From Antiphilosophy to Worlds and from Beckett to Wittgenstein, the 110 entries in this dictionary provide detailed explanations and engagements with Badious's key concepts and major interlocutors.

New Blackfriars, Volume 102, Issue 1101, Page 694-705, September 2021.

A literary and historical analysis of the structure and meaning of recurrent symbols, images, and actions employed in Plato’s dialogues. In this book, Clinton DeBevoise Corcoran examines the use of place in Plato’s dialogues. Corcoran argues that spatial representations, such as walls, caves, and roads, as well as the creation of eternal patterns and chaotic images in the particular spaces, times, characterizations, and actions of the dialogues, provide clues to Plato’s philosophic project. Throughout the dialogues, the Good serves (...) as an overarching ordering principle for the construction of place and the proper limit of spaces, whether they be here in the world, deep in the underworld, or in the nonspatial ideal realm of the Forms. The Good, since it escapes the limits of space and time, equips Plato with a powerful mythopoetic tool to create settings, frames, and arguments that superimpose different dimensions of reality, allowing worlds to overlap that would otherwise be incommensurable. The Good also serves as a powerful ethical tool for evaluating the order of different spaces. Corcoran explores how Plato uses wrestling and war as metaphors for the mixing of the nonspatial, eternal forms in the world and history, and how he uses spatial images throughout the dialogues to critique Athens’s tragic overreach in the Peloponnesian War. Far from merely an incidental backdrop in the dialogues, place etches the tragic intersection of the mortal and the immortal, good and evil, and Athens’s past, present, and future. (shrink)

The work of creating the future is being done now ─ and much of it is unsustainable in terms of natural and cultural resources. How will the next generation of leadership for environmental sustainability be raised up? Can we imagine sustainable futures, and can we enable transformative leadership to help us realize them? How can we best ensure that the several generations share their particular knowledge? What are the ethical frameworks, methodologies, curricula, and tools necessary for advancing and strengthening education (...) for intergenerational sustainability learning and leadership? In this book, 82 authors from 26 countries across 6 continents seek answers in 32 essays to the many questions related to the intergenerational collaboration that holds promise for creating sustainable futures. The authors themselves represent a diversity of geography, gender, and generation ─ and include the institutions comprising the emerging International Intergenerational Net-work of Centers. They speak to key principles, perspectives, and praxes at the intersection of intergenerational learning and transformative leadership in the context of education for sustainability. A visionary tour de force, this book explores the challenges and complexities of future learning models beyond the UN Decade of Education for Sustainable Development. It provides a timely discourse encompassing intergenerational and cultural dimensions, including ethics. Contributors articulate a deeper understanding of leadership in the post-Enlightenment era. Chapters in the book offer examples cutting across a diverse range of experiences worldwide, making this volume not only refreshing for practitioners, but also invaluable to policy-makers. ─ Dzulkifli Abdul Razak President, International Association of Universities Former Vice-Chancellor, Universiti Sains Malaysia These are critical times for the global community. Big issues are on the table: security, climate change, environmental destabilization, disease, hunger, and others. Actions we take today must embrace lessons from the past, present realities, and the rights and interests of future generations. Indeed, intergenerational learning is vital to education for sustainable development. This book makes a very timely and urgent call, challenging our traditional learning approaches and inviting us to dig deeper, stretch our minds wider, and see farther. ─ Akpezi Ogbuigwe Former Head of Environmental Education and Training, United Nations Environment Programme Executive Chairperson, Anpez Center for Environment and Development, Port Harcourt, Nigeria Achieving sustainable development requires generations, young and old, to engage with and learn from each other, and for leadership to embrace change. That is one of the key messages of this remarkable book. I wish I had this book when I took up the leadership role of my university two decades ago. I wish I could have given this book to my sons as they started their university education. ─ Goolam Mohamedbhai Former Secretary-General, Association of African Universities Honorary President, International Association of Universities. (shrink)

