Beziau Logica Universalis

Towards a General Theory of Logic
2. Auflage 2007
ISBN: 978-3-7643-8354-1
Verlag: Springer
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 246 Seiten

Universal Logic is not a new logic, but a general theory of logics, considered as mathematical structures. The name was introduced about ten years ago, but the subject is as old as the beginning of modern logic. It was revived after the flowering of thousands of new logics during the last thirty years: there was a need for a systematic theory of logics to put some order in this chaotic multiplicity.
The present book contains recent works on universal logic by first-class researchers from all around the world. The book is full of new and challenging ideas that will guide the future of this exciting subject. It will be of interest for people who want to better understand what logic is. It will help those who are lost in the jungle of heterogeneous logical systems to find a way. Tools and concepts are provided here for those who want to study classes of already existing logics or want to design and build new ones.

Autoren/Hrsg.

Beziau, Jean-Yves

Inhaltsverzeichnis

1;Contents;5
2;Preface;7
2.1;References;10
3;Preface to the Second Edition;11
4;From Consequence Operator to Universal Logic: A Survey of General Abstract Logic;13
4.1;1. Introduction;13
4.2;2. Tarski’s consequence operator;14
4.3;3. Hertz and Gentzen’s proof systems;16
4.4;4. Matrix theory and abstract logic;19
4.5;5. Algebraic logic;20
4.6;6. Da Costa’s theory of valuation;22
4.7;7. Universal logic;23
4.8;References;25
5;Abstract Model Theory as a Framework for Universal Logic;28
5.1;1. Introduction;28
5.2;2. Abstract Model Theory;29
5.3;3. Lindstr¨ om Theorems;34
5.4;4. Abstract logic without negation;37
5.5;5. Logic frames;38
5.6;References;40
6;A Topological Approach to Universal Logic: Model- Theoretical Abstract Logics;43
6.1;1. Introduction;43
6.2;2. Model-theoretical abstract logics;45
6.3;3. Two examples;67
6.4;4. Future work;68
6.5;References;69
7;Selfextensional Logics with Implication;70
7.1;1. Introduction;70
7.2;2. Preliminaries;73
7.3;3. Hilbert-based deductive systems;77
7.4;4. Fregean logics with a deduction-detachment theorem;83
7.5;5. Selfextensional logics with a deduction-detachment theorem and Gentzen calculi;86
7.6;References;92
8;Logic without Self-Deductibility;94
8.1;1. The law of self-deductibility;94
8.2;2. A model for a logic without self-deductibility;95
8.3;3. Category-theoretic aspects of the subject;97
8.4;References;100
9;Equipollent Logical Systems;102
9.1;1. Introduction;102
9.2;2. From logical systems to theory spaces;104
9.3;3. Isomorphisms;106
9.4;4. Equipollence;110
9.5;5. Conclusion;113
9.6;References;114
10;What is a Logic?;115
10.1;1. Introduction;115
10.2;2. Institutions and Logics;117
10.3;3. Equivalence of Institutions;120
10.4;4. Model-Theoretic Invariants of Institutions;123
10.5;5. Proof Theoretic Invariants;127
10.6;6.;132
10.7;institutions;132
10.8;7. Conclusions;134
10.9;References;135
11;What is a Logic, and What is a Proof ?;138
11.1;1. Introduction;138
11.2;2. What is a logic ?;139
11.3;3. What is a proof ?;143
11.4;4. Summary;152
11.5;References;153
12;Non-deterministic Matrices and Modular Semantics of Rules;157
12.1;1. Introduction;157
12.2;2. Preliminaries;158
12.3;3. Semantics for the Basic System;164
12.4;4. Semantics for the Extensions of B Induced by;167
12.5;5. Applications;169
12.6;6. Other Plausible Extensions of the Basic System;170
12.7;7. Conclusions and Further Research;173
12.8;References;174
13;Two’s Company: The Humbug of Many Logical Values”;176
13.1;1. Suszko’s Thesis;177
13.2;2. Reductive results;178
13.3;3. Separating truth-values;180
13.4;4. Dyadic semantics;183
13.5;5. From finite matrices to dyadic valuations;185
13.6;6. Some Illustrations;187
13.7;7. Application: tableaux for logics with dyadic semantics;191
13.8;8. Conclusions;193
13.9;References;194
14;Friendliness and Sympathy in Logic†;196
14.1;1. Friendliness;196
14.2;2. Links with Familiar Notions;207
14.3;3. From Friendliness to Sympathy;215
14.4;4. Open Questions;221
14.5;5. Appendix;222
14.6;References;224
15;Logical Discrimination;226
15.1;1. Introduction;226
15.2;2. Discrimination In Logics as Sets of Formulas;230
15.3;3. Discrimination in Logics as Consequence Relations;236
15.4;References;245