Logica Universalis (eBook)

Contents 5
Preface 7
References 10
Preface to the Second Edition 11
From Consequence Operator to Universal Logic: A Survey of General Abstract Logic 13
1. Introduction 13
2. Tarski’s consequence operator 14
3. Hertz and Gentzen’s proof systems 16
4. Matrix theory and abstract logic 19
5. Algebraic logic 20
6. Da Costa’s theory of valuation 22
7. Universal logic 23
References 25
Abstract Model Theory as a Framework for Universal Logic 28
1. Introduction 28
2. Abstract Model Theory 29
3. Lindstr¨ om Theorems 34
4. Abstract logic without negation 37
5. Logic frames 38
References 40
A Topological Approach to Universal Logic: Model- Theoretical Abstract Logics 43
1. Introduction 43
2. Model-theoretical abstract logics 45
3. Two examples 67
4. Future work 68
References 69
Selfextensional Logics with Implication 70
1. Introduction 70
2. Preliminaries 73
3. Hilbert-based deductive systems 77
4. Fregean logics with a deduction-detachment theorem 83
5. Selfextensional logics with a deduction-detachment theorem and Gentzen calculi 86
References 92
Logic without Self-Deductibility 94
1. The law of self-deductibility 94
2. A model for a logic without self-deductibility 95
3. Category-theoretic aspects of the subject 97
References 100
Equipollent Logical Systems 102
1. Introduction 102
2. From logical systems to theory spaces 104
3. Isomorphisms 106
4. Equipollence 110
5. Conclusion 113
References 114
What is a Logic? 115
1. Introduction 115
2. Institutions and Logics 117
3. Equivalence of Institutions 120
4. Model-Theoretic Invariants of Institutions 123
5. Proof Theoretic Invariants 127
6. 132
institutions 132
7. Conclusions 134
References 135
What is a Logic, and What is a Proof ? 138
1. Introduction 138
2. What is a logic ? 139
3. What is a proof ? 143
4. Summary 152
References 153
Non-deterministic Matrices and Modular Semantics of Rules 157
1. Introduction 157
2. Preliminaries 158
3. Semantics for the Basic System 164
4. Semantics for the Extensions of B Induced by 167
5. Applications 169
6. Other Plausible Extensions of the Basic System 170
7. Conclusions and Further Research 173
References 174
Two’s Company: The Humbug of Many Logical Values” 176
1. Suszko’s Thesis 177
2. Reductive results 178
3. Separating truth-values 180
4. Dyadic semantics 183
5. From finite matrices to dyadic valuations 185
6. Some Illustrations 187
7. Application: tableaux for logics with dyadic semantics 191
8. Conclusions 193
References 194
Friendliness and Sympathy in Logic† 196
1. Friendliness 196
2. Links with Familiar Notions 207
3. From Friendliness to Sympathy 215
4. Open Questions 221
5. Appendix 222
References 224
Logical Discrimination 226
1. Introduction 226
2. Discrimination In Logics as Sets of Formulas 230
3. Discrimination in Logics as Consequence Relations 236
References 245