Universal Logic and the Geography of Thought – Reflections on Logical
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Deduction Systems Based on Resolution, We Limit the Following Considerations to Transition Systems Based on Analytic Calculi
Author’s Address Norbert Eisinger European Computer–Industry Research Centre, ECRC Arabellastr. 17 D-8000 M¨unchen 81 F. R. Germany [email protected] and Hans J¨urgen Ohlbach Max–Planck–Institut f¨ur Informatik Im Stadtwald D-6600 Saarbr¨ucken 11 F. R. Germany [email protected] Publication Notes This report appears as chapter 4 in Dov Gabbay (ed.): ‘Handbook of Logic in Artificial Intelligence and Logic Programming, Volume I: Logical Foundations’. It will be published by Oxford University Press, 1992. Fragments of the material already appeared in chapter two of Bl¨asis & B¨urckert: Deduction Systems in Artificial Intelligence, Ellis Horwood Series in Artificial Intelligence, 1989. A draft version has been published as SEKI Report SR-90-12. The report is also published as an internal technical report of ECRC, Munich. Acknowledgements The writing of the chapters for the handbook has been a highly coordinated effort of all the people involved. We want to express our gratitude for their many helpful contributions, which unfortunately are impossible to list exhaustively. Special thanks for reading earlier drafts and giving us detailed feedback, go to our second reader, Bob Kowalski, and to Wolfgang Bibel, Elmar Eder, Melvin Fitting, Donald W. Loveland, David Plaisted, and J¨org Siekmann. Work on this chapter started when both of us were members of the Markgraf Karl group at the Universit¨at Kaiserslautern, Germany. With our former colleagues there we had countless fruitful discus- sions, which, again, cannot be credited in detail. During that time this research was supported by the “Sonderforschungsbereich 314, K¨unstliche Intelligenz” of the Deutsche Forschungsgemeinschaft (DFG). -
A Survey of Abstract Algebraic Logic 15 Bras
J. M. Font A Survey of Abstract Algebraic R. Jansana D. Pigozzi Logic Contents Introduction 14 1. The First Steps 16 1.1. Consequence operations and logics . 17 1.2. Logical matrices . 20 1.3. Lindenbaum-Tarski algebras . 22 2. The Lindenbaum-Tarski Process Fully Generalized 24 2.1. Frege's principles . 24 2.2. Algebras and matrices canonically associated with a logic . 26 3. The Core Theory of Abstract Algebraic Logic 29 3.1. Elements of the general theory of matrices . 29 3.2. Protoalgebraic logics . 33 3.3. Algebraizable logics . 38 3.4. The Leibniz hierarchy . 43 4. Extensions of the Core Theory 50 4.1. k-deductive systems and universal algebra . 50 4.2. Gentzen systems and their generalizations . 55 4.3. Equality-free universal Horn logic . 61 4.4. Algebras of Logic . 64 5. Generalized Matrix Semantics and Full Models 71 6. Extensions to More Complex Notions of Logic 79 6.1. The semantics-based approach . 80 6.2. The categorial approach . 85 References 88 Special Issue on Abstract Algebraic Logic { Part II Edited by Josep Maria Font, Ramon Jansana, and Don Pigozzi Studia Logica 74: 13{97, 2003. c 2003 Kluwer Academic Publishers. Printed in the Netherlands. 14 J. M. Font, R. Jansana, and D. Pigozzi Introduction Algebraic logic was born in the XIXth century with the work of Boole, De Morgan, Peirce, Schr¨oder, etc. on classical logic, see [12, 16]. They took logical equivalence rather than truth as the primitive logical predicate, and, exploiting the similarity between logical equivalence and equality, they devel- oped logical systems in which metalogical investigations take on a distinctly algebraic character. -
Domain Theory and the Logic of Observable Properties
