Logic, Language and Games

152 Pages·2015·6.689 MB·

by  Gabriel Sandu

ACTA
PHILOSOPHICA
FENNICA
VOL. 91 2015

LOGIC, LANGUAGE AND GAMES
GABRIEL SANDU

SOCIETAS PHILOSOPHICA FENNICA
ACTA PHILOSOPHICA FENNICA
Editor:
Ilkka Niiniluoto
Editorial Board:
Timo Airaksinen
Leila Haaparanta
Olh Koistinen
Simo Knuuttila
Gabriel Sandu
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ACTA PHILOSOPHICA FENNICA
Vol. 91

LOGIC, LANGUAGE AND
GAMES
GABRIEL SANDU
HELSINKI 2015
Copyright © 2015 The Philosophical Society of Finland
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A KOLLEGIALT GRANSKAD
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www.tsv.fi/tunnus
ISBN 978-951-9264-81-3
ISSN 0355-1792
Hakapaino Oy
Helsinki 2015
Contents
Introduction 1
1 Dependence and independence of quantifiers 3
1.1 Quantifiers . . . .. ... ... e 3
1.2 Dependence of quantifiers . . . .. ... ... ... ........ 4
1.3 Independence of quantifiers (Sher) . . .. ... ... ... .. .. 5
1.4 Independence of quantifiers (Henkin, Barwise) . . . . . . . .. .. 7
1.5 Game-theoretical interpretation of dependence and independence 9
1.5.1 Semantic games for classical logic . . . .. ... ... ... 9
1.5.2 Game-theoretical account of independence . . . . . . . .. 11
1.5.3 Independence as uniformity in extensive games . . . . . . 11
1.5.4 Independence as Nash equilibria in strategic games . . . . 12
2 Introduction to game theory 15
2.1 Extensive games with perfect information . . .. ... ... ... 15
2.2 Strategies . . . . . . ... e e e e e e 16
2.3 Solution concepts: backward induction . . . . . ... ... ..., 18
2.4 Backward Induction and Common knowledge of rationality . . . 20
25 Win-losegames . . . . .. ... ... ... oo 22
2.6 Determinacy of games: Zermelo’s theorem . . . . . ... .. ... 23
2.7 Extensive games with imperfect information . . . . . . ... ... 24
3 Strategic games 29
3.1 Basicnotions . . . . . ... . . . .. .. 29
3.2 Solution concepts: The elimination of dominated strategies . . . 30
3.3 Solution concepts: The iterative elimination of dominated strategies 32
3.4 Solution concepts: Equilibria and best response strategies . . .. 34
3.5 Equilibria and players’ beliefs: rationalization . . . . . . ... .. 37
4 Mixed strategies 41
4.1 Mixed strategies and expected utilities . . . . . . ... ... ... 41
4.2 Mixed strategy equilibria . . . . ... ... .000oL , 44
4.3 Identifying mixed strategy equilibria . . . . . .. ... ... ... 47
CONTENTS
5 Signaling games 53
5.1 The Stag Hunt and the Prisoner’s Dilemma . . . ... ... ... 93
5.2 Stenius: The Garden Game . . ... .. ... ... ........ 59
5.3 David Lewis: Signaling systems . . . . .. ... .......... 56
5.4 Signaling systems as extensive games with incomplete information 57
5.5 Incomplete vs. Imperfect information. . . . . . ... .. ... .. 58
5.6 Sender-Receivergames . . . . ... ... ... ........... 60
5.6.1 Strategies in Sender-Receiver games . . . ... ... ... 61
5.6.2 Utilities and Bayesian equilibria . . . . . ... ... ... 62
9.7 First example of a Sender-Receivergame . . . . . . ... ... .. 64
5.8 Second example of Sender-Receiver games: Lewis’s signaling sys-
LEIMNS . . . . . e e e e e e e e e e e e e e e e e e e e 67
9.9 Lewis’ signaling systems: strategic cheap talk games . .. .. .. 69
5.10 Conclusions . . . . . . . . . . . ... . ... e 71
Game-theoretical semantics 73
6.1 Introduction. . .. ... .. ... ... . ... ... . . ..., 73
6.2 First-order languagesand models . . . . . .. ... ........ 74
6.2.1 Languages. . . . . . . . . . . . e 74
6.22 Models ... ... ... . . ... .. e 75
6.2.3 Assignments . ... .......... ..... .... , 75
6.2.4 Satisfaction of atomic formulas . . ... ... ... .. .. 76
6.3 Semantical games: extensive games of perfect information . ... 76
6.4 Winning strategies: Game-theoretic truth and falsity . . . . . . . 78
6.5 Skolem functions and counter-examples . ... ... ... .. .. 80
6.6 Logical equivalences as recipes for transforming strategies . . .. 83
6.7 Game-theoretical negation . . . . . .. ... ... ......... 86
6.8 Equivalence of Tarskian and GTS truth (satisfaction) ... ... 91
6.9 Semantic games for FOL: strategic form . .. ... ... ... .. 92
Independence-friendly logic 95
7.1 Introduction. . . . .. .. .. ... ... . ... e 95
72 Thesyntax . .. .. .. . . . . .. i 96
7.3 Semantical games: extensive games of imperfect information . . . 97
7.4 Indeterminacy and Signaling in IF logic . . . ... ........ 98
7.5 NegationinIF logic ... ...................... 101
7.6 IF prefixes which are not first-order . . . .. ... ... ..... 103
7.7 Expressinginfinity . ... ... ... ... ... ... . ... 106
7.8 The incompleteness of IF logic . . ... .. ............ 108
7.9 Equivalent IF prefixes . . ... ... ... ... .......... 109
7.10 Playability of games: Perfect Recall . ... ... ... ...... 113
CONTENTS iii
8 Probabilistic IF logic 115
8.1 Indeterminacy. . ... ... .. .. ... .. ... ..., 115
8.2 Strategic IF games: pure strategies . . . . .. ... ... ..... 116
8.3 Strategic IF games: mixed strategies . . .. ... ... ... ... 117
8.4 Equilibrium semantics . . ... ... ... ... .. 000, 117
8.5 Equilibrium semantics: Case studies . . . . .. ... ... .... 121
8.6 Lewis’s signaling gamesin IF logic . .. ... ... ........ 122
8.7 Infinity on finite structures . . . . .. ... ... ... ... ... 127
8.8 What kind of probabilistic logic is Probabilistic IF logic? . . . . . 129
8.8.1 Probabilistic semantics: degrees of rational belief . . . . . 130
8.8.2 Probabilistic semantics: statistical knowledge (relative fre-
QUENCY) -« v v v v e e e e e e e e e e e e e e e e e e e 132
8.8.3 IF probabilities: some comparisons . . . . ... ... ... 133
884 Conclusions . . ... ... ... ... .. .. . 0. 136
References 137

Introduction
The present study is based on existing publications and lecture notes organized
in a new way. It centers around game-theoretical semantics for first-order logic
and its extension, Independence-friendly logic (IF logic). IF logic and its latest
development, probabilistic IF logic rely heavily on game-theoretical results (von
Neumann’s minimax theorem). It is tailored made to model various game-
theoretical phenomena like indeterminacy and signaling. For this reason a great
part of the book contains introductory material to game theory and some of its
solution concepts, which lead, in the end, to (Nash) equilibrium semantics.