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Logic and Information. a Unifying Approach to Semantic Information Theory

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Logic and Information. a Unifying Approach to Semantic Information Theory Aus dem Departement f¨urInformatik Universit¨atFreiburg (Schweiz) Logic and Information A Unifying Approach to Semantic Information Theory Inaugural{Dissertation zur Erlangung der W¨urdeeines Doctor scientiarum informaticarum der Mathematisch-Naturwissenschaftlichen Fakult¨at der Universit¨atFreiburg in der Schweiz vorgelegt von Jutta Langel aus Deutschland Nr. 1657 UniPrint, Freiburg 2009 Pragmatically speaking, a question sets up a choice-situation between a set of propositions, namely, those propositions which count as answers to it. Charles Leonard Hamblin (1922 - 1985) Questions in Montague English v Acknowledgements During the last five years and especially while writing this thesis, I was supported by many people. They have all contributed to the successful completion of this work. First and foremost, I am grateful to Prof. J¨urgKohlas, the director of my thesis, who put confidence in me and motivated me to start this work. He supported me in finding the subject of this thesis and had time for many discussions to improve my work with ideas and constructive feedback. Many thanks go further to the (former) tcs and run research group members of the Universities of Fribourg and Berne for their various support and the friendly and stimulating working ambiance they created: Bernhard Anrig, Christian Eichenberger, Rolf Haenni, Jacek Jonczy, Reto Kohlas, Norbert Lehmann, Marc Pouly, Cesar Schneuwly and Michael Wachter. I am also thankful to Prof. J´eromeMengin and to Prof. Ulrich Ultes-Nitsche who accepted to act as referees for my thesis. Also, I am greatly indebted to Christian Eichenberger who proof-read important parts of this thesis and who helped to improve it by pointing out lots of details. Furthermore, I thank Andrea Stengel and Heidemarie Langel for linguistic advice. Chocolat Villars also merits recognition for confecting chocolate and coffee support- ing me constantly during the last five years. My friends Ale, Andrea, Anita, Christian, Christian, David, Lionel, Manuel, Markus, Moritz, Nadine, Nathalie, Nicole, Sarah, Susa, Ursula and Val´eriemade my life beyond the thesis enjoyable and I am very thankful for that. Last but not least, my cordial thanks are addressed to my family. Alexis, Heidemarie and Ulrich, Helma, Laurent and Lena were always there for me and backed me up in every situation. vi vii Abstract The commonly used information theory, going back to Shannon, is almost exclusively concerned with the measure of information. However, by measuring information, one does not get very much to know about its nature, about what information actually is. Therefore, this thesis has two main goals: • The first is to provide an adequate definition of the concept of information, by a semantic interpretation of the abstract, axiomatic information algebra framework. This leads to the formulation of an algebraic theory of semantic information, which is exemplified by logics. • The second is to validate this theory, by comparison with already established semantic information theories (which also apply to logics) of other disciplines. The Algebraic Theory of Semantic Information An information algebra is a two-sorted algebra, consisting of a set of possible pieces of information, of a lattice of questions and of two operations (combination and focusing), that satisfy a set of five axioms. The following properties of informa- tion and questions are formalized by the information algebra framework: pieces of information refer to related questions; pieces of information can be focused, in or- der to extract information relative to some specific question of interest; pieces of information may be combined (aggregated). Information can be perceived in two ways. One can look at how information is represented, or one can examine what information expresses. Information repre- sentation involves a (formal) language. In order to understand the meaning of the information, semantics is needed. In this thesis, we have chosen the latter approach. A semantic interpretation of the information algebra framework allows to draw conclusions about the nature of information and questions: A semantic piece of information is a set of possibilities, which may be interpreted in different ways. A piece of information is perceived as an answer to a question. A question, in turn, is semantically given by its possible answers. These results constitute the algebraic theory of semantic information, which applies to many instances, including logics. Propositional logic and predicate logic are shown to be information algebra instances. Since the meaning of information matters, the proofs are given on the semantic level. A Unifying Approach In disciplines which are also dealing with logics, like philosophy or linguistics, se- mantic information theories can be found, too. Three of them are presented in this thesis. viii 1. Carnap and Bar-Hillel's theory of semantic information from the early 1950s provides a very basic framework for semantic information and its measure. Information is perceived as a set of excluded possibilities. The theory is in- stantiated by a restricted monadic predicate logic language. 