Domain Theory and the Logic of Observable Properties Samson Abramsky Submitted for the degree of Doctor of Philosophy Queen Mary College University of London October 31st 1987 Abstract The mathematical framework of Stone duality is used to synthesize a number of hitherto separate developments in Theoretical Computer Science: • Domain Theory, the mathematical theory of computation introduced by Scott as a foundation for denotational semantics. • The theory of concurrency and systems behaviour developed by Milner, Hennessy et al. based on operational semantics. • Logics of programs. Stone duality provides a junction between semantics (spaces of points = denotations of computational processes) and logics (lattices of properties of processes). Moreover, the underlying logic is geometric, which can be com- putationally interpreted as the logic of observable properties—i.e. properties which can be determined to hold of a process on the basis of a finite amount of information about its execution. These ideas lead to the following programme: 1. A metalanguage is introduced, comprising • types = universes of discourse for various computational situa- tions. • terms = programs = syntactic intensions for models or points. 2. A standard denotational interpretation of the metalanguage is given, assigning domains to types and domain elements to terms. 3. The metalanguage is also given a logical interpretation, in which types are interpreted as propositional theories and terms are interpreted via a program logic, which axiomatizes the properties they satisfy. 2 4. The two interpretations are related by showing that they are Stone duals of each other. Hence, semantics and logic are guaranteed to be in harmony with each other, and in fact each determines the other up to isomorphism. -
Bi-Directional Transformation Between Normalized Systems Elements and Domain Ontologies in OWL
Bi-directional Transformation between Normalized Systems Elements and Domain Ontologies in OWL Marek Suchanek´ 1 a, Herwig Mannaert2, Peter Uhnak´ 3 b and Robert Pergl1 c 1Faculty of Information Technology, Czech Technical University in Prague, Thakurova´ 9, Prague, Czech Republic 2Normalized Systems Institute, University of Antwerp, Prinsstraat 13, Antwerp, Belgium 3NSX bvba, Wetenschapspark Universiteit Antwerpen, Galileilaan 15, 2845 Niel, Belgium Keywords: Ontology, Normalized Systems, Transformation, Model-driven Development, Ontology Engineering, Software Modelling. Abstract: Knowledge representation in OWL ontologies gained a lot of popularity with the development of Big Data, Artificial Intelligence, Semantic Web, and Linked Open Data. OWL ontologies are very versatile, and there are many tools for analysis, design, documentation, and mapping. They can capture concepts and categories, their properties and relations. Normalized Systems (NS) provide a way of code generation from a model of so-called NS Elements resulting in an information system with proven evolvability. The model used in NS contains domain-specific knowledge that can be represented in an OWL ontology. This work clarifies the potential advantages of having OWL representation of the NS model, discusses the design of a bi-directional transformation between NS models and domain ontologies in OWL, and describes its implementation. It shows how the resulting ontology enables further work on the analytical level and leverages the system design. Moreover, due to the fact that NS metamodel is metacircular, the transformation can generate ontology of NS metamodel itself. It is expected that the results of this work will help with the design of larger real-world applications as well as the metamodel and that the transformation tool will be further extended with additional features which we proposed. -
What Is Abstract Algebraic Logic?
