Book contents
- Frontmatter
- Contents
- List of Figures
- List of Tables
- Abbreviations
- Symbols
- Acknowledgements
- 1 Introduction
- I Logical Preliminaries - Hybrid Logics, Decidability, Deductive Systems
- 2 Modal logic, decidability and complexity
- 3 Deductive systems
- 4 Hybrid logic
- 5 Logic M(En)
- 6 Remarks on description logics contributions
- II Deductive Systems for Hybrid Logics
- Bibliography
- Index
3 - Deductive systems
from I - Logical Preliminaries - Hybrid Logics, Decidability, Deductive Systems
Published online by Cambridge University Press: 05 January 2015
- Frontmatter
- Contents
- List of Figures
- List of Tables
- Abbreviations
- Symbols
- Acknowledgements
- 1 Introduction
- I Logical Preliminaries - Hybrid Logics, Decidability, Deductive Systems
- 2 Modal logic, decidability and complexity
- 3 Deductive systems
- 4 Hybrid logic
- 5 Logic M(En)
- 6 Remarks on description logics contributions
- II Deductive Systems for Hybrid Logics
- Bibliography
- Index
Summary
Since a major part of the book is devoted to concrete deductive systems for hybrid logics, providing a formal definition of a deductive system is in place. We use this notion in a similar way to Indrzejczak in [53], therefore, we repeat his very general definition of a deductive system.
Definition 3.1 (Deductive system ([53])). Every deductive system may be characterized on two elementary levels of description:
· the calculus,
· the realization.
By a calculus we understand a non-empty set of schemata of rules of the form:
X1, …,Xn/Y1, …,Ym, n ≥ 0, m ≥ 1,
with a possible list of side conditions. Symbols Xi, denote some data structures being transformed into data structures Yi.
By a realization we usually understand the set of instructions of how to apply rules to perform a derivation/construct a proof.
In the book, we describe instantiations of two basic types of deductive systems: tableau calculi and sequent calculi. We dedicate to each of these types a short, introductory note. We skip all historical remarks as hardly relevant for our considerations. We also waive the idea of a profound exposition of Hilbert-style calculi, even though several such calculi are presented in this book. We assume that the notion of Hilbert-style calculus is rather familiar to most readers and we consider its thorough examination as lying outside the scope of the book. Considerations in this section are based on [53, 52].
SEQUENT CALCULI
Originally invented by Gentzen in the 1930s, sequent calculi are finite sets of primitive rules of the form:
S1, …,Sn/Sn + 1,
where Si is a sequent schema.
By a sequent we understand an ordered pair of finite sequences of formulas with a symbol ⇒ between these sequences. Every sequent is therefore of the form:
φ1, …,φn ⇒ ψ1, …,ψmn, m ≥ 0.
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- Information
- Publisher: Jagiellonian University PressPrint publication year: 2014