Young people have an enormous stake in the present and future state of Earth. Almost half of the human population is under the age of 25. If young people’s resources of energy, time, and knowledge are misdirected towards violence, terrorism, socially-isolating technologies, and unsustainable consumption, civilization risks destabilization. Yet, there is a powerful opportunity for society if young people can participate positively in all aspects of sustainable development. In order to do so, young people need education, political support, resources, skills, (...) and hope. This volume offers a global perspective on education initiatives by and for young people that promote a transition to sustainability. It includes 38 essays co-authored by 68 contributors from 25 nations, representing a diversity of geography, gender, and generation. "The development of youth leadership has been a central concern of my professional work. Young people have a lot to offer to sustainable development and should participate in planning for our planet’s future. This rich collection of theoretical and practical approaches captures the growing response of young people to this challenge. I am particularly pleased with the attention paid to the role of often-underrepresented regions of the world, to the role of women, and to the Earth Charter in the creating of a peaceful and sustainable future." Julia Marton-Lefèvre, Director General of IUCN (International Union for Conversation of Nature), and former Rector of the University for Peace "This book is a rich, global tapestry depicting the inter-linkages among youth, education and sustainable development. What is particularly interesting is that it shows how education, at all levels, can be a powerful engine for promoting sustainability. This work is an important contribution to the United Nations Decade of Education for Sustainable Development." Goolam Mohamedbhai, Secretary-General, Association of African Universities, and Past President, International Association of Universities "This important and skillfully-prepared book comes at a critical time.... A great strength of this volume is that it blends together theoretical and practical insights regarding education for sustainable development by and for young people. It is written from diverse cultural perspectives from all world regions. Readers, from many fields, especially young people and their teachers, will find it timely and relevant. I hope that it will inspire the younger generation to get involved in seeking solutions to the challenges we face." James Gustave Speth, Former Administrator of the United Nations Development Programme, and Dean Emeritus of the School of Forestry and Environmental Studies at Yale University "It is my hope that you are inspired, as I have been, by Young People, Education, and Sustainable Development: Exploring Principles, Perspectives, and Praxis, for the stories in this volume are full of the possibilities that emerge when we honor and support young people." Ruud Lubbers, Former Prime Minister of The Netherlands, and past United Nations High Commissioner for Refugees. (shrink)

It is one thing for a given proposition to follow or to not follow from a given set of propositions and it is quite another thing for it to be shown either that the given proposition follows or that it does not follow.* Using a formal deduction to show that a conclusion follows and using a countermodel to show that a conclusion does not follow are both traditional practices recognized by Aristotle and used down through the history of logic. These (...) practices presuppose, respectively, a criterion of validity and a criterion of invalidity each of which has been extended and refined by modern logicians: deductions are studied in formal syntax (proof theory) and coun¬termodels are studied in formal semantics (model theory). The purpose of this paper is to compare these two criteria to the corresponding criteria employed in Boole’s first logical work, The Mathematical Analysis of Logic (1847). In particular, this paper presents a detailed study of the relevant metalogical passages and an analysis of Boole’s symbolic derivations. It is well known, of course, that Boole’s logical analysis of compound terms (involving ‘not’, ‘and’, ‘or’, ‘except’, etc.) contributed to the enlargement of the class of propositions and arguments formally treatable in logic. The present study shows, in addition, that Boole made significant contributions to the study of deduc¬tive reasoning. He identified the role of logical axioms (as opposed to inference rules) in formal deductions, he conceived of the idea of an axiomatic deductive sys¬tem (which yields logical truths by itself and which yields consequences when ap¬plied to arbitrary premises). Nevertheless, surprisingly, Boole’s attempt to imple¬ment his idea of an axiomatic deductive system involved striking omissions: Boole does not use his own formal deductions to establish validity. Boole does give symbolic derivations, several of which are vitiated by “Boole’s Solutions Fallacy”: the fallacy of supposing that a solution to an equation is necessarily a logical consequence of the equation. This fallacy seems to have led Boole to confuse equational calculi (i.e., methods for gen-erating solutions) with deduction procedures (i.e., methods for generating consequences). The methodological confusion is closely related to the fact, shown in detail below, that Boole had adopted an unsound criterion of validity. It is also shown that Boole totally ignored the countermodel criterion of invalid¬ity. Careful examination of the text does not reveal with certainty a test for invalidity which was adopted by Boole. However, we have isolated a test that he seems to use in this way and we show that this test is ineffectual in the sense that it does not serve to identify invalid arguments. We go beyond the simple goal stated above. Besides comparing Boole’s earliest criteria of validity and invalidity with those traditionally (and still generally) employed, this paper also investigates the framework and details of THE MATHEMATICAL ANALYSIS OF LOGIC. (shrink)