What is Abstract Algebraic Logic? Umberto Rivieccio University of Genoa Department of Philosophy Via Balbi 4 - Genoa, Italy email [email protected] Abstract Our aim is to answer the question of the title, explaining which kind of problems form the nucleus of the branch of algebraic logic that has come to be known as Abstract Algebraic Logic. We give a concise account of the birth and evolution of the field, that led from the discovery of the connections between certain logical systems and certain algebraic structures to a general theory of algebraization of logics. We consider first the so-called algebraizable logics, which may be regarded as a paradigmatic case in Abstract Algebraic Logic; then we shift our attention to some wider classes of logics and to the more general framework in which they are studied. Finally, we review some other topics in Abstract Algebraic Logic that, though equally interesting and promising for future research, could not be treated here in detail. 1 Introduction Algebraic logic can be described in very general terms as the study of the relations between algebra and logic. One of the main reasons that motivate such a study is the possibility to treat logical problems with algebraic methods. This means that, in order to solve a logical problem, we may try to translate it into algebraic terms, find the solution using algebraic techniques, and then translate back the result into logic. This strategy proved to be fruitful, since algebra is an old and well{established field of mathematics, that offers powerful tools to treat problems that would possibly be much harder to solve using the logical machinery alone. -
What Is a Logic Translation? in Memoriam Joseph Goguen
, 1{29 c 2009 Birkh¨auserVerlag Basel/Switzerland What is a Logic Translation? In memoriam Joseph Goguen Till Mossakowski, R˘azvan Diaconescu and Andrzej Tarlecki Abstract. We study logic translations from an abstract perspective, with- out any commitment to the structure of sentences and the nature of logical entailment, which also means that we cover both proof-theoretic and model- theoretic entailment. We show how logic translations induce notions of logical expressiveness, consistency strength and sublogic, leading to an explanation of paradoxes that have been described in the literature. Connectives and quanti- fiers, although not present in the definition of logic and logic translation, can be recovered by their abstract properties and are preserved and reflected by translations under suitable conditions. 1. Introduction The development of a notion of logic translation presupposes the development of a notion of logic. Logic has been characterised as the study of sound reasoning, of what follows from what. Hence, the notion of logical consequence is central. It can be obtained in two complementary ways: via the model-theoretic notion of satisfaction in a model and via the proof-theoretic notion of derivation according to proof rules. These notions can be captured in a completely abstract manner, avoiding any particular commitment to the nature of models, sentences, or rules. The paper [40], which is the predecessor of our current work, explains the notion of logic along these lines. Similarly, an abstract notion of logic translation can be developed, both at the model-theoretic and at the proof-theoretic level. A central question concern- ing such translations is their interaction with logical structure, such as given by logical connectives, and logical properties. -
Explanations
© Copyright, Princeton University Press. No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means without prior written permission of the publisher. CHAPTER 1 Explanations 1.1 SALLIES Language is an instrument of Logic, but not an indispensable instrument. Boole 1847a, 118 We know that mathematicians care no more for logic than logicians for mathematics. The two eyes of exact science are mathematics and logic; the mathematical sect puts out the logical eye, the logical sect puts out the mathematical eye; each believing that it sees better with one eye than with two. De Morgan 1868a,71 That which is provable, ought not to be believed in science without proof. Dedekind 1888a, preface If I compare arithmetic with a tree that unfolds upwards in a multitude of techniques and theorems whilst the root drives into the depthswx . Frege 1893a, xiii Arithmetic must be discovered in just the same sense in which Columbus discovered the West Indies, and we no more create numbers than he created the Indians. Russell 1903a, 451 1.2 SCOPE AND LIMITS OF THE BOOK 1.2.1 An outline history. The story told here from §3 onwards is re- garded as well known. It begins with the emergence of set theory in the 1870s under the inspiration of Georg Cantor, and the contemporary development of mathematical logic by Gottlob Frege andŽ. especially Giuseppe Peano. A cumulation of these and some related movements was achieved in the 1900s with the philosophy of mathematics proposed by Alfred North Whitehead and Bertrand Russell. -
