Using labelled formulae, a cut-free sequent calculus for intuitionistic propositional logic is presented, together with an easy cut-admissibility proof; both extend to cover, in a uniform fashion, all intermediate logics characterised by frames satisfying conditions expressible by one or more geometric implications. Each of these logics is embedded by the Gödel–McKinsey–Tarski translation into an extension of S4. Faithfulness of the embedding is proved in a simple and general way by constructive proof-theoretic methods, without appeal to semantics other than in the (...) explanation of the rules. (shrink)

In the proof-theoretic semantics approach to meaning, harmony , requiring a balance between introduction-rules (I-rules) and elimination rules (E-rules) within a meaning conferring natural-deduction proof-system, is a central notion. In this paper, we consider two notions of harmony that were proposed in the literature: 1. GE-harmony , requiring a certain form of the E-rules, given the form of the I-rules. 2. Local intrinsic harmony : imposes the existence of certain transformations of derivations, known as reduction and expansion . We propose (...) a construction of the E-rules (in GE-form) from given I-rules, and prove that the constructed rules satisfy also local intrinsic harmony. The construction is based on a classification of I-rules, and constitute an implementation to Gentzen’s (and Pawitz’) remark, that E-rules can be “read off” I-rules. (shrink)

That every first-order theory has a coherent conservative extension is regarded by some as obvious, even trivial, and by others as not at all obvious, but instead remarkable and valuable; the result is in any case neither sufficiently well-known nor easily found in the literature. Various approaches to the result are presented and discussed in detail, including one inspired by a problem in the proof theory of intermediate logics that led us to the proof of the present paper. It can (...) be seen as a modification of Skolem’s argument from 1920 for his “Normal Form” theorem. “Geometric” being the infinitary version of “coherent”, it is further shown that every infinitary first-order theory, suitably restricted, has a geometric conservative extension, hence the title. The results are applied to simplify methods used in reasoning in and about modal and intermediate logics. We include also a new algorithm to generate special coherent implications from an axiom, designed to preserve the structure of formulae with relatively little use of normal forms. (shrink)

The paper briefly surveys the sentential proof-theoretic semantics for fragment of English. Then, appealing to a version of Frege’s context-principle (specified to fit type-logical grammar), a method is presented for deriving proof-theoretic meanings for sub-sentential phrases, down to lexical units (words). The sentential meaning is decomposed according to the function-argument structure as determined by the type-logical grammar. In doing so, the paper presents a novel proof-theoretic interpretation of simple type, replacing Montague’s model-theoretic type interpretation (in arbitrary Henkin models). The domains (...) of derivations are collections of derivations in the associated “dedicated” natural-deduction proof-system, and functions therein (with no appeal to models, truth-values and elements of a domain). The compositionality of the semantics is analyzed. (shrink)

The paper presents a proof-theoretic semantics (PTS) for a fragment of natural language, providing an alternative to the traditional model-theoretic (Montagovian) semantics (MTS), whereby meanings are truth-condition (in arbitrary models). Instead, meanings are taken as derivability-conditions in a dedicated natural-deduction (ND) proof-system. This semantics is effective (algorithmically decidable), adhering to the meaning as use paradigm, not suffering from several of the criticisms formulated by philosophers of language against MTS as a theory of meaning. In particular, Dummett’s manifestation argument does not (...) obtain, and assertions are always warranted, having grounds of assertion. The proof system is shown to satisfy Dummett’s harmony property, justifying the ND rules as meaning conferring. The semantics is suitable for incorporation into computational linguistics grammars, formulated in type-logical grammar. (shrink)

By considering the new notion of theinversesof syllogisms such asBarbaraandCelarent, we show how the rule ofIndirect Proof, in the form (no multiple or vacuous discharges) used by Aristotle, may be dispensed with, in a system comprising four basic rules of subalternation or conversion and six basic syllogisms.