Published with the aid of a grant from the National Endowment for the Humanities. Contains the only complete English-language text of The Concept of Truth in Formalized Languages. Tarski made extensive corrections and revisions of the original translations for this edition, along with new historical remarks. It includes a new preface and a new analytical index for use by philosophers and linguists as well as by historians of mathematics and philosophy.

This is a theoretical treatment of the term "Sha Zhu Pan" (杀猪盘) in Chinese, which translates to “Pig-Butchering” in English. The article critically examines the propagation and validation of "Pig Butchering," an animal metaphor, and its implications for the dehumanisation of victims of online fraud across various discourses. The study provides background information about this type of fraud before investigating its theoretical foundations and linking its emergence to the dehumanisation of fraud victims. The analysis highlights the disparity between academic literature, (...) subjected to rigorous peer-review processes, and sensationalised narratives prevalent in the media. While academic works subject "pig butchering" to critical scrutiny and refrain from endorsing derogatory terms to depict fraud victims, numerous media outlets employ this term uncritically, further worsening the predicament of these victims. "Pig butchering" is firmly rooted in the concept of dehumanisation, and this article underscores how language moulds perceptions of fraud and behaviour, extending to the development of preventive strategies. The role of law enforcement agencies in generating and disseminating materials is also a central theme, emphasising their responsibility as trusted sources of information. We suggest that these agencies should adopt non-victim shaming language to encourage victims to report crimes and alleviate the stigma attached to victimisation. Additionally, the article offers valuable cross-cultural insights by comparing metaphors from Chinese and Nigerian contexts. This comparative analysis enriches our comprehension of the global dimensions of online fraud and its cultural diversities, highlighting the substantial impact of language on perceptions and behaviours. We advocate for a departure from victim-blaming tendencies perpetuated by select media outlets, urging a more compassionate and accurate portrayal of those affected by online fraud. We, therefore, call for a more empathetic and accurate portrayal of individuals affected by online fraud, aligning with the broader objective of promoting understanding and support for these victims. (shrink)

This paper develops a modal, Sentential logic having "not", "if...Then" and necessity as logical constants. The semantics (system of meanings) of the logic is the most obvious generalization of the usual truth-Functional semantics for sentential logic and its deductive system (system of demonstrations) is an obvious generalization of a suitable (jaskowski-Type) natural deductive system for sentential logic. Let a be a set of sentences and p a sentence. "p is a logical consequence of a" is defined relative to the semantics (...) and "p is demonstrable from a" is defined relative to the deductive system. Main meta-Theorem: p is demonstrable from a if and only if p is a logical consequence of a. Henkin-Type methods are used. The theorems of the logic are exactly those of s5. The deductive system is rigorously developed as a system of linear strings. (shrink)

Based on the analysis of 267 tokens derived from editorial columns primarily drawn from two Persian newspapers, following on earlier studies by Chafe, Jahani, Lazard, Dahl, Adel, and Dafouze, and inspired by a series of Hyland’s studies on metadiscourse signals, this study has aimed at investigating evidential markers in these columns. In order to come to grips with the types of evidentials, first, we classify them into two major types – inferential and reportative; the reportative evidentials are further classified into (...) four types. The reportative classification is based, in the first place, on whether the source of information comes from an individual or from a government body, hence institutional. The further classifications are based on their identifiability/specificity. Results show that inferential evidentials comprise about 15% of all the tokens. Among the four reportative types, those whose source is individual and identified/specified and those that are institutional and unidentified/unspecified, coded as TYPE 1 and TYPE 4, respectively, have the highest frequency. The results overall show that Persian editorials in these two papers feature a high frequency of attribution of information to identified sources when the source is individual, but to unidentified sources when the source is institutional. The results also support other authors who claim that the imperfective aspect marker mi-, which is frequent in Persian, is a marker worthy of consideration in evidentiality. (shrink)

Ferrarese develops our thinking about the social conditions of caring for others, while arguing for an understanding of morality that is materialist and political - always-already political.