Duality Theory and Abstract Algebraic Logic
Duality Theory and Abstract Algebraic Logic María Esteban ADVERTIMENT. La consulta d’aquesta tesi queda condicionada a l’acceptació de les següents condicions d'ús: La difusió d’aquesta tesi per mitjà del servei TDX (www.tdx.cat) i a través del Dipòsit Digital de la UB (diposit.ub.edu) ha estat autoritzada pels titulars dels drets de propietat intel·lectual únicament per a usos privats emmarcats en activitats d’investigació i docència. No s’autoritza la seva reproducció amb finalitats de lucre ni la seva difusió i posada a disposició des d’un lloc aliè al servei TDX ni al Dipòsit Digital de la UB. No s’autoritza la presentació del seu contingut en una finestra o marc aliè a TDX o al Dipòsit Digital de la UB (framing). Aquesta reserva de drets afecta tant al resum de presentació de la tesi com als seus continguts. En la utilització o cita de parts de la tesi és obligat indicar el nom de la persona autora. ADVERTENCIA. La consulta de esta tesis queda condicionada a la aceptación de las siguientes condiciones de uso: La difusión de esta tesis por medio del servicio TDR (www.tdx.cat) y a través del Repositorio Digital de la UB (diposit.ub.edu) ha sido autorizada por los titulares de los derechos de propiedad intelectual únicamente para usos privados enmarcados en actividades de investigación y docencia. No se autoriza su reproducción con finalidades de lucro ni su difusión y puesta a disposición desde un sitio ajeno al servicio TDR o al Repositorio Digital de la UB. -
Data Models for Home Services
__________________________________________PROCEEDING OF THE 13TH CONFERENCE OF FRUCT ASSOCIATION Data Models for Home Services Vadym Kramar, Markku Korhonen, Yury Sergeev Oulu University of Applied Sciences, School of Engineering Raahe, Finland {vadym.kramar, markku.korhonen, yury.sergeev}@oamk.fi Abstract An ultimate penetration of communication technologies allowing web access has enriched a conception of smart homes with new paradigms of home services. Modern home services range far beyond such notions as Home Automation or Use of Internet. The services expose their ubiquitous nature by being integrated into smart environments, and provisioned through a variety of end-user devices. Computational intelligence require a use of knowledge technologies, and within a given domain, such requirement as a compliance with modern web architecture is essential. This is where Semantic Web technologies excel. A given work presents an overview of important terms, vocabularies, and data models that may be utilised in data and knowledge engineering with respect to home services. Index Terms: Context, Data engineering, Data models, Knowledge engineering, Semantic Web, Smart homes, Ubiquitous computing. I. INTRODUCTION In recent years, a use of Semantic Web technologies to build a giant information space has shown certain benefits. Rapid development of Web 3.0 and a use of its principle in web applications is the best evidence of such benefits. A traditional database design in still and will be widely used in web applications. One of the most important reason for that is a vast number of databases developed over years and used in a variety of applications varying from simple web services to enterprise portals. In accordance to Forrester Research though a growing number of document, or knowledge bases, such as NoSQL is not a hype anymore [1]. -
Are Dogs Logical Animals ? Jean-Yves Beziau University of Brazil, Rio De Janeiro (UFRJ) Brazilian Research Council (Cnpq) Brazilian Academy of Philosophy (ABF)
Are Dogs Logical Animals ? Jean-Yves Beziau University of Brazil, Rio de Janeiro (UFRJ) Brazilian Research Council (CNPq) Brazilian Academy of Philosophy (ABF) Autre être fidèle du tonneau sans loques malgré ses apparitions souvent loufoques le chien naît pas con platement idiot ne se laisse pas si facilement mener en rat d’eau il ne les comprend certes pas forcément toutes à la première coupe ne sait pas sans l’ombre deux doutes la fière différance antre un loup et une loupe tout de foi à la croisade des parchemins il saura au non du hasard dès trousser la chienlit et non en vain reconnaître celle qui l’Amen à Rhum sans faim Baron Jean Bon de Chambourcy 1. Human beings and other animals In Ancient Greece human beings have been characterized as logical animals. This traditional catching is a bit lost or/and nebulous. What this precisely means is not necessarily completely clear because the word Logos has four important aspects: language, science, reasoning, relation. The Latinized version of “logical”, i.e. “rational”, does not fully preserve these four meanings. Moreover later on human beings were qualified as “homo sapiens” (see Beziau 2017). In this paper we will try to clarify the meaning of “logical animals” by examining if dogs can also be considered logical animals. An upside question of “Are dogs logical animals?” is: “Are human beings the only rational animals?”