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Towards a Framework for Semantic Information Towards a Framework for Semantic Information Simon D'Alfonso Submitted in total fulfilment of the requirements of the degree of Doctor of Philosophy Department of Philosophy The University of Melbourne September 2012 Produced on archival quality paper ii Contents Abstract ix Declaration xi Preface xiii Acknowledgements xv 1 An Introductory Overview of Information 1 1.1 What is Information? . 2 1.1.1 Data . 4 1.1.2 The Mathematical Theory of Communication . 8 1.1.3 Moving Beyond Data . 16 1.2 Philosophy and Information . 18 1.2.1 The Philosophy of Information . 19 1.3 Semantic Information and Environmental Information . 20 1.3.1 Gricean Meaning . 20 1.3.2 Semantic Information . 21 iii iv CONTENTS 1.3.3 Environmental Information . 27 1.4 The Alethic Nature of Semantic Information . 30 1.5 Conclusion . 44 2 Quantifying Semantic Information 45 2.1 Bar-Hillel and Carnap's Theory of Semantic Information . 47 2.1.1 Some Comments on the Theory of Classical Semantic Information . 50 2.1.2 The Bar-Hillel-Carnap Paradox and Paraconsistent Logic . 51 2.2 CSI and Truth . 54 2.2.1 Epistemic Utility . 54 2.2.2 CSI, Truth and Scoring Rules . 55 2.3 Floridi's Theory of Strongly Semantic Information . 56 2.3.1 Some Comments on Floridi's Theory . 61 2.4 Information Quantification via Truthlikeness . 62 2.4.1 The Basic Feature Approach to Truthlikeness . 63 2.4.2 The Tichy/Oddie Approach to Truthlikeness . 65 2.4.3 Truthlikeness Adequacy Conditions and Information Conditions . 78 2.4.4 Niiniluoto on Truthlikeness . 79 2.4.5 An Interpretation of the Tichy/Oddie Measure . 80 2.4.6 Adjusting State Description Utility Values . 87 2.4.7 Another Interpretation of the Tichy/Oddie Measure . 88 2.5 Another Method to Quantify Semantic Information . 89 CONTENTS v 2.5.1 Adequacy Conditions . 91 2.5.2 Adjusting Utilities . 96 2.5.3 Misinformation . 97 2.6 Estimated Information . 99 2.7 Combining Measures . 102 2.8 Conclusion . 105 3 Agent-Relative Informativeness 107 3.1 AGM Belief Change . 108 3.2 Combining Information Measurement and Belief Revision . 113 3.2.1 True Database Content and True Input . 113 3.2.2 False Database Content and True Input . 115 3.2.3 True Inputs Guaranteed to Increase Information Yield . 116 3.3 Agent-Relative Informativeness . 125 3.3.1 Adding Information to Information . 126 3.3.2 Adding Information to Information/Misinformation . 128 3.3.3 Adding Misinformation/Information to Misinformation/Information . 131 3.4 Dealing with Uncertainty and Conflicting Input . 133 3.4.1 Paraconsistent Approaches . 137 3.5 Applications . 152 3.5.1 Extension to Other Spaces . 152 3.5.2 The Value of Information . 153 vi CONTENTS 3.5.3 Lottery-Style Scenarios . 159 3.5.4 The Conjunction Fallacy . 163 3.5.5 Quantifying Epistemic and Doxastic Content . 164 3.5.6 Agent-Oriented Relevance . 165 3.6 Conclusion . 170 4 Environmental Information and Information Flow 173 4.1 Dretske on Information . 174 4.1.1 Dretske's Account and Properties of Information Flow . 186 4.2 Probabilistic Information . 188 4.3 A Counterfactual Theory of Information . 194 4.3.1 The Logic of Counterfactuals and Information Flow Properties . 201 4.3.2 Transitivity, Monotonicity and Contraposition . 203 4.4 A Modal Logical Account of Information Flow . 208 4.4.1 Variable Information Flow Contexts . 211 4.4.2 Agent-Relative Information Flow . 218 4.4.3 Information Closure . 219 4.4.4 Variable Relevant Alternatives and the Other Information Flow Prop- erties . 243 4.5 Another Requirement on Semantic Information? . 251 4.6 Conclusion . 254 5 Information and Knowledge 257 CONTENTS vii 5.1 Some Background . 257 5.2 Dretske on Knowledge . 261 5.2.1 Testing and Supplementing Dretske's Informational Epistemology . 266 5.2.2 How reliable does an information source have to be? . 284 5.2.3 Dealing with Knowledge of Necessary Truths . 293 5.2.4 Knowledge, Information and Testimony . 296 5.3 Epistemic Logics for Informational Epistemologies . 302 5.3.1 Alternative Sets of Relevant Alternatives and Multi-Modal Logic . 303 5.3.2 `Knows that' as a semi-penetrating operator . 306 5.3.3 One Approach to Epistemic Logic and Relevant Alternatives . 306 5.3.4 Going Non-normal . 309 5.3.5 A Logic for Dretskean Epistemology . 315 5.4 The Value of Information and Knowledge . 318 5.4.1 Knowledge and True Belief Generation . 320 5.4.2 The Value of Knowledge . 326 5.5 Conclusion . 339 A Quantifying Semantic Information 345 A.1 Adequacy Condition Proofs for the Value Aggregate Method . 347 A.2 Translation Invariance . 350 A.3 Formula-Based Approaches . 353 Appendices 345 viii CONTENTS B Agent-Relative Informativeness 357 B.1 Combining Information Measurement and Belief Revision . 357 B.1.1 True Content and False Input . 357 B.1.2 False Content and False Input . 357 B.2 True Inputs Guaranteed to Increase Information Yield . 358 B.3 Paraconsistent Approaches . 361 C Environmental Information and Information Flow 367 C.1 Probabilistic Information . 367 C.2 The Arrow of Information Flow . 