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Modalities for Reasoning about Knowledge and Quantities Wiebe van der Hoek Modalities for Reasoning about Knowledge and Quantities Wiebe van der Hoek © W. van der Hoek, Amsterdam, 1992 ISBN 90-9004930-4 VRDE UNIVERSITEIT Modalities for Reasoning about Knowledge and Quantities ACADEMISCH PROEFSCHRIFT ter verkrijging van de graad van debtor aan de Vrije Universiteit van Amsterdam, op gezag van de rector magnificus dr. C. Datema, hoogleraar aan de faculteit der letteren, in het openbaar te verdedigen ten overstaan van de promotiecommissie van de faculteit der wiskunde en informatica op donderdag 12 maart 1992 te 13.30 uur in het hoofdgebouw van de universiteit, De Boelelaan 1105 door Wiebe van der Hoek geboren te Luxwoude Elinkwijk, Utrecht 1992 Promotor: prof.dr. J.-J.Ch. Meyer Referent: prof.dr. J.F.A.K. van Benthem Acknowledgements This dissertation had probably not been written without the continual support of John-Jules Meyer, my promotor. He encouraged me and taught me how to work out my often sketchy ideas about a topic into a coherent paper. Moreover, I thank him for putting his confidence in me and my work, from the very start. I always enjoyed our co-operation—which has been expanded beyond the scope of this thesis by now—not in the least because of his cheerfulness. This dissertation would not have been written the way it is without the stimulating support of Johan van Benthem. In fact, his role in this thesis goes beyond that of a referent: he supervised my master’s thesis, which laid the foundation of Chapter 4 of this dissertation, which, on its turn, affected several other chapters. Moreover, he kept his interest in my work during the last four years, and provided me with several valuable suggestions. This work received a surplus value through the co-operation with Maarten de Rijke. His contribution is most obvious in Chapter 6, which was written in a joint project. He also read this whole dissertation very carefully, and was willing to explain me (sometimes repeatedly, I have to admit) various subtleties concerning modal logic: we became true ‘electronic pen-pals’. Moreover, when printing this work almost failed the last minute before its final submission, he kept his head cool and used his Macintosh-skills to avert a drama. This thesis would not have been on its present subject, if Roland Backhouse had not encouraged me to follow my own interests, for which I like to thank him at this place. Several occasions and people stimulated my work. Beside the Free University, which supported me on several occasions, I thank NWO and Shell for their financial support for my travels to Tbilisi and St. Petersburg, respectively. I enjoyed the meetings of the Colloquium on ‘Niet-Monotoon Redeneren’ and, in particular, I like to thank the co-organizers Yao-Hua Tan and Cees Witteveen for the pleasant collaboration. Roel de Vrijer and Gerard Vreeswijk always showed interest in my work; they read and commented upon several tentative versions of my papers. Moreover, they are members of our group ‘Theoretical Computer Science’, of which I enjoyed being a member during the last four years. Apart from the members already quoted, I thank Jaco de Bakker, Franck van Breugel, Erik Hamoen, Jan-Willem Klop, Aart Middeldorp, Vincent van Oostrom and Erik de Vink for the animated discussions we had. Finally, I thank Betsy and Tom for their support. Tom contributed invaluably to the joy of my everyday life and Betsy made me sensible of so many things outside the scope of my work. I am aware that their friendship contributed more to my work on this thesis than the other way around. Contents Acknowledgements v 1 A Guide to this Thesis 1 1 Introduction.................................................................................................................1 2 Modalities, Knowledge and Quantities.............................................................2 2.1 Modalities 2 2.2 Knowledge 7 2.3 Quantities 10 3 Modal logic and this thesis........................................................................................13 4 Organisation of the thesis..........................................................................................17 2 Systems for Knowledge and Beliefs 19 1 Introduction...............................................................................................................19 2 The system KBCD as a basis for knowledge and belief..........................................20 3 Kripke semantics for KB..........................................................................................26 4 Some correspondence results...................................................................................33 5 Conscious Beliefs and Believed Consciousness.......................................................37 6 Conclusions and problems.......................................................................................46 3 Making Some Issues of Implicit Knowledge Explicit 47 1 Introduction..............................................................................................................47 2 Some correspondence results for intersection and union.........................................49 3 Splitting worlds.........................................................................................................54 4 Adding special properties..........................................................................................56 5 Applications to implicit knowledge...........................................................................68 6 (Truncated) graded modalities.........................................................................73 7 Explicit definitions for intersection...........................................................................77 4 On the Semantics of Graded Modalities 81 1 Introduction..............................................................................................................81 2 Language and semantics...........................................................................................82 3 Elementary model theory: preservation....................................................................85 4 Expressive power 1: graded modal equivalence.......................................................89 5 Expressive power 2: correspondence.......................................................................96 5.1 Definability of first order properties 97 5.2 First order definability of modal principles 101 6 Filtration..................................................................................................................107 5 Graded Modalities in Epistemic Logic 113 1 Introduction.............................................................................................................113 2 The system Gr(S5)..................................................................................................115 3 Epistemic reading.....................................................................................................118 4 Examples.................................................................................................................121 5 Conclusion..............................................................................................................123 