## Modalities and MultimodalitiesIn the last two decades modal logic has undergone an explosive growth, to thepointthatacompletebibliographyofthisbranchoflogic,supposingthat someone were capable to compile it, would ?ll itself a ponderous volume. What is impressive in the growth of modal logic has not been so much the quick accumulation of results but the richness of its thematic dev- opments. In the 1960s, when Kripke semantics gave new credibility to the logic of modalities? which was already known and appreciated in the Ancient and Medieval times? no one could have foreseen that in a short time modal logic would become a lively source of ideas and methods for analytical philosophers,historians of philosophy,linguists, epistemologists and computer scientists. The aim which oriented the composition of this book was not to write a new manual of modal logic (there are a lot of excellent textbooks on the market, and the expert reader will realize how much we bene?ted from manyofthem)buttoo?ertoeveryreader,evenwithnospeci?cbackground in logic, a conceptually linear path in the labyrinth of the current panorama of modal logic. The notion which in our opinion looked suitable to work as a compass in this enterprise was the notion of multimodality, or, more speci?cally, the basic idea of grounding systems on languages admitting more than one primitive modal operator. |

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### Indice

1 | |

4 | |

13 The semantical analysis of PC | 9 |

14 Constructive completeness of PC | 12 |

15 Decidability of PC | 14 |

16 Postcompleteness and other properties of PC | 17 |

17 Exercises | 19 |

18 Further reading | 22 |

64 Other temporal systems | 166 |

65 USlogics metric tense logics and hybrid logics | 174 |

66 Exercises | 178 |

67 Further reading | 180 |

Epistemic logic knowledge and belief | 183 |

72 Knowledge belief and agents | 187 |

73 The minimal logic of knowledge | 189 |

74 The systems KᴹKTᴹS4ᴹ and S5ᴹ | 194 |

The syntax of normal modal systems | 25 |

22 Minimal properties of modal systems | 31 |

23 Systems between K and S5 | 34 |

24 Modalities in S5 | 43 |

25 Exercises | 46 |

26 Further reading | 48 |

The semantics of normal modal systems | 49 |

32 Carnapian models and relational models | 53 |

33 Correspondence theory and bisimulations | 64 |

34 The method of relational tableaux | 72 |

35 Exercises | 81 |

36 Further reading | 84 |

Completeness and canonicity | 87 |

42 Completeness by Henkins method | 92 |

models versus frames | 104 |

44 The logic of arithmetical provability | 107 |

45 Exercises | 114 |

46 Further reading | 115 |

Incompleteness and finite models | 117 |

52 Finite model property and filtrations | 123 |

53 Exercises | 136 |

54 Further reading | 138 |

Temporal logics | 141 |

62 Completeness and incompleteness of PFlogics | 156 |

63 Monomodal fragments of PFlogics | 162 |

75 Common knowledge and implicit knowledge | 196 |

76 The logic of belief | 200 |

77 Exercises | 202 |

78 Further reading | 203 |

Multimodal logics | 205 |

82 Multimodal languages | 206 |

83 The elementary multimodal systems | 209 |

84 Axioms for multimodal logics | 213 |

85 Multimodal systems and strict implication | 220 |

86 Multimodal models and completeness | 222 |

87 Exercises | 235 |

88 Further reading | 237 |

Towards quantified modal logic | 240 |

92 Necessary and contingent identities | 250 |

93 The problem of completeness in firstorder modal logic | 256 |

94 Inclusive domains and arbitrary domains | 263 |

95 Quantification and multimodalities | 267 |

96 Exercises | 270 |

97 Further reading | 271 |

273 | |

289 | |

293 | |

297 | |

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accessibility relations affirmative agent algebra arbitrary assignment atomic variables axiom schemas axiomatized Barcan Formula bisimulation bridge axioms called canonical frame canonical model Chapter characterized class of frames complete with respect contradiction deﬁnable deﬁned Deﬁnition derived diagram diﬀerent diﬃculties disjunction Dugundji’s epistemic logic equivalent example Exercise exists fact false ﬁnd ﬁnite number ﬁrst ﬁrst-order frame F hence iﬀ implies inﬁnite instance introduced irreﬂexive knowledge language Lemma maximal consistent extension maximal consistent set modal algebra modal formulas modal logic modal operators modal parameters monomodal multi-agent frames multimodal logics multimodal systems negation normal modal system notion obtained Peano arithmetic possible worlds procedure proof properties propositional variables provable prove quantiﬁers Reductio reﬂexive result rules S-consistent satisﬁes Section semantics Springer Science+Business Media subformulas Suppose syntactical tableau tautology temporal temporal logic theorem of S5 transitive true truth-values valid α α