This paper shows that, for the Hertz–Gentzen Systems of 1933, extended by a classical rule T1 and using certain axioms, all derivations are analytic: every cut formula occurs as a subformula in the cut’s conclusion. Since the Stoic cut rules are instances of Gentzen’s Cut rule of 1933, from this we infer the decidability of the propositional logic of the Stoics. We infer the correctness for this logic of a “relevance criterion” and of two “balance criteria”, and hence that a (...) particular derivable sequent has no derivation that is “normal” in the sense that the first premiss of each cut is cut-free. We also infer that Cut is not admissible in the Stoic system, based on the standard Stoic axioms, the T1 rule and the instances of Cut with just two antecedent formulae in the first premiss. OPEN ACCESS. (shrink)

We describe a sequent calculus, based on work of Herbelin, of which the cut-free derivations are in 1-1 correspondence with the normal natural deduction proofs of intuitionistic logic. We present a simple proof of Herbelin's strong cut-elimination theorem for the calculus, using the recursive path ordering theorem of Dershowitz.

We give a direct proof of admissibility of cut and contraction for the contraction-free sequent calculus G4ip for intuitionistic propositional logic and for a corresponding multi-succedent calculus: this proof extends easily in the presence of quantifiers, in contrast to other, indirect, proofs. i.e., those which use induction on sequent weight or appeal to admissibility of rules in other calculi.

We give a direct proof of admissibility of cut and contraction for the contraction-free sequent calculus G4ip for intuitionistic propositional logic and for a corresponding multi-succedent calculus: this proof extends easily in the presence of quantifiers, in contrast to other, indirect, proofs. i.e., those which use induction on sequent weight or appeal to admissibility of rules in other calculi.

The decision problem for positively quantified formulae in the theory of linearly ordered Heyting algebras is known, as a special case of work of Kreisel, to be solvable; a simple solution is here presented, inspired by related ideas in Gödel-Dummett logic.

"The papers in this volume are extended versions of presentations at the fourth International Workshop on Extensions of Logic Programming, held at the University of St Andrews, March/April 1993. Among the topics covered in the volume are: defintional reflection and completion, modules in lambda-Prolog, representation of logics as partial inductive definitions, non-procedural logic programming, knowledge representation, contradiction avoidance, disjunctive databases, strong negation, linear logic programming, proof theory and regular search spaces, finite sets and constraint logic programming, search-space pruning and universal (...) algebra, and implementation on transputer networks."--PUBLISHER'S WEBSITE. (shrink)

A major strategy to reduce transport congestion and other social costs of transport is to ensure that travellers make the best decisions, based on real time information. A wide range of technological systems have been developed to provide this information, but little is known about how travellers actually respond to it. This book offers an overview of various transport telematics options and provides an appropriate methodological framework, followed by a presentation of results from actual applications of these telematics systems from (...) a range of European countries in various transport sectors. The empirical results are supplemented by analytical models and geographic information systems representations with a view on generalizing these findings and identifying the key parameters which determine user response. (shrink)

The eclipse riddle -- Seeing surfaces -- The disappearing act -- Spinning shadows -- Berkeley's shadow -- Para-reflections -- Para-refractions : shadowgrams and the black drop -- Goethe's colored shadows -- Filtows -- Holes in the light -- Black and blue -- Seeing in black and white -- We see in the dark -- Hearing silence.

Can God create a stone too heavy for him to lift? Can time have a beginning? Which came first, the chicken or the egg? Riddles, paradoxes, conundrums--for millennia the human mind has found such knotty logical problems both perplexing and irresistible. Now Roy Sorensen offers the first narrative history of paradoxes, a fascinating and eye-opening account that extends from the ancient Greeks, through the Middle Ages, the Enlightenment, and into the twentieth century. When Augustine asked what God was doing before (...) He made the world, he was told: "Preparing hell for people who ask questions like that." A Brief History of the Paradox takes a close look at "questions like that" and the philosophers who have asked them, beginning with the folk riddles that inspired Anaximander to erect the first metaphysical system and ending with such thinkers as Lewis Carroll, Ludwig Wittgenstein, and W.V. Quine. Organized chronologically, the book is divided into twenty-four chapters, each of which pairs a philosopher with a major paradox, allowing for extended consideration and putting a human face on the strategies that have been taken toward these puzzles. Readers get to follow the minds of Zeno, Socrates, Aquinas, Ockham, Pascal, Kant, Hegel, and many other major philosophers deep inside the tangles of paradox, looking for, and sometimes finding, a way out. Filled with illuminating anecdotes and vividly written, A Brief History of the Paradox will appeal to anyone who finds trying to answer unanswerable questions a paradoxically pleasant endeavor. (shrink)