6 Generalized Quantifiers and Modal Logic 125 1 Introduction.............................................................................................................125 2 The systems QUANT and QUANT^ .....................................................................126 2.1 Basic definitions and examples 126 2.2 Normal forms 129 2.3 Connections with other formalisms 130 3 Completeness and complexity........................................................................133 3.1 Prerequisites 133 3.2 Completeness of QUANT^ 134 3 Complexity 137 4 Semantic constraints and inferential patterns.............................................................138 4.1 Semantic constraints 139 4.2 Inferential patterns 142 5 Beyond the first order boundary..............................................................................146 5.1 Axioms and notation 147 5.2 Completeness 149 5.3 Normal forms and semantic constraints 150 5.4 Other higher order quantifiers 155 6 Further directions; concluding remarks...................................................................157 7 Qualitative Modalities 159 1 Introduction.............................................................................................................159 2 The System QM...............................................................................................161 3 Probabilistic Kripke models....................................................................................165 4 The scheme B(m).....................................................................................................169 5 Correspondence and expressiveness.......................................................................174 6 Conclusion..............................................................................................................181 8 PFD: A Logic Combining Modalities and Probabilities 183 1 Introduction.............................................................................................................183 2 The logic PFD..................................................................................................187 3 A semantics for PFD and some of its properties.....................................................191 4 Conclusion and directions for further research..............................................199 A References 201 В Samenvatting 209 1 A Guide to this Thesis 1 Introduction In this chapter, I will informally introduce the notions that are brought together in this thesis: modalities, (reasoning about) knowledge and quantities. I will also indicate how they are related and that it makes sense to certify them as one subject of a dissertation. To be more precise, in section 2,1 will argue that modal logic provides a natural setting for reasoning about knowledge and quantities. Here, ‘modal logic’ is to be understood as the logic of ‘philosophical modalities’; studied since Aristotle. Although I think that ‘Modalities for Reasoning about Knowledge and Quantities’ make sense in several contexts (after all, it brings together notions from the fields of logic, philosophy and psychology), in section 2 I will focus on their use and applicability in the fields of computer science and artificial intelligence. The fact that the logic of ‘philosophical modalities’ may be of help to study reasoning about knowledge and quantities indicates that they share some general structure—in fact, as will be mentioned in section 2, this structure crops up in diverse, sometimes unexpected areas (with sometimes unlooked for ‘interpretations’ of the modality); each trying to formalize some specific kind of reasoning. This observation justifies the study of general patterns behind the specific behaviour of the interpretations. In this sense, modal logic becomes the formal study of the (properties of) logic(s) of modalities. This thesis aims to contribute to modal logic (in the latter sense): for instance, questions that arise in the field of (the representation of) knowledge are lifted to a more abstract level and then studied from a general modal logical point of view. As a consequence, several chapters in this thesis (rely on and) develop techniques and results that are relevant for modal logic—in the latter sense. In section 3 of this chapter, I will become more specific about which typical modal themes will be addressed in this thesis. Although making a distinction between the two readings of ‘modal logic’ (i.e. the study of 2 A Guide to this Thesis formalising the classical modalities versus the mathematical analyses of the formal modal systems) may be fruitful in an introductory chapter (1 will emphasize the two meanings in sections 2 and 3, respectively) such a distinction is not always well defined or useful. In fact, in the rest of this thesis (from chapter 2 onwards) I will not explicitly distinguish between the two meanings. Whereas section 3 may be considered to contain a subject-oriented description of this dissertation, in section 4, adopting a chapter-oriented approach, I will attempt to prepare the reader for the several routes he can take through this thesis and mention several criss-crosses (s)he may choose between the several chapters. As the reader may expect by now, the three topics united in the title of this dissertation will then re-emerge, since they will be important landmarks on his or her journey. 2 Modalities, Knowledge and Quantities 2.1 Modalities As the title of this thesis suggests, modalities play a crucial role in this study. With modalities, I refer to their formal definition and use in the area of (philosophical) logic. Modal logic is the logic of ‘must be’ and ‘may be’ (according to one of the standard introductions ([HugCre68]) to this field); the logic of necessity and possibility, (as another standard introduction ([Che80]) puts it). The ‘standard’, or ‘neutral’ symbols for necessity and possibility are ‘D’ (“box”) and ‘0’ (“diamond”), respectively. They are considered ‘dual’ in the following sense; Dp= —10—ip, i.e. “necessarily p” is equivalent to “not possibly —ip”. This is indeed the standard, or ‘philosophical’ (cf. [Gam82]) notion of modality, which gives us a way to distinguish between facts that are ‘accidentally’ true—like “it is raining” (r)—and those that are necessarily true—like “necessarily, all events have a cause” (Dc). This standard modality has been studied since Aristotle, and was fully recognized by Kant in his ‘Kritik der Reinen Vemunft’; however, the subject of modalities was rigourously expelled from the field of formal logic by Frege, using the argument that when we say that a proposition is necessarily true, we only give an impression of the reasons for our judgement. And, according to Frege, it is only the content of a judgement that is logically relevant. As is clearly exposed in [Gam82], the ‘philosophical’ modalities were re-introduced through the backdoor to cope with the problems that several logicians had with the fact that, around 1925, the material implication was declared to be the implication for formal logic. Logicians (but also logicians-to-be, as may be confirmed by everyone who has taught an introductory course in logic) have had problems with the paradoxes of this material implication which come