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Propositional Quantification and Comparison in Modal Logic By Propositional Quantification and Comparison in Modal Logic by Yifeng Ding A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Logic and the Methodology of Science in the Graduate Division of the University of California, Berkeley Committee in charge: Associate Professor Wesley H. Holliday, Chair Professor Shamik Dasgupta Professor Paolo Mancosu Professor Dana S. Scott Professor Seth Yalcin Spring 2021 Propositional Quantification and Comparison in Modal Logic Copyright 2021 by Yifeng Ding 1 Abstract Propositional Quantification and Comparison in Modal Logic by Yifeng Ding Doctor of Philosophy in Logic and the Methodology of Science University of California, Berkeley Associate Professor Wesley H. Holliday, Chair We make the following contributions to modal logics with propositional quantifiers and modal logics with comparative operators in this dissertation: • We define a general notion of normal modal logics with propositional quantifiers. We call them normal Π-logics. Then, as was done by Scrogg's theorem on extensions of the modal logic S5, we study in general the normal Π-logics extending S5. We show that they are all complete with respect to their algebraic semantics based on complete simple monadic algebras. We also show that the lattice formed by these logics is isomorphic to the lattice of the open sets of the disjoint union of two copies of the one-point compactification of N with the natural order topology. Further, we show how to determine the computability of normal Π-logics extending S5Π. A corollary is that they can be of arbitrarily high Turing-degree. • Regarding the normal Π-logics extending the modal logic KD45, we identify two im- 8 portant axioms: Immod : 8p(p ! p) and 4 : 8p' ! 8p'. We argue that when is interpreted as the belief operator, we should not accept Immod in the logic of belief while 48 is desirable in settings with full introspection. Then we provide al- gebraic semantics based on complete well-connected pseudo monadic algebras for and 8p and show that with respect to these algebras, normal Π-logics extending KD45 and 48 with a finite list of formulas are complete. We also give a general completeness theorem for atomic complete well-connected pseudo monadic algebras and a sufficient condition for the decidability of logics obtained by classes of these algebras, atomic or not. A special case of these general theorems is that the normal Π-logic of serial, transitive, and Euclidean Kripke frames, that is, the Kripke frames validating KD45, 8 is axiomatized by KD45, 4 , Immod, and 9p(p ^ 8q(q ! (p ! q))) together with the usual axioms and rules for propositional quantifiers, and this logic is decidable. 2 Other than completeness and decidability, we also show that 48 is not in the smallest normal Π-logic extending KD45, using a countermodel based on a possible-world frame with propositional contingency, and that 48 is valid in any complete Boolean algebra expansion validating KD45. • For modal logics with comparative operators, we first provide an axiomatization of the logic of comparing the cardinality of sets, as defined by Cantor. The main technical contribution is the observation that in a purely comparative language, we can define finiteness well enough so that an axiomatization can be done by combining the logic of comparative cardinality for finite sets and the logic of comparative cardinality for infinite sets. Note that these two logics are very different: the former contains the axiom of qualitative additivity: jAj ≥ jBj iff jA n Bj ≥ jB n Aj but not the axiom of absorption: if jAj ≥ jBj and jAj ≥ jCj then jAj ≥ jB [ Cj, while the latter does the opposite. • Then we consider modal logics for comparative imprecise probability. In these logics, comparisons are made according to a set of probability measures and can be intuitively read as either \at least as likely as" (symbolized by %) or \more likely than" (symbol- ized by ). We first disambiguate two interpretations of \more likely than" based on a set of probability measures and show that the stronger interpretation is not definable from \at least as likely as" while the weaker sense is. Then, we go on to axiomatize the logic of imprecise probability in a sequence of languages obtained by adding to the language with just \at least as likely as", one by one, the comparative operator for \more likely than" (in the stronger sense), a unary operator ♦ for \possibly", and a binary operator h'i for \possibly ', and after learning the truth of ', ". We also comment on the expressivity of these languages and the decidability of the logics in these languages. In particular, we show that many distinctive features of the imprecise probability approach of representing the doxastic states of agents, such as the problem of dilation, are observable at this purely comparative level. Finally, we add a pair of + − operators Ip ' and Ip ', intuitively read, respectively, as \after learning the existence of an actually true new proposition, now named by p, '", and \after learning the ex- istence of an actually false new proposition, now named by p, '". We show that this pair of operators allow us to formalize a common kind of information dynamics and will boost the expressivity of the language to a quantitative level. i Contents Contents i List of Figures iii 1 Introduction 1 1.1 Modal Logics with Propositional Quantifiers .................................. 2 1.2 Modal Logics for Comparing Propositions ..................................... 10 1.3 Comparison with the Published Versions ...................................... 17 2 Logics with Propositional Quantifiers Extending S5Π 19 2.1 Introduction ...................................................................... 20 2.2 Preliminaries ..................................................................... 22 2.3 Semantical and syntactical reduction........................................... 26 2.4 Types and type space ........................................................... 28 2.5 Main results ...................................................................... 31 2.6 Conclusion........................................................................ 35 3 Logics of Belief and Propositional Quantifiers 36 3.1 Introduction ...................................................................... 37 3.1.1 Dubious principles and possible-world semantics...................... 37 3.1.2 Axiomatizability for modal logics with propositional quantifiers..... 41 3.1.3 Organization ............................................................. 43 3.2 Syntax, semantics, logics, and the problem of Immod ......................... 44 3.3 Soundness of 48 on complete KD45 algebras .................................. 53 3.4 Completeness of KD485Π with respect to complete proper filter algebras ... 61 3.4.1 Auxiliary languages, semantics, and translations...................... 63 3.4.2 Logics in auxiliary languages and completeness proof ................ 67 3.4.3 Syntactical Reduction ................................................... 73 3.4.4 Quotients of complete Boolean algebras ............................... 80 3.5 Stronger logics and decidability................................................. 87 3.6 Conclusion........................................................................ 95 4 The Logic of Comparative Cardinality 98 ii 4.1 Introduction ...................................................................... 99 4.2 Formal setup and statement of main result .................................... 100 4.3 Polarizability rule and finite cancellation axiom schema ...................... 106 4.4 Other types of models ........................................................... 114 4.4.1 Measure algebra models ................................................. 114 4.4.2 Comparison algebra models ............................................. 116 4.4.3 Effective finite model property ......................................... 120 4.5 Canonical comparison algebra models.......................................... 122 4.6 Completeness with predicates for infinite and finite sets...................... 124 4.7 Completeness without predicates for infinite and finite sets .................. 125 4.7.1 Flexible models .......................................................... 126 4.7.2 Representation using axioms of the language.......................... 130 4.7.3 Completeness ............................................................ 132 4.8 Open problems ................................................................... 133 5 Logics of Imprecise Comparative Probability 136 5.1 Introduction ...................................................................... 137 5.2 Representation ................................................................... 142 5.3 The Logic IP(%) ................................................................. 147 5.4 The Logic IP(%; ) .............................................................. 150 5.4.1 Logic...................................................................... 151 5.4.2 Expressivity .............................................................. 155 5.5 The Logic IP(%; ; ♦) ........................................................... 156 5.5.1 Logic...................................................................... 157 5.5.2 Complexity ............................................................... 161 5.5.3 Expressivity .............................................................. 161 5.6 Dynamics ........................................................................
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