We live in turbulent times, our world is changing at accelerating speed. Information is everywhere, but wisdom appears in short supply when trying to address key inter-related challenges of our time such as; runaway climate change, the loss of biodiversity, the depletion of natural resources, the on-going homogenization of culture, and rising inequity. Living in such times has implications for education and learning. This book explores the possibilities of designing and facilitating learning-based change and transitions towards sustainability. In 31 chapters (...) contributors from across the world discuss (re)emerging forms of learning that not only assist in breaking down unsustainable routines, forms of governance, production and consumption, but also can help create ones that are more sustainable. The book has been divided into three parts: re-orienting science and society, re-connecting people and planet and re-imagining education and learning. This is essential reading for educators, educational designers, change agents, researchers, students, policymakers and entrepreneurs alike, who are concerned about the well-being of the planet and convinced of our ability to do better. The content and related issues can be discussed on the blog by editor Arjen Wals: Transformative learning. 'We are living in times of incertitude, complexity, and contestation, but also of connectivity, responsibility, and new opportunities. This book analyses the consequences of these times for learning in formal, non-formal, and informal education. It explores the possibilities offered by the concept of sustainability as a central category of a holistic paradigm which harmonizes human beings with Earth. To change people and to change the world are interdependent processes - this book contributes to both.' Moacir Gadotti, Director of Paulo Freire Institute, São Paulo, Brazil 'I hope you share my excitement about the innovations for sustainability that this book catalogues and analyses. While the ecological news is grim, the human news is not. Even in a time of accelerating change, people are showing their enormous capacities to learn, adapt, restore and protect.' From the Foreword by Juliet Schor, author of ‘True Wealth: how and why millions of Americans are creating a time-rich, ecologically-light, small-scale high-satisfaction economy’ 'This book implies a ‘culture of critical commitment’ in educational thinking and practice - engaged enough to make a real difference to social-ecological resilience and sustainability but reflexively critical enough to learn constantly from experience and to keep options open in working for a sustainability transformation.' From the Afterword by Stephen Sterling, Professor of Sustainability Education, Centre for Sustainable Futures, Plymouth University, United Kingdom. (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)

In lieu of an abstract, here is a brief excerpt of the content:Logic of the SiteAlain Badiou (bio)Translated by Steve Corcoran (bio) and Bruno Bosteels (bio)The Commune Is a Site 1. Ontology of the CommuneTake any world whatsoever. A multiple that is an object of this world—whose elements are indexed by the transcendental of this world—is a site, if it happens to count itself within the referential field of its own indexation. Or again: a site is a multiple that (...) happens to behave in the world with regard to itself as with regard to its elements, in such a way as to be the support of being of its own appearance.Even if the idea is still obscure, we can begin to see its content: a site is a singularity, because it convokes its being in the appearing of its own multiple composition. It makes itself, in the world, the being-there of its being. Among other consequences, the site gives itself an intensity of existence. A site is a being that happens to exist by itself.We will ask: can we give a more concrete idea of what a site is? Is there a site?Let us consider the world "Paris at the end of the Franco-Prussian War of 1870." We are in the month of March 1871. After a semblance of resistance, and shot through with fear of revolutionary and worker Paris, the interim government of bourgeois "Republicans" capitulates to Bismarck's Prussians. In order to consolidate this political "victory"—very comparable to Petain's reactionary revenge in 1940 (where preferring an arrangement with the external enemy to exposure to the internal enemy)—it has an assembly with a royalist majority hastily elected by a frightened rural world, an assembly that sits in Bordeaux.Led by Thiers, the government hopes to take advantage of the circumstances to annihilate the political capacity of the workers. But on the Parisian front, the proletariat is armed in the form of a National Guard, owing to its having been mobilized during the siege on Paris. In theory the Parisian proletariat has many hundreds of cannons at its disposition. The "military" organism of the Parisians is the Central Committee, at which assemble the delegates of the various battalions of the National Guard, battalions [End Page 141] that are in turn linked to the great working-class quartiers of Paris—Montmartre, Belleville, and so forth.Thus we have a divided world whose logical organization—what in philosophical jargon could be called its transcendental organisation—reconciles intensities of political existence according to two sets of antagonistic criteria. Concerning the representative, electoral, and legal dispositions, one cannot but observe the preeminence of the Assembly of traditionalist Rurals,1 Thiers's capitulard government, and the officers of the regular army, who, having been licked without much of a fight by the Prussian soldiers, dream of doing battle with the Parisian workers. That is where the power is, especially as it is the only power recognized by the occupier. On the side of resistance, political intervention, and French revolutionary history, there is the fecund disorder of Parisian worker organizations, which intermingles with the Central Committee of the twenty quartiers, the Federation of Syndicate Chambers, a few members of the International, local military committees. In truth, the historical consistency of this world, which had been separated and disbanded [délié] owing to the war, is held together only by the majority conviction that no kind of worker capability for government exists. For the vast majority of people, including often the workers themselves, the politicized workers of Paris are simply incomprehensible. These workers are the inexistent aspect [l'inexistant propre] of the term "political capacity" in the uncertain world of the spring of 1871. But for the bourgeoisie they are still too existent, at least physically. The government receives threats from the stock exchange of this sort: "You will never have financial operations if you do not get rid of these reprobates." First up then, an imperative task, and a seemingly easy one to carry out: disarm the workers and, in particular, recuperate the cannons spread throughout working-class Paris by the military committees of the National Guard. It is this initiative that will make... (shrink)