. It is one thing to claim that human beings are logical animals, but another thing is to super claim that human beings are the only logical animals. -
Integrating Fuzzy Logic in Ontologies
INTEGRATING FUZZY LOGIC IN ONTOLOGIES Silvia Calegari and Davide Ciucci Dipartimento di Informatica, Sistemistica e Comunicazione, Universita` degli Studi di Milano Bicocca, via Bicocca degli Arcimboldi 8, 20126 Milano, Italy Keywords: Concept modifiers, fuzzy logics, fuzzy ontologies, membership modifiers, KAON, ontology editor. Abstract: Ontologies have proved to be very useful in sharing concepts across applications in an unambiguous way. Nowadays, in ontology-based applications information is often vague and imprecise. This is a well-known problem especially for semantics-based applications, such as e-commerce, knowledge management, web por- tals, etc. In computer-aided reasoning, the predominant paradigm to manage vague knowledge is fuzzy set theory. This paper presents an enrichment of classical computational ontologies with fuzzy logic to create fuzzy ontologies. So, it is a step towards facing the nuances of natural languages with ontologies. Our pro- posal is developed in the KAON ontology editor, that allows to handle ontology concepts in an high-level environment. 1 INTRODUCTION A possible solution to handle uncertain data and, hence, to tackle these problems, is to incorporate fuzzy logic into ontologies. The aim of fuzzy set An ontology is a formal conceptualization of a partic- theory (Klir and Yuan, 1995) introduced by L. A. ular domain of interest shared among heterogeneous Zadeh (Zadeh, 1965) is to describe vague concepts applications. It consists of entities, attributes, rela- through a generalized notion of set, according to tionships and axioms to provide a common under- which an object may belong to a certain degree (typ- standing of the real world (Lammari and Mtais, 2004; ically a real number from the interval [0,1]) to a set. -
Maurice Finocchiaro Discusses the Lessons and the Cultural Repercussions of Galileo’S Telescopic Discoveries.” Physics World, Vol
MAURICE A. FINOCCHIARO: CURRICULUM VITAE CONTENTS: §0. Summary and Highlights; §1. Miscellaneous Details; §2. Teaching Experience; §3. Major Awards and Honors; §4. Publications: Books; §5. Publications: Articles, Chapters, and Discussions; §6. Publications: Book Reviews; §7. Publications: Proceedings, Abstracts, Translations, Reprints, Popular Media, etc.; §8. Major Lectures at Scholarly Meetings: Keynote, Invited, Funded, Honorarium, etc.; §9. Other Lectures at Scholarly Meetings; §10. Public Lectures; §11. Research Activities: Out-of-Town Libraries, Archives, and Universities; §12. Professional Service: Journal Editorial Boards; §13. Professional Service: Refereeing; §14. Professional Service: Miscellaneous; §15. Community Service §0. SUMMARY AND HIGHLIGHTS Address: Department of Philosophy; University of Nevada, Las Vegas; Box 455028; Las Vegas, NV 89154-5028. Education: B.S., 1964, Massachusetts Institute of Technology; Ph.D., 1969, University of California, Berkeley. Position: Distinguished Professor of Philosophy, Emeritus; University of Nevada, Las Vegas. Previous Positions: UNLV: Assistant Professor, 1970-74; Associate Professor, 1974-77; Full Professor, 1977-91; Distinguished Professor, 1991-2003; Department Chair, 1989-2000. Major Awards and Honors: 1976-77 National Science Foundation; one-year grant; project “Galileo and the Art of Reasoning.” 1983-84 National Endowment for the Humanities, one-year Fellowship for College Teachers; project “Gramsci and the History of Dialectical Thought.” 1987 Delivered the Fourth Evert Willem Beth Lecture, sponsored by the Evert Willem Beth Foundation, a committee of the Royal Netherlands Academy of Sciences, at the Universities of Amsterdam and of Groningen. 1991-92 American Council of Learned Societies; one-year fellowship; project “Democratic Elitism in Mosca and Gramsci.” 1992-95 NEH; 3-year grant; project “Galileo on the World Systems.” 1993 State of Nevada, Board of Regents’ Researcher Award.