## Analysis and Synthesis of Logics: How to Cut and Paste Reasoning SystemsStarting with simple examples showing the relevance of cutting and pasting logics, the monograph develops a mathematical theory of combining and decomposing logics, ranging from propositional and first-order based logics to higher-order based logics as well as to non-truth functional logics. The theory covers mechanisms for combining semantic structures and deductive systems either of the same or different nature (for instance, two Hilbert calculi or a Hilbert calculus and a tableau calculus). The important issue of preservation of properties is extensively addressed. For instance, sufficient conditions are provided for a combined logic to be sound and complete when the original component logics are known to be sound and complete. The book brings the reader to the front line of current research in the field by showing both recent achievements and directions of future investigations (in particular, multiple open problems). It also provides examples of potential applications in emergent fields like security protocols, quantum computing, networks and argumentation theory, besides discussing more classical applications like software specification, knowledge representation, computational linguistics and modular automated reasoning. This monograph will be of interest to researchers and graduate students in mathematical logic, theory of computation and philosophical logic with no previous knowledge of the subject of combining and decomposing logics, but with a working knowledge of first-order logic. The book will also be relevant for people involved in research projects where logic is used as a tool and the need for working with several logics at the same time is mandatory (for instance, temporal, epistemic and probabilistic logics). |

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### Contents

1 | |

3 | |

10 | |

12 | |

15 | |

17 | |

124 GodelLob modal logic and Peano arithmetic | 19 |

13 Algebraic fibring | 22 |

662 Completeness | 256 |

67 Final remarks | 260 |

Fibring higherorder logics | 263 |

71 Higherorder signatures | 265 |

72 Higherorder Hilbert calculi | 269 |

73 Higherorder interpretation systems | 275 |

74 Higherorder logic systems | 288 |

75 A general completeness theorem | 291 |

14 Possibletranslations semantics | 32 |

Splicing logics Syntactic fibring | 37 |

21 Language | 39 |

22 Hilbert calculi | 45 |

23 Preservation results | 55 |

232 Metatheorems | 59 |

233 Interpolation | 70 |

24 Final remarks | 88 |

Splicing logics Semantic fibring | 91 |

31 Interpretation systems | 92 |

32 Logic systems | 110 |

33 Preservation results | 113 |

332 Soundness | 116 |

333 Completeness | 119 |

34 Relationship with fibring by functions | 125 |

35 Final remarks | 136 |

Heterogeneous fibring | 139 |

41 Fibring consequence systems | 140 |

412 Fibring of consequence systems | 150 |

42 Fibring abstract proof systems | 160 |

422 Induced proof systems | 162 |

423 Fibring | 167 |

424 Proof systems vs consequence systems | 174 |

43 Final remarks | 177 |

Fibring nontruth functional logics | 179 |

51 Specifying valuation semantics | 180 |

52 Fibring nontruth functional logics | 195 |

53 Nontruth functional logic systems | 198 |

54 Preservation results | 201 |

542 Preservation of completeness by fibring | 208 |

55 Selffibring and nontruth functionality | 211 |

56 Final remarks | 213 |

Fibring ﬁrstorder logics | 215 |

61 Firstorder signatures | 216 |

62 Interpretation systems | 221 |

63 Hilbert calculi | 231 |

64 Firstorder logic systems | 240 |

65 Fibring | 242 |

66 Preservation results | 246 |

76 Fibring higherorder logic systems | 299 |

761 Preservation of soundness | 314 |

762 Preservation of completeness | 315 |

77 Final remarks | 322 |

Modulated fibring | 323 |

81 Language | 325 |

82 Modulated interpretation systems | 327 |

83 Modulated Hilbert calculi | 353 |

84 Modulated logic systems | 371 |

85 Preservation results | 376 |

852 Completeness | 379 |

86 Final remarks | 387 |

Splitting logics | 389 |

91 Basic notions | 391 |

92 Possibletranslations semantics | 400 |

93 Plain fibring of matrices | 419 |

94 Final remarks | 432 |

New trends Network fibring | 435 |

101 Introduction | 436 |

102 Integrating ﬂows of information | 439 |

103 Input output networks | 457 |

104 Fibring neural networks | 466 |

105 Fibring Bayesian networks | 476 |

106 Selffibring networks | 497 |

107 Final remarks | 515 |

Summingup and outlook | 519 |

112 Knowledge representation and agent modeling | 523 |

113 Argumentation theory | 527 |

114 Software specification | 541 |

1142 Synchronization | 546 |

1143 Specifications on institutions | 547 |

115 Emergent applications | 550 |

116 Outlook | 557 |

559 | |

579 | |

Table of symbols | 591 |

List of Figures | 595 |

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### Common terms and phrases

assignment Assume axiom Bayesian network calculus H causal Chapter classical logic component condition connectives consequence system consider corresponding Craig interpolation defined as follows Definition denote fibring by functions Figure first-order logic graph Hence Heyting algebra higher-order Hilbert calculus HLog implication induced inference rules input interpretation structure interpretation system presentation introduced intuitionistic logic Kripke frames Kripke structures Lemma Let H matrix logics metatheorem modal logic modulated fibring modulated Hilbert modulated interpretation system morphism h neuron nodes non-truth functional notion obtained operator original logics output paraconsistent paraconsistent logic presented in Example preservation results preserved by fibring proof systems properties propositional logic prove proviso pushout Recall recursive Bayesian network relation respect schema variables Section sequent sequent calculus set of formulas sharing signature morphism substitution system morphism Theorem translation truth values unconstrained fibring valuation