Emergency declarations are a vital legal authority that can activate funds, personnel, and material and change the legal landscape to aid in the response to a public health threat. Traditionally, declarations have been used against immediate and unforeseen threats such as hurricanes, tornadoes, wildfires, and pandemic influenza. Recently, however, states have used emergency declarations to address public health issues that have existed in communities for months and years and have risk factors such as poverty and substance misuse. Leaders in these (...) states have chosen to use emergency powers that are normally reserved for sudden catastrophes to address these enduring public health issues. This article will explore emergency declarations as a legal mechanism for response; describe recent declarations to address hepatitis A and the opioid overdose epidemic; and seek to answer the question of whether it is appropriate to use emergency powers to address public health issues that are not traditionally the basis for an emergency declaration. (shrink)

Lynne Baker was a trenchant critic of reductionist and physicalist conceptions of the universe, as well as the foremost defender of the constitution view of human persons. Baker was a staunch defender of a kind of practical realism, or what she sometimes called a metaphysics of everyday life. And it was this general “common sense” philosophical outlook that underwrote her non-reductionist, constitution view of reality. Whereas most of her contemporaries were given to metaphysical reductionism and eliminativism, born of a penchant (...) for so-called Quinean desert landscapes, Baker was unapologetic and philosophically deft in her defense of ontological pluralism. This volume honors Baker’s work by bringing together 16 critical essays by some of her students, colleagues, interlocutors, and friends. The essays fall into four areas, each an area to which Baker made unique and influential contributions: Practical Realism about the Mind, The Constitution View of Human Persons, The First Person Perspective, and God, Christianity and Naturalism. (shrink)

What is the function of cognition? On one influential account, cognition evolved to co-ordinate behaviour with environmental change or complexity. Liberal interpretations of this view ascribe cognition to an extraordinarily broad set of biological systems—even bacteria, which modulate their activity in response to salient external cues, would seem to qualify as cognitive agents. However, equating cognition with adaptive flexibility per se glosses over important distinctions in the way biological organisms deal with environmental complexity. Drawing on contemporary advances in theoretical biology (...) and computational neuroscience, we cash these distinctions out in terms of different kinds of generative models, and the representational and uncertainty-resolving capacities they afford. This analysis leads us to propose a formal criterion for delineating cognition from other, more pervasive forms of adaptive plasticity. On this view, biological cognition is rooted in a particular kind of functional organisation; namely, that which enables the agent to detach from the present and engage in counterfactual inference. (shrink)

This presentation of Aristotle's natural deduction system supplements earlier presentations and gives more historical evidence. Some fine-tunings resulted from conversations with Timothy Smiley, Charles Kahn, Josiah Gould, John Kearns,John Glanvillle, and William Parry.The criticism of Aristotle's theory of propositions found at the end of this 1974 presentation was retracted in Corcoran's 2009 HPL article "Aristotle's demonstrative logic".