2. Groenendijk and Stokhof's theory of the semantics of questions and the prag- matics of answers provides a framework for questions. This theory has been developed in the early 1980s and has been extended by van Rooij in the first decade of the 21st century. As questions are identified with their possible answers, a detailed description of the nature of answers is also given. Two instances, propositional and predicate logic, are considered. 3. Barwise and Seligman's theory of information flow, which came up in the late 1990s, provides a framework for the representation of information and the computation with it in distributed systems. As to the representation, informa- tion is seen from a dual perspective, taking into account syntax and semantics. This dual representation is identified in this thesis with the approach of formal concept analysis, which was introduced in the 1980s. Barwise and Seligman mainly exemplify their theory by predicate logic. All these three theories fit into the information algebra framework. Therefore, the algebraic theory of semantic information encompasses these theories. ix Zusammenfassung Die renommierte Informationstheorie der Informatik, die auf Shannon zuruck¨ geht, besch¨aftigt sich fast ausschliesslich mit dem Messen des Informationsgehalts von Nachrichten. Das blosse Messen sagt jedoch wenig uber¨ die Beschaffenheit von In- formation aus. Die Frage, was Information wirklich ist, bleibt offen. Deshalb verfolgt diese Dissertation zwei Hauptziele: • Zum einen soll eine ad¨aquate Definition von Information auf konzeptueller Ebene gegeben werden. Diese Definition beruht auf einem abstrakten, axio- matischen Framework, genannt Informationsalgebra, das aus einem semanti- schen Blickwinkel betrachtet wird. Daraus resultiert eine algebraische Theorie semantischer Information, die von Logik veranschaulicht wird. • Zum anderen soll die algebraische Theorie semantischer Information validiert werden. Dazu wurden drei semantische Informationstheorien aus anderen Dis- ziplinen ausgew¨ahlt, dargestellt und mit der algebraischen Theorie semanti- scher Information verglichen. Logik dient als Beispiel fur¨ alle vier Theorien. Die algebraische Theorie semantischer Information Eine Informationsalgebra ist eine zweisortige Struktur, die aus einer Menge von m¨oglichen Informationen und einem Verband von Fragen besteht sowie aus zwei Operationen (Kombination und Fokussierung), die funf¨ Axiome erfullen.¨ Informati- on hat Eigenschaften, die mit Hilfe des Informationsalgebra-Frameworks formalisiert werden k¨onnen: Information bezieht sich auf Fragen, Fragen wiederum stehen unter- einander in einer gewissen Beziehung. Um Information zu extrahieren, wird sie auf eine bestimmten Frage fokussiert. Da Information meist aus verschiedenen Quel- len stammt und teilweise unvollst¨andig ist, muss sie kombiniert werden um einen Gesamteindruck zu vermitteln. Information kann auf zwei verschiedene Arten betrachtet werden. Das Interesse kann entweder auf die Darstellung oder die Bedeutung der Information gerichtet sein. Zur Darstellung der Information wird eine (formale) Sprache eingesetzt. Die Auseinan- dersetzung mit der Bedeutung der Information bedarf jedoch der Semantik, die Ge- genstand dieser Dissertation ist. Die semantische Auslegung des Informationsalgebra- Framework erlaubt folgende Ruckschl¨ usse¨ uber¨ die Beschaffenheit von Informati- on und Fragen: Semantische Information ist durch eine Menge von verschiedenen M¨oglichkeiten gegeben. Information wird als Antwort auf eine (implizite) Frage wahrgenommen. Eine Frage wird semantisch durch m¨ogliche Antworten beschrie- ben. Diese Erkenntnisse fuhren¨ zur algebraischen Theorie semantischer Information, die auf viele Formalismen, u. a. auf Logik, angewandt werden kann. Es wird gezeigt, dass Aussagenlogik und Pr¨adikatenlogik Informationsalgebren bilden. Da das Ge- wicht auf der Bedeutung von Information liegt, werden die dazugeh¨origen Beweise auf semantischer Ebene ausgefuhrt.¨ x Ein allumfassender Ansatz Auch in anderen Disziplinen, die Logik einsetzen, wie z. B. in der Philosophie oder der Sprachwissenschaft, finden sich semantische Informationstheorien. Drei dieser Theorien werden in dieser Dissertation vorgestellt. 1. In den fruhen¨ 50er Jahren des 20. Jahrhunderts entwickelten Carnap und Bar- Hillel eine semantische Informationstheorie, die sich sehr grundlegend mit se- mantischer Information und ihrer Messung
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