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Analysis and Synthesis of Logics: How to Cut and Paste Reasoning Systems

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Starting 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).

Author(s): Walter Carnielli, Marcelo Coniglio, Dov M. Gabbay, Paula Gouveia, Cristina Sernadas
Series: Applied Logic Series 35
Publisher: Springer
Year: 2008

Language: English
Pages: 612

140206781X......Page 1
Contents......Page 6
Preface......Page 9
1. Introductory overview......Page 17
1.1 Consequence systems......Page 19
1.2 Splicing and splitting......Page 26
1.2.1 Fusion of modal logics......Page 28
1.2.2 Product of modal logics......Page 31
1.2.3 Fibring by functions......Page 33
1.2.4 Gödel-Löb modal logic and Peano arithmetic......Page 35
1.3 Algebraic fibring......Page 38
1.4 Possible-translations semantics......Page 48
2. Splicing logics: Syntactic fibring......Page 53
2.1 Language......Page 55
2.2 Hilbert calculi......Page 61
2.3.1 Global and local derivation......Page 71
2.3.2 Metatheorems......Page 75
2.3.3 Interpolation......Page 86
2.4 Final remarks......Page 104
3. Splicing logics: Semantic fibring......Page 106
3.1 Interpretation systems......Page 107
3.2 Logic systems......Page 125
3.3.1 Global and local entailment......Page 128
3.3.2 Soundness......Page 131
3.3.3 Completeness......Page 134
3.4 Relationship with fibring by functions......Page 140
3.5 Final remarks......Page 151
4. Heterogeneous fibring......Page 153
4.1.1 Induced consequence systems......Page 154
4.1.2 Fibring of consequence systems......Page 164
4.2.1 Abstract proof systems......Page 174
4.2.2 Induced proof systems......Page 176
4.2.3 Fibring......Page 181
4.2.4 Proof systems vs consequence systems......Page 188
4.3 Final remarks......Page 191
5. Fibring non-truth functional logics......Page 193
5.1 Specifying valuation semantics......Page 194
5.2 Fibring non-truth functional logics......Page 209
5.3 Non-truth functional logic systems......Page 212
5.4.1 Encoding CEQ in the object Hilbert calculus......Page 215
5.4.2 Preservation of completeness by fibring......Page 222
5.5 Self-fibring and non-truth functionality......Page 225
5.6 Final remarks......Page 227
6. Fibring first-order logics......Page 229
6.1 First-order signatures......Page 230
6.2 Interpretation systems......Page 235
6.3 Hilbert calculi......Page 245
6.4 First-order logic systems......Page 254
6.5 Fibring......Page 256
6.6.1 Metatheorems......Page 260
6.6.2 Completeness......Page 270
6.7 Final remarks......Page 274
7. Fibring higher-order logics......Page 276
7.1 Higher-order signatures......Page 278
7.2 Higher-order Hilbert calculi......Page 282
7.3 Higher-order interpretation systems......Page 288
7.4 Higher-order logic systems......Page 301
7.5 A general completeness theorem......Page 304
7.6 Fibring higher-order logic systems......Page 312
7.6.1 Preservation of soundness......Page 327
7.6.2 Preservation of completeness......Page 328
7.7 Final remarks......Page 335
8. Modulated fibring......Page 336
8.1 Language......Page 338
8.2 Modulated interpretation systems......Page 340
8.3 Modulated Hilbert calculi......Page 366
8.4 Modulated logic systems......Page 384
8.5.1 Soundness......Page 389
8.5.2 Completeness......Page 392
8.6 Final remarks......Page 400
9. Splitting logics......Page 402
9.1 Basic notions......Page 404
9.2 Possible-translations semantics......Page 413
9.3 Plain fibring of matrices......Page 432
9.4 Final remarks......Page 445
10. New trends: Network fibring......Page 447
10.1 Introduction......Page 448
10.2 Integrating flows of information......Page 451
10.3 Input output networks......Page 469
10.4 Fibring neural networks......Page 478
10.5 Fibring Bayesian networks......Page 488
10.6 Self-fibring networks......Page 509
10.7 Final remarks......Page 527
11.1 Synthesis......Page 531
11.2 Knowledge representation and agent modeling......Page 535
11.3 Argumentation theory......Page 539
11.4.1 Temporalization and parameterization......Page 553
11.4.2 Synchronization......Page 558
11.4.3 Specifications on institutions......Page 559
11.5 Emergent applications......Page 562
11.6 Outlook......Page 569
Bibliography......Page 571
B......Page 591
C......Page 592
E......Page 593
H......Page 594
I......Page 595
L......Page 596
M......Page 597
P......Page 598
R......Page 600
W......Page 601
Table of symbols......Page 602
List of figures......Page 605