Proof theory and algebra in logic:
Saved in:
| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
Singapore, Singapore
Springer
[2019]
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| Series: | Short textbooks in logic
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | viii, 160 Seiten Diagramme |
| ISBN: | 9789811379963 |
Staff View
MARC
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Record in the Search Index
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| adam_text | Contents Part I Proof Theory 1 Sequent Systems......................................................................................... 1.1 Prologue............................................................................................. 1.2 Sequent Systems LK for Classical Logic...................................... 1.3 Completeness and Cut Elimination................................................. 1.4 Sequent System LJ for Intuitionistic Logic.................................... 3 3 7 17 19 2 Cut Elimination for Sequent Systems.................................................... 2.1 Basic Idea......................................................................................... 2.2 Cut Elimination................................................................................ 2.3 Subformula Property......................................................................... 23 23 26 32 3 Proof-Theoretic Analysis of Logical Properties.................................... 3.1 Decidability of Intuitionistic Logic.................................................. 3.2 Disjunction Property........................................................................ 3.3 Craig’s Interpolation Property......................................................... 3.4 Glivenko’s Theorem . . .................................................................... 35 35 38 39 46 4 Modal and Substructural Logics............................................................. 4.1 Standard Sequent Systems for Normal Modal Logics................. 4.2 Roles of Structural
Rules.................................................................. 4.3 Sequent Systems for Basic Substructural Logics........................... 47 48 52 54 5 Deducibility and Axiomatic Extensions.................................................. 5.1 Deducibility and Deduction Theorem............................................. 5.2 Local Deduction Theorems.............................................................. 5.3 Axiomatic Extensions.......................................... 5.4 Framework for Substructural Logics and Modal Logics............... 5.5 A View of Substructural Logics....................................................... 61 61 64 66 69 71 vii
Contents viii Part II Algebra in Logic 6 From Algebra to Logic............................................................................. 6.1 Lattices and Boolean Algebras......................................................... 6.2 Subalgebras, Homomorphisme and Direct Products...................... 6.3 Representations of Boolean Algebras.............................................. 6.4 Algebraic Completeness of Classical Logic.................................. 6.5 Many-Valued Chains and the Law of Residuation...................... 77 77 83 86 88 89 7 Basics of Algebraic Logic........................ 7.1 Heyting Algebras.............................................................................. 7.2 Lindenbaum-Tarski Algebras........................................................... 7.3 Locally Finite Algebras..................................................................... 7.4 Finite Embeddability Property and Finite ModelProperty............ 7.5 Canonical Extensions of Heyting Algebras.................................... 97 97 99 102 105 107 8 Logics and Varieties.................................................................................... 8.1 Lattice Structure of Superintuitionistic Logics............................... 8.2 The Variety HA of All Heyting Algebras...................................... 8.3 Subvarieties of HA and Superintuitionistic Logics....................... 8.4 Subdirect Representation Theorem.................................................. 8.5 Algebraic Aspects of Logical Properties......................................... 113 113
116 119 123 125 9 Residuated Structures................................................................................. 9.1 Residuated Lattices and FL֊Algebras............................................. 9.2 FL-Algebras and Substructural Logics........................................... 9.3 Residuations Over the Unit Interval................................................ 129 129 133 135 10 Modal Algebras............................................................................................. 10.1 Modal Algebras................................... 10.2 Canonical Extensions and Jónsson-Tarski Theorem...................... 10.3 Kripke Semantics from Algebraic Viewpoint............................... 10.4 Gödel Translation.............................................................................. 139 139 142 143 147 References............................................................................................................. Further Reading for Part 1............................................................................ Further Reading for Part Π......................................................................... 151 155 156 Index 157
Short Texthooks in Logic. Proof Theory and Algebra in Logic This book offers a concise introduction to both proof-theory and algebraic metho the core of the syntactic and semantic study of logic respectively. The importanc combining these two has been increasingly recognized in recent years. It highlig contrasts between the deep, concrete results using the former and the general, a ones using the latter. Covering modal logics, many-valued logics, superintuitionis and substructural logics, together with their algebraic semantics, the book also p an introduction to nonclassical logic for undergraduate or graduate level courses The book is divided into two parts: Proof Theory in Part I and Algebra in Logic in II. Part I presents sequent systems and discusses cut elimination and its applica in detail. It also provides simplified proof of cut elimination, making the topic more accessible. The last chapter of Part I is devoted to clarification of the classes of l that are discussed in the second part. Part II focuses on algebraic semantics for logics. At the same time, it is a gentle introduction to the basics of algebraic logic universal algebra with many examples of their applications in logic. Part II can be independently of Part I, with only minimum knowledge required, and as such is s able as a textbook for short introductory courses on algebra in logic.