In previous articles, it has been shown that the deductive system developed by Aristotle in his "second logic" is a natural deduction system and not an axiomatic system as previously had been thought. It was also stated that Aristotle's logic is self-sufficient in two senses: First, that it presupposed no other logical concepts, not even those of propositional logic; second, that it is (strongly) complete in the sense that every valid argument expressible in the language of the system is deducible (...) by means of a formal deduction in the system. Review of the system makes the first point obvious. The purpose of the present article is to prove the second. Strong completeness is demonstrated for the Aristotelian system. (shrink)

For each positive n , two alternative axiomatizations of the theory of strings over n alphabetic characters are presented. One class of axiomatizations derives from Tarski's system of the Wahrheitsbegriff and uses the n characters and concatenation as primitives. The other class involves using n character-prefixing operators as primitives and derives from Hermes' Semiotik. All underlying logics are second order. It is shown that, for each n, the two theories are definitionally equivalent [or synonymous in the sense of deBouvere]. It (...) is further shown that each member of one class is synonymous with each member of the other class; thus that all of the theories are definitionally equivalent with each other and with Peano arithmetic. Categoricity of Peano arithmetic then implies categoricity of each of the above theories. (shrink)

Argumentations are at the heart of the deductive and the hypothetico-deductive methods, which are involved in attempts to reduce currently open problems to problems already solved. These two methods span the entire spectrum of problem-oriented reasoning from the simplest and most practical to the most complex and most theoretical, thereby uniting all objective thought whether ancient or contemporary, whether humanistic or scientific, whether normative or descriptive, whether concrete or abstract. Analysis, synthesis, evaluation, and function of argumentations are described. Perennial philosophic (...) problems, epistemic and ontic, related to argumentations are put in perspective. So much of what has been regarded as logic is seen to be involved in the study of argumentations that logic may be usefully defined as the systematic study of argumentations, which is virtually identical to the quest of objective understanding of objectivity. (shrink)

After a short preface, the first of the three sections of this paper is devoted to historical and philosophic aspects of categoricity. The second section is a self-contained exposition, including detailed definitions, of a proof that every mathematical system whose domain is the closure of its set of distinguished individuals under its distinguished functions is categorically characterized by its induction principle together with its true atoms (atomic sentences and negations of atomic sentences). The third section deals with applications especially those (...) involving the distinction between characterizing a system and axiomatizing the truths of a system. (shrink)

This collection brings together cutting-edge research on the metaphysics of human nature and soul-body dualism.Kevin Corcoran's collection, Soul, Body, and ...

As the new Cell-free DNA (Cf-DNA) prenatal screening test for Down syndrome was being introduced into the UK’s fetal anomaly screening program, Down syndrome charities had an opportunity to participate. An experience of co-production where we were the minority voice then followed. This paper explores that process and our experience as a charity. Institutional and societal structures meant that it was difficult to be heard and a significant amount of bias was noted within the program. Consequently, our viewpoints were often (...) considered and then dismissed. However, at times we were listened to, and feel that there were some valuable changes made resulting from our involvement. The end product was far from reflective of all that we stand for, and there are still lessons to be learned in England about the need to place a higher value on minority voices of lived experience in a co-production exercise. (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 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)

