Modalities for Reasoning about Knowledge and Quantities [PhD Thesis]

221 Pages·1992·7.32 MB·English

by  Wiebe van der Hoek

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