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work. This work is in the public domain (...) in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

The purpose of this article is to analyse the mathematical practices leading to Rafael Bombelli’s L’algebra (1572). The context for the analysis is the Italian algebra practiced by abbacus masters and Renaissance mathematicians of the fourteenth to sixteenth centuries. We will focus here on the semiotic aspects of algebraic practices and on the organisation of knowledge. Our purpose is to show how symbols that stand for underdetermined meanings combine with shifting principles of organisation to change the character of algebra.

An entertaining history of the idea of nothing - including absences, omissions, and shadows - from the Ancient Greeks through the 20th century How can nothing cause something? The absence of something might seem to indicate a null or a void, an emptiness as ineffectual as a shadow. In fact, 'nothing' is one of the most powerful ideas the human mind has ever conceived. This short and entertaining book by Roy Sorensen is a lively tour of the history and philosophy (...) of nothing, explaining how various thinkers throughout history have conceived and grappled with the mysterious power of absence -- and how these ideas about shadows, gaps, and holes have in turned played a very positive role in the development of some of humankind's most important ideas. Filled with Sorensen's characteristically entertaining mix of anecdotes, puzzles, curiosities, and philosophical speculation, the book is ordered chronologically, starting with the Taoists, the Buddhists, and the ancient Greeks, moving forward to the middle ages and the early modern period, then up to the existentialists and present day philosophy. The result is a diverting tour through the history of human thought as seen from a novel and unusual perspective. (shrink)

This book examines the intricate dynamics of when individuals adhere to laws, taking into account the context in which laws attempt to shape human behavior. While existing literature touches upon various reasons why people comply with laws, the book focuses on a critical question which has been missing from the discussion: when do people obey laws? By treating law as a form of social communication, it develops an integrated framework to answer this question. It explores how social, psychological, and institutional (...) conditions shape compliance decisions of individuals. What does a law signify? When does the compulsion to obey arise? When do individuals comply out of a fear for legal sanctions or social repercussions? Why do some laws have high symbolic values and others fail despite harsh punishments? The book unveils the contextual intricacies that underlie obedience to law. It challenges conventional wisdom and offers a fresh perspective on the power and limitations of law in shaping human behavior. For scholars and academics seeking a deeper understanding of legal compliance and role of law in shaping behaviors, this book will be an indispensable resource. (shrink)

In ‘A New Route to the Necessity of Origin’, Rohbraugh and deRosset offer an argument for the Necessity of Origin appealing neither to Suffciency of Origin nor to a branching-times model of necessity. What is doing the crucial work in their argument is instead the thesis they name ‘Locality of Prevention’. In this response, we object that their argument is question-begging by showing, first, that the locality of prevention thesis is not strong enough to satisfactorily derive from it the intended (...) conclusion, and, second, that the argument is not sound unless the Necessity of Origin is operating as an implicit premiss. (shrink)

Cet essai est une tentative pour penser autrement I liberté en ses actes. Une de ses originalités est de procéder par changements d'échelle, c'est un microscope philosophique. Sous la grande idée de liberté, il cerne d'imperceptibles appuis qui la font tenir debout. L'enjeu est donc de taille. Son développement prend la forme d'un parcours progressif par étapes, traversant des conceptions très différentes de la liberté et des domaines d'application divers. Il épouse ainsi les contours d'une mise en variation du concept (...) d'appui, au point que celui-ci pourra être identifié chez certains philosophes alors qu'il y passait auparavant inaperçu. Il s'agit aussi de montrer de quels problèmes ce concept tire sa nécessité. Ces problèmes sont tout autant théoriques que pratiques, spirituels que corporels, tout autant moraux que politiques. Bien plus, en justifiant pourquoi un appui, n'est pas un soutien, cet essai donne une certaine allure à la liberté : celle de la spirale, de l'hélice. Il montre que, sous de nouvelles manières d'agir ou de penser, peuvent pousser des spirales de la liberté"--Page 4 of cover. (shrink)

The history of contemporary metaphysics shows there are two schools of thought. One for whom the analysis of categories assumes primarily an epistemological character and on the other hand some believe that the field to be investigated is rather to be approached in primarily ontological terms. Strawson's conception of descriptive metaphysics is a clear example of this approach.