Prior Analytics by the Greek philosopher Aristotle (384 – 322 BCE) and Laws of Thought by the English mathematician George Boole (1815 – 1864) are the two most important surviving original logical works from before the advent of modern logic. This article has a single goal: to compare Aristotle’s system with the system that Boole constructed over twenty-two centuries later intending to extend and perfect what Aristotle had started. This comparison merits an article itself. Accordingly, this article does not discuss (...) many other historically and philosophically important aspects of Boole’s book, e.g. his confused attempt to apply differential calculus to logic, his misguided effort to make his system of ‘class logic’ serve as a kind of ‘truth-functional logic’, his now almost forgotten foray into probability theory, or his blindness to the fact that a truth-functional combination of equations that follows from a given truth-functional combination of equations need not follow truth-functionally. One of the main conclusions is that Boole’s contribution widened logic and changed its nature to such an extent that he fully deserves to share with Aristotle the status of being a founding figure in logic. By setting forth in clear and systematic fashion the basic methods for establishing validity and for establishing invalidity, Aristotle became the founder of logic as formal epistemology. By making the first unmistakable steps toward opening logic to the study of ‘laws of thought’—tautologies and laws such as excluded middle and non-contradiction—Boole became the founder of logic as formal ontology. (shrink)

Demonstrative logic, the study of demonstration as opposed to persuasion, is the subject of Aristotle's two-volume Analytics. Many examples are geometrical. Demonstration produces knowledge (of the truth of propositions). Persuasion merely produces opinion. Aristotle presented a general truth-and-consequence conception of demonstration meant to apply to all demonstrations. According to him, a demonstration, which normally proves a conclusion not previously known to be true, is an extended argumentation beginning with premises known to be truths and containing a chain of reasoning showing (...) by deductively evident steps that its conclusion is a consequence of its premises. In particular, a demonstration is a deduction whose premises are known to be true. Aristotle's general theory of demonstration required a prior general theory of deduction presented in the Prior Analytics. His general immediate-deduction-chaining conception of deduction was meant to apply to all deductions. According to him, any deduction that is not immediately evident is an extended argumentation that involves a chaining of intermediate immediately evident steps that shows its final conclusion to follow logically from its premises. To illustrate his general theory of deduction, he presented an ingeniously simple and mathematically precise special case traditionally known as the categorical syllogistic. (shrink)

Corcoran’s 27 entries in the 1999 second edition of Robert Audi’s Cambridge Dictionary of Philosophy [Cambridge: Cambridge UP]. -/- ancestral, axiomatic method, borderline case, categoricity, Church (Alonzo), conditional, convention T, converse (outer and inner), corresponding conditional, degenerate case, domain, De Morgan, ellipsis, laws of thought, limiting case, logical form, logical subject, material adequacy, mathematical analysis, omega, proof by recursion, recursive function theory, scheme, scope, Tarski (Alfred), tautology, universe of discourse. -/- The entire work is available online free at more (...) than one website. Paste the whole URL. http://archive.org/stream/RobertiAudi_The.Cambridge.Dictionary.of.Philosophy/Robert.Audi_The.Cambrid ge.Dictionary.of.Philosophy -/- The 2015 third edition will be available soon. Before you think of buying it read some reviews on Amazon and read reviews of its competition: For example, my review of the 2008 Oxford Companion to Philosophy, History and Philosophy of Logic,29:3,291-292. URL: http://dx.doi.org/10.1080/01445340701300429 -/- Some of the entries have already been found to be flawed. For example, Tarski’s expression ‘materially adequate’ was misinterpreted in at least one article and it was misused in another where ‘materially correct’ should have been used. The discussion provides an opportunity to bring more flaws to light. -/- Acknowledgements: Each of these entries was presented at meetings of The Buffalo Logic Dictionary Project sponsored by The Buffalo Logic Colloquium. The members of the colloquium read drafts before the meetings and were generous with corrections, objections, and suggestions. Usually one 90-minute meeting was devoted to one entry although in some cases, for example, “axiomatic method”, took more than one meeting. Moreover, about half of the entries are rewrites of similarly named entries in the 1995 first edition. Besides the help received from people in Buffalo, help from elsewhere was received by email. We gratefully acknowledge the following: José Miguel Sagüillo, John Zeis, Stewart Shapiro, Davis Plache, Joseph Ernst, Richard Hull, Concha Martinez, Laura Arcila, James Gasser, Barry Smith, Randall Dipert, Stanley Ziewacz, Gerald Rising, Leonard Jacuzzo, George Boger, William Demopolous, David Hitchcock, John Dawson, Daniel Halpern, William Lawvere, John Kearns, Ky Herreid, Nicolas Goodman, William Parry, Charles Lambros, Harvey Friedman, George Weaver, Hughes Leblanc, James Munz, Herbert Bohnert, Robert Tragesser, David Levin, Sriram Nambiar, and others. -/- . (shrink)

