Trends in Logic

By: Francesca Poggiolesi

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Hardcover | 2 December 2010

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Hardcover
236 Pages

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23.5 x 15.88 x 1.91

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The book is about Gentzen calculi for (the main systems of) modal logic. It is divided into three parts. In the first part we introduce and discuss the main philosophical ideas related to proof theory, and we try to identify criteria for distinguishing good sequent calculi. In the second part we present the several attempts made from the 50's until today to provide modal logic with Gentzen calculi. In the third and and final part we analyse new calculi for modal logics, called tree-hypersequent calculi, which were recently introduced by the author. We show in a precise and clear way the main results that can be proved with and about them.

Industry Reviews

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"This book is mainly devoted to an extensive presentation of the author's original formalisation of modal logics called tree-hypersequent calculus but other approaches are also discussed at length. ... This book is a valuable contribution to the subject of proof techniques for modal logics. The presentation is clear with a detailed but readable account of technical details." (Andrzej B. Indrzejczak, Mathematical Reviews, August, 2013)

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Non-Fiction Mathematics Calculus & Mathematical Analysis Calculus Philosophy Philosophy & Logic Mathematical Foundation Mathematical Logic Computing & I.T.Computer Science Artificial Intelligence Computer Vision

Language & Linguistics Linguistics

An Overview of the Sequent Calculus
What Is a Good Sequent Calculus?	p. 3
The Sequent Calculus Gc1	p. 3
Formal Remarks	p. 7
Philosophical Remarks	p. 12
Analyticity	p. 12
Logicality	p. 14
From Logicality to Inferentialism	p. 16
Harmony	p. 18
Inferentialism	p. 20
Concluding Remarks	p. 21
Subformula Property	p. 22
Admissibility of the Structural Rules	p. 24
Operational vs Global Meaning	p. 26
Admissibility of the Logical Rules	p. 28
Explicitness, Separation and Symmetry	p. 28
Uniqueness	p. 29
Syntactic Purity	p. 29
Dosen's Principle Redefined	p. 31
Modularity	p. 34
Sequent Calculi for Modal Logic
Modal Logic and Ordinary Sequent Calculi	p. 39
Normal Modal Logic	p. 39
Ordinary Sequent Calculi for Modal Logic	p. 45
The Idea of Generalising the Gentzen Calculus	p. 51
Purely Syntactic Methods	p. 55
Multiple Sequent Calculi	p. 56
Higher-Arity Sequent Calculi	p. 62
Display Sequent Calculi	p. 66
Semantic Methods	p. 75
Semantic Modal Sequent Calculi	p. 76
Indexed Sequent Calculi	p. 84
Internalised Forcing (Relation) Sequent Calculi	p. 92
Comparing the Different Generalisations of the Sequent Calculus	p. 101
From Multiple Sequent Calculi to Display Sequent Calculi	p. 101
From Higher-Arity Sequent Calculi to Display Sequent Calculi	p. 104
From Indexed Sequent Calculi to Internalised Forcing Sequent Calculi	p. 107
From Indexed Sequent Calculi to Semantic Modal Sequent Calculi and Vice Versa	p. 112
From Display Sequent Calculi to Internalised Forcing Sequent Calculi	p. 113
Tree-Hypersequent Calculi
On the Tree-Hypersequent Calculi	p. 119
The Calculi Thsk*	p. 122
Logical Variant of the Tree-Hypersequent Calculi	p. 126
Adequacy of the Tree-Hypersequent Calculi	p. 138
Syntactic Cut-Admissibility and Decidability	p. 143
Cut-Admissibility in the Tree-Hypersequent Calculi	p. 143
Decidability of the Tree-Hypersequent Calculi	p. 156
Semantic Adequacy	p. 165
Semantic Validity of the Tree-Hypersequent Calculi	p. 165
Semantic Completeness of the Tree-Hypersequent Calculi	p. 170
A Hypersequent Calculus for the System S5	p. 175
The Calculus ThS5_L	p. 176
Admissibility of the Structural Rules in ThS5_L	p. 177
Adequacy of ThS5_L	p. 180
Cut-Admissibility in ThS5_L	p. 181
Decidability of ThS5_L	p. 183
A Tree-Hypersequent Calculus for the Modal Logic of Provability	p. 187
The Calculus Thsgl_L	p. 187
Admissibility of the Structural Rules in Thsgl_L	p. 188
Adequacy of Thsgl_L	p. 193
Cut-Admissibility in Thsgl_L	p. 196
Further Results on Tree-Hypersequent Calculi	p. 203
Tree-Hypersequent Calculi and Other Calculi	p. 203
Tree-Hypersequent Calculi and Other Logics	p. 206
Tree-Hypersequent Calculi and Further Developments	p. 207
References	p. 209
Symbols and Notations	p. 215
Index	p. 219
Table of Contents provided by Ingram. All Rights Reserved.

Trends in Logic

At a Glance

Hardcover

Industry Reviews

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