This book provides a new understanding of art education, connecting art, necessity, and pedagogy. Plenty has been written about art education, but its potential contribution to inner autonomy and existential emancipation due to the re-articulation of time and space in art has not been adequately explored in a pedagogic context. This book explores art as an affective continuum--not a plaything of culture, but rather a mode of alignment of the existential Eros with our ontological truth.

S(zp,zp) performs an innovative analysis of one of modern logic's most celebrated cornerstones: the proof of Gödel's first incompleteness theorem. The book applies the semiotic theories of French post- structuralists such as Julia Kristeva, Jacques Derrida and Gilles Deleuze to shed new light on a fundamental question: how do mathematical signs produce meaning and make sense? S(zp,zp) analyses the text of the proof of Gödel's result, and shows that mathematical language, like other forms of language, enjoys the full complexity of (...) language as a process, with its embodied genesis, constitutive paradoxical forces and unbounded shifts of meaning. These effects do not infringe on the logico-mathematical validity of Gödel's proof. Rather, they belong to a mathematical unconscious that enables the successful function of mathematical texts for a variety of different readers. S(zp,zp) breaks new ground by synthesising mathematical logic and post-structural semiotics into a new form of philosophical fabric, and offers an original way of bridging the gap between the "two cultures". (shrink)

The Next Step in Religion: An Essay toward the Coming Renaissance is a classic religious essay by Roy Wood Sellars that examines christianity and humanism includes the following excerpt: More than people are consciously aware, a new view of the universe and of man's place in it is forming. It is forming in the laboratories of scientists, the studies of thinkers, the congresses of social workers, the assemblies of reformers, the studios of artists and, even more quietly, in the circles (...) of many homes. This new view is growing beneath the old as a bud grows beneath its covering, and is slowly pushing it aside. (shrink)

This book engages with the dynamic intersection of several domains such as philosophy, psychology, sociology, and pedagogy, in order to critically analyze and reinvent our understanding of curriculum. The chapters raise important questions such as: what are the conditions of possibility for a living curriculum in which Eros and intellect (or reason and intuition) are not separated? How is it possible to escape ideology that keeps us bound to defunct categories? What are the ingredients of an inquiry that is able (...) to grasp curriculum as an expanding interpersonal movement? How do the teacher-learner ensemble get creatively constituted beyond obstructive dualities? How can we reinvent meaning in curriculum without totalization? Which indigenous understandings can be recovered in order to reinvent curriculum with greater relevance for diverse peoples? This volume addresses elements of reason, nonreason, becoming, dissipation, violence, uncertainty, transcendence, love, and death in order to come to a critical understanding of the relationship between knowledge and knower from these multiple perspectives. (shrink)

Social Thought in Indic Civilization presents an interdisciplinary perspective on the pre-colonial social thought. It draws on the methodologies and research traditions of history, political science and sociology to look at major themes and social processes to provide a comprehensive understanding of the society in the historical setting contextualized in the social and political relations of the time. The arguments, facts, themes and interpretations presented in the book are usually not found in the mainstream academic narratives. This book explores a (...) range of key themes such as non-violence in religious praxis; dharma in Indic social traditions; medicinal concepts and institutions; ideas and praxis of Shastrarth; knowledge traditions and institutions; music traditions; and Stritva in texts and praxis. This study highlights the impact of colonial rule on the 'construction of knowledge' from a Western (colonial) perspective and how it ignored the importance of Indian political thought of the pre-colonial period. (shrink)

In this book, Roy Bhaskar sets out to revindicate ontology, critiquing the reduction of being in favor of knowledge, which he calls the "epistemic fallacy".