frustrated with the costs of traditional logic textbooks (though many of them deserve high praise for their accuracy and depth), the move to open source has become more (...) and more attractive. We're happy to provide it free of charge for educational use. With that being said, there are always improvements to be made here and we would be most grateful for constructive feedback and criticism. We have chosen to write this text in LaTex and have adopted certain conventions with symbols. Certainly many important aspects of critical thinking and logic have been omitted here, including historical developments and key logicians, and for that we apologize. Our goal was to create a textbook that could be provided to students free of charge and still contain some of the more important elements of critical thinking and introductory logic. To that end, an additional benefit of providing this textbook as a Open Education Resource (OER) is that we will be able to provide newer updated versions of this text more frequently, and without any concern about increased charges each time. We are particularly looking forward to expanding our examples, and adding student exercises. We will additionally aim to continually improve the quality and accessibility of our text for students and faculty alike. We have included a bibliography that includes many admirable textbooks, all of which we have benefited from. The interested reader is encouraged to consult these texts for further study and clarification. These texts have been a great inspiration for us and provide features to students that this concise textbook does not. We would both like to thank the philosophy students at numerous schools in the Puget Sound region for their patience and helpful suggestions. In particular, we would like to thank our colleagues at Green River College, who have helped us immensely in numerous different ways. Please feel free to contact us with comments and suggestions. We will strive to correct errors when pointed out, add necessary material, and make other additional and needed changes as they arise. Please check back for the most up to date version. (shrink)

Resumo -/- A partir da experiência de produção de uma videoaula de Lógica em Libras (Testa, Moraes, Bizio e Caló, 2021) para o IFSP FILOLIBRAS, inserida no contexto do projeto ‘O Ensino de Filosofia para Surdos: elaboração de material didático em uma perspectiva de inclusão escolar’ (Moraes e Bizio, 2021), levantamos algumas questões relativas ao arcabouço teórico do projeto. Após introduzirmos as motivações do projeto, explicamos como sua metodologia foi tratada no contexto da aula de Lógica, expondo as principais dificuldades (...) e soluções encontradas. Através da análise desta experiência, pretendemos levantar subsídios teóricos para pesquisas futuras no tema, incluindo a produção de um curso inteiramente voltado para o ensino de Lógica em Libras, não apenas enquanto área da Filosofia, mas também enquanto área de estudo multi e transdisciplinar, propedêutica ao fomento do Pensamento Crítico. -/- Palavras-chave: Ensino de Lógica, Educação Especial, Pedagogia Bilíngue, Libras. -/- Abstract -/- Based on the experience of producing a video lesson on Logic in Libras (Testa, Moraes, Bizio and Caló, 2021) to the IFSP FILOLIBRAS, inserted in the context of the project ‘O Ensino de Filosofia para Surdos: elaboração de material didático em uma perspectiva de inclusão escolar’ (‘Teaching Philosophy for Deaf students: elaboration of didactic material in a perspective of inclusive education’, in a free translation) (Moraes and Bizio, 2021), we raise some questions regarding the theoretical framework of the project. After introducing the motivations of the project, we explain how the chosen methodology was considered in the context of the Logic class, exposing the main difficulties and solutions found. Through the analysis of this experience, we intend to raise theoretical subsidies for future research on the subject, including the production of a course entirely focused on the teaching of Logic in Libras, not only as an area of Philosophy but also as a multi and transdisciplinary area of study, propaedeutic to fostering Critical Thinking. -/- Keywords: Teaching Logic, Special Education, Bilingual Pedagogy, Libras. (shrink)

En diciembre de 1996 fui invitado a impartir una sesión a profesores de lógica en los estudios institucionales del Studium Generale de la Prelatura del Opus Dei. En aquella ocasión preparé concienzudamente un texto escrito que pasé a mi querido y admirado colega Ángel Luis González para su revisión. Pocos días después Ángel Luis me lo devolvió con unas pocas correcciones y sugerencias y un alentador “¡Mucho ánimo!” en su encabezamiento. Durante muchos años conservé ese texto con sus anotaciones manuscritas. (...) Por este motivo, me ha parecido que podría ser adecuado reproducir en este volumen en homenaje de Ángel Luis aquella exposición en forma abreviada con unas pocas correcciones y actualizaciones de detalle. Mi exposición se divide en tres partes: 1) Situación de la lógica en la filosofía contemporánea; 2) El papel de la lógica en los estudios institucionales; y termina con 3) A modo de conclusión, una reflexión más personal. (shrink)