Logica Universalis (eBook)

Contents 6
Preface 8
References 11
Part I Universal Logic: Frameworks and Structures 12
From Consequence Operator to Universal Logic: A Survey of General Abstract Logic 14
1. Introduction 14
2. Tarski’s consequence operator 15
3. Hertz and Gentzen’s proof systems 17
4. Matrix theory and abstract logic 20
5. Algebraic logic 21
6. Da Costa’s theory of valuation 23
7. Universal logic 24
References 26
Abstract Model Theory as a Framework for Universal Logic 30
1. Introduction 30
2. Abstract Model Theory 31
3. Lindstr¨ om Theorems 36
4. Abstract logic without negation 39
5. Logic frames 40
References 42
A Topological Approach to Universal Logic: Model-Theoretical Abstract Logics 46
1. Introduction 47
2. Model-theoretical abstract logics 48
3. Two examples 72
4. Further work 73
References 74
Selfextensional Logics with Implication 76
1. Introduction 76
2. Preliminaries 79
3. Hilbert-based deductive systems 83
4. Fregean logics with a deduction-detachment theorem 89
5. Selfextensional logics with a deduction-detachment theorem and Gentzen calculi 92
References 98
Logic without Self-Deductibility 100
1. The law of self-deductibility 100
2. A model for a logic without self-deductibility 101
3. Category-theoretic aspects of the subject 103
References 106
Part II Identity and Nature of Logical Structures 108
Equipollent Logical Systems 110
1. Introduction 110
2. From logical systems to theory spaces 112
3. Isomorphisms 114
4. Equipollence 118
5. Conclusion 121
References 122
What is a Logic? 124
1. Introduction 124
2. Institutions and Logics 126
3. Equivalence of Institutions 129
4. Model-Theoretic Invariants of Institutions 132
5. Proof Theoretic Invariants 136
6. Conclusions 141
References 142
What is a Logic, and What is a Proof ? 146
1. Introduction 146
2. What is a logic ? 147
3. What is a proof ? 151
References 154
Part III Tools and Concepts for Universal Logic 158
Non-deterministic Matrices and Modular Semantics of Rules 160
1. Introduction 160
2. Preliminaries 161
3. Semantics for the Basic System 166
4. Semantics for the Extensions of B Induced by 169
5. Applications 171
6. Other Plausible Extensions of the Basic System 173
7. Conclusions and Further Research 176
References 177
Two’s Company: The Humbug of Many Logical Values” 180
1. Suszko’s Thesis 181
2. Reductive results 182
3. Separating truth-values 184
4. Dyadic semantics 187
5. From .nite matrices to dyadic valuations 189
6. Some Illustrations 191
7. Application: tableaux for logics with dyadic semantics 195
8. Conclusions 197
References 198
Logical Friendliness and Sympathy 202
1. Friendliness 202
2. From Friendliness to Sympathy 210
3. Open Questions 214
Appendix 214
Acknowledgements 216
Logical Discrimination 218
1. Introduction 218
2. Discrimination In Logics as Sets of Formulas 222
3. Discrimination in Logics as Consequence Relations 228
References 237

A Topological Approach to Universal Logic: Model-Theoretical Abstract Logics (p. 35-36)

Steffen Lewitzka

Abstract. In this paper we present a model-theoretic approach to a general theory of logics. We de.ne a model-theoretical abstract logic as a structure consisting of a set of expressions, a class of interpretations and a satisfaction relation between interpretations (models) and expressions. The main idea is to use the observation that there exist in some sense pre-topological structures on the set of theories and on the class of interpretations. For example, if the logic has conjunction, then these structures turn out to be topological spaces. Properties of a given abstract logic now are reflected in topological properties of these spaces. One of the aim of this research is to investigate relationships between abstract logics. We introduce the concept of a logichomomorphism between abstract logics by means of topological terms. This leads in a natural way to the notion of a logic-isomorphism, a mapping that preserves all structural properties of a logic. We study in detail variations of logic-homomorphisms and their properties. One of the main results is that logic-homomorphisms with a special property satisfy a condition which has the same form as the satisfaction axiom of institutions. This fact can serve in future work to investigate possible connections between (classes of) modeltheoretical abstract logics and a resulting institution. At the end of the paper we sketch out this idea. Furthermore, we outline two examples of modeltheoretical abstract logics and the respective logic-homomorphisms. However, a systematic study of relevant logics as abstract logics, together with their logic-homomorphisms and further relationships, remains to be done in future work.

Mathematics Subject Classification (2000). Primary 03B22, Secondary 03G30, 03C95.
Keywords. Abstract logic, universal logic, general logic theory, topology, institutions.

1. Introduction
The term "abstract logic" perhaps could serve as a synonyme for notions as "general logic" or "universal logic". The aim is to develop a general theory that allows to investigate in an uniform way a big class of specific logics and the relationships between them. It seems that in some sense one could compare this attempt with the situation that we have in the established mathematical field that studies algebraic structures: Universal Algebra denotes the area in which concepts, methods and results of specific algebraic structures are generalized and unified. However, a precise de.nition of a notion of "logic" depends on what one considers as the central components (consequence relation, calculus, model-theoretic semantics, category-theoretic components etc.). So the nature of a general theory of logics depends on the choice between these di.erent alternatives.