368 Abstract This thesis addresses some important questions regarding an account of semantic informa- tion. Starting with the contention that semantic information is to be understood as truthful meaningful data, several key elements for an account of semantic information are developed. After an introductory overview of information, the thesis is developed over four chapters. `Quantifying Semantic Information' looks at the quantification of semantic information as represented in terms of propositional logic. The main objective is to investigate how tradi- tional inverse probabilistic approaches to quantifying semantic information can be replaced with approaches based on the notion of truthlikeness. In `Agent-Relative Informativeness' the results of the previous chapter are combined with belief revision in order to construct a formal framework in which to, amongst other things, measure agent-relative informative- ness; how informative some piece of information is relative to a given agent. `Environmental Information and Information Flow' analyses several existing accounts of environmental in- formation and information flow before using this investigation to develop a better account of and explicate these notions. Finally, `Information and Knowledge' contributes towards the case for an informational epistemology, based on Fred Dretske's information-theoretic account of knowledge. ix x ABSTRACT Declaration This is to certify that i the thesis comprises only my original work towards the PhD except where indicated in the Preface, ii due acknowledgement has been made in the text to all other material used, iii the thesis is fewer than 100,000 words in length, exclusive of tables, maps, bibliographies and appendices. Simon D'Alfonso xi xii DECLARATION Preface This PhD thesis is the result of research conducted over the last three or so years. Whilst searching for a thesis topic, I became interested in philosophical work on information after coming across some of Luciano Floridi's work in the philosophy of information. This discovery led me to several other pieces of literature in the field that served as a starting point for my research. A notable mention goes to the introductory text Information and Information Flow [22], which introduced me to and provided an accessible overview of the areas that came to form my research agenda. Whilst `information' is a term that everyone is familiar with, the notion of information is one that I had not thought much about. Struck by the richness of the simple question `what is information?', my investigation was initiated by the curiosity that it raised. The novelty of and my interest in this general question has remained a driving factor in my research. The notion of information presents a vast conceptual labyrinth and there is a plethora of research avenues one could take in researching it. My background and preliminary readings resulted in my focus on semantic conceptions of information and the investigation carried out in this thesis. Thus the aim of this thesis is basically to establish a definition of semantic information, address some questions regarding semantic information that have been raised and develop several key elements for an account of semantic information. Not too long after Claude Shannon introduced in the middle of the twentieth century what is known as the mathematical theory of communication or information theory, philosophers started taking a serious interest in information. By the end of the first decade of the twenty- first century a substantial body of work on the philosophy of information has accumulated. This thesis makes a modest, yet I hope worthwhile, contribution to this field of research by expanding upon and adding to this body of work. After an introductory overview of information, this thesis is developed over four chapters. `Quantifying Semantic Information' investigates ways to quantitatively measure semantic in- formation as represented in terms of propositional logic. In `Agent-Relative Informativeness' xiii xiv PREFACE the results of the previous chapter are combined with belief revision to construct a for- mal account of measuring how informative some piece of information is to a given agent. `Environmental Information and Information Flow' analyses several existing accounts of en- vironmental information and information flow before using this investigation to develop a better account of and explicate these notions. Finally, with contributions from some of the previous chapter's results, `Information and Knowledge' contributes towards the case for an informational epistemology. Beyond their relevance to the philosophy of information, some of the results in this the- sis will be of particular relevance to and potentially find applicability in other areas. Two examples worth mentioning are the chapter on information and knowledge, which provides re- sponses to some general questions in epistemology and the chapter on agent-relative informa- tiveness, which deals with topics that overlap with formal epistemology and belief/database revision.
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