Since the time of Aristotle's students, interpreters have considered Prior Analytics to be a treatise about deductive reasoning, more generally, about methods of determining the validity and invalidity of premise-conclusion arguments. People studied Prior Analytics in order to learn more about deductive reasoning and to improve their own reasoning skills. These interpreters understood Aristotle to be focusing on two epistemic processes: first, the process of establishing knowledge that a conclusion follows necessarily from a set of premises (that is, on the (...) epistemic process of extracting information implicit in explicitly given information) and, second, the process of establishing knowledge that a conclusion does not follow. Despite the overwhelming tendency to interpret the syllogistic as formal epistemology, it was not until the early 1970s that it occurred to anyone to think that Aristotle may have developed a theory of deductive reasoning with a well worked-out system of deductions comparable in rigor and precision with systems such as propositional logic or equational logic familiar from mathematical logic. When modern logicians in the 1920s and 1930s first turned their attention to the problem of understanding Aristotle's contribution to logic in modern terms, they were guided both by the Frege-Russell conception of logic as formal ontology and at the same time by a desire to protect Aristotle from possible charges of psychologism. They thought they saw Aristotle applying the informal axiomatic method to formal ontology, not as making the first steps into formal epistemology. They did not notice Aristotle's description of deductive reasoning. Ironically, the formal axiomatic method (in which one explicitly presents not merely the substantive axioms but also the deductive processes used to derive theorems from the axioms) is incipient in Aristotle's presentation. Partly in opposition to the axiomatic, ontically-oriented approach to Aristotle's logic and partly as a result of attempting to increase the degree of fit between interpretation and text, logicians in the 1970s working independently came to remarkably similar conclusions to the effect that Aristotle indeed had produced the first system of formal deductions. They concluded that Aristotle had analyzed the process of deduction and that his achievement included a semantically complete system of natural deductions including both direct and indirect deductions. Where the interpretations of the 1920s and 1930s attribute to Aristotle a system of propositions organized deductively, the interpretations of the 1970s attribute to Aristotle a system of deductions, or extended deductive discourses, organized epistemically. The logicians of the 1920s and 1930s take Aristotle to be deducing laws of logic from axiomatic origins; the logicians of the 1970s take Aristotle to be describing the process of deduction and in particular to be describing deductions themselves, both those deductions that are proofs based on axiomatic premises and those deductions that, though deductively cogent, do not establish the truth of the conclusion but only that the conclusion is implied by the premise-set. Thus, two very different and opposed interpretations had emerged, interestingly both products of modern logicians equipped with the theoretical apparatus of mathematical logic. The issue at stake between these two interpretations is the historical question of Aristotle's place in the history of logic and of his orientation in philosophy of logic. This paper affirms Aristotle's place as the founder of logic taken as formal epistemology, including the study of deductive reasoning. A by-product of this study of Aristotle's accomplishments in logic is a clarification of a distinction implicit in discourses among logicians--that between logic as formal ontology and logic as formal epistemology. (shrink)

Presents a new way of looking at what it means to be human, offering a convincing case that humans are more than immaterial souls or "biological computers".

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)

In the present article we attempt to show that Aristotle's syllogistic is an underlying logiC which includes a natural deductive system and that it isn't an axiomatic theory as had previously been thought. We construct a mathematical model which reflects certain structural aspects of Aristotle's logic. We examine the relation of the model to the system of logic envisaged in scattered parts of Prior and Posterior Analytics. Our interpretation restores Aristotle's reputation as a logician of consummate imagination and skill. Several (...) attributions of shortcomings and logical errors to Aristotle are shown to be without merit. Aristotle's logic is found to be self-sufficient in several senses: his theory of deduction is logically sound in every detail. (His indirect deductions have been criticized, but incorrectly on our account.) Aristotle's logic presupposes no other logical concepts, not even those of propositional logic. The Aristotelian system is seen to be complete in the sense that every valid argument expressible in his system admits of a deduction within his deductive system: every semantically valid argument is deducible. (shrink)