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

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Springer Science & Business Media, Jan 22, 2008 - Mathematics - 620 pages
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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).
 

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Contents

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
Bibliography
559
Subject index
579
Table of symbols
591
List of Figures
595
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