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| any_adam_object | 1 |
| author | Ono, Hiroakira |
| author_GND | (DE-588)1159674604 |
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| building | Verbundindex |
| bvnumber | BV046155275 |
| classification_rvk | CC 2600 SK 130 |
| ctrlnum | (OCoLC)1123208422 (DE-599)BVBBV046155275 |
| dewey-full | 160 |
| dewey-hundreds | 100 - Philosophy & psychology |
| dewey-ones | 160 - Philosophical logic |
| dewey-raw | 160 |
| dewey-search | 160 |
| dewey-sort | 3160 |
| dewey-tens | 160 - Philosophical logic |
| discipline | Mathematik Philosophie |
| format | Book |
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| indexdate | 2024-07-10T08:36:46Z |
| institution | BVB |
| isbn | 9789811379963 |
| language | English |
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| spelling | Ono, Hiroakira (DE-588)1159674604 aut Proof theory and algebra in logic Hiroakira Ono Singapore, Singapore Springer [2019] © 2019 viii, 160 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Short textbooks in logic Logic Mathematical Logic and Formal Languages Mathematical Logic and Foundations Order, Lattices, Ordered Algebraic Structures Computer science Logic, Symbolic and mathematical Algebra Algebra (DE-588)4001156-2 gnd rswk-swf Logik (DE-588)4036202-4 gnd rswk-swf Beweistheorie (DE-588)4145177-6 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Logik (DE-588)4036202-4 s Beweistheorie (DE-588)4145177-6 s Algebra (DE-588)4001156-2 s DE-604 Erscheint auch als Online-Ausgabe 978-981-13-7997-0 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=031535303&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=031535303&sequence=000003&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Ono, Hiroakira Proof theory and algebra in logic Logic Mathematical Logic and Formal Languages Mathematical Logic and Foundations Order, Lattices, Ordered Algebraic Structures Computer science Logic, Symbolic and mathematical Algebra Algebra (DE-588)4001156-2 gnd Logik (DE-588)4036202-4 gnd Beweistheorie (DE-588)4145177-6 gnd |
| subject_GND | (DE-588)4001156-2 (DE-588)4036202-4 (DE-588)4145177-6 (DE-588)4123623-3 |
| title | Proof theory and algebra in logic |
| title_auth | Proof theory and algebra in logic |
| title_exact_search | Proof theory and algebra in logic |
| title_full | Proof theory and algebra in logic Hiroakira Ono |
| title_fullStr | Proof theory and algebra in logic Hiroakira Ono |
| title_full_unstemmed | Proof theory and algebra in logic Hiroakira Ono |
| title_short | Proof theory and algebra in logic |
| title_sort | proof theory and algebra in logic |
| topic | Logic Mathematical Logic and Formal Languages Mathematical Logic and Foundations Order, Lattices, Ordered Algebraic Structures Computer science Logic, Symbolic and mathematical Algebra Algebra (DE-588)4001156-2 gnd Logik (DE-588)4036202-4 gnd Beweistheorie (DE-588)4145177-6 gnd |
| topic_facet | Logic Mathematical Logic and Formal Languages Mathematical Logic and Foundations Order, Lattices, Ordered Algebraic Structures Computer science Logic, Symbolic and mathematical Algebra Logik Beweistheorie Lehrbuch |
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