LOGIC: lecture notes for philosophy, mathematics, and computer science:
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| Main Author: | |
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| Format: | Book |
| Language: | English |
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Cham, Switzerland
Springer
[2021]
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| Series: | Springer undergraduate texts in philosophy
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | x, 227 Seiten |
| ISBN: | 9783030648138 9783030648107 |
| ISSN: | 2569-8737 |
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Record in the Search Index
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| adam_text | Contents 1 Basic Notions................................................................................................. 1.1 What Is Logic?................................................................................... 1.2 Arguments and Their Formulation................................................... 1.3 Complex Reasoning............................................................................ 1.4 Truth and Falsity................................................................................ 1.5 Bivalence............................................................................................. 1 1 3 4 6 8 2 Validity........................................................................................................... 2.1 Some Set-Theoretical Notions........................................................... 2.2 True Premises...................................................................................... 2.3 Validity as Necessary Truth Preservation......................................... 2.4 Other Logical Properties and Relations............................................ 2.5 Important Facts About Validity.......................................................... 2.6 Validity Is Not Everything................................................................. 11 11 12 13 15 16 20 3 Formality....................................................................................................... 3.1 Formal Validity.................................................................................... 3.2 Formal
Invalidity................................................................................. 3.3 Formal Language................................................................................ 3.4 Formal System..................................................................................... 3.5 Object Language and Metalanguage............................... 3.6 Further Set-Theoretical Notions....................................................... 25 25 28 29 30 31 32 4 The Symbols of Propositional Logic.......................................................... 4.1 Sentence Letters.................................................................................. 4.2 Sentential Connectives........................................................................ 4.3 Brackets........................................ 4.4 Expressive Completeness................................................................... 4.5 Truth-Functionality and Substitutivity .............................................. 4.6 Formalization in a PropositionalLanguage....................................... 35 35 36 38 39 41 41 5 The Language L............................................................................................ 5.1 Formation Rules..................................................................... 45 45 vii
viii Contents Syntactic Trees.................................................................................... Scope.................................................................................................... Interpretation....................................................................................... Truth Tables ........................................................................................ 46 47 48 49 б Logical Consequence in L........................................................................... 6.1 Definition of Logical Consequence.........................-........................ 6.2 Other Logical Properties and Relations ........................................... 6.3 Important Facts About Logical Consequence................................... 6.4 Logical Consequence as a Test for Validity...................................... 6.5 Effective Computability..................................................................... 53 53 54 55 56 57 7 The System G,............................................................................................... 7.1 Derivation............................................................................................ 7.2 Rules for ......................................................................................... 7.3 Rules for D ......................................................................................... 7.4 Rules for л ......................................................................................... 7.5 Rules for v .................................. 61 61 62 64 65 68 8
Derivability in G.......................................................................................... 8.1 Derivability and Related Notions........................ 8.2 Important Facts About Derivability................................................. 8.3 Some Tips......................... 8.4 Derived Rules..................................................................................... 8.5 Other Natural Deduction Systems.................................................... 71 71 72 73 75 76 9 The System L................................................................................................. 9.1 Axioms and Inference Rule.............................................................. 9.2 Deduction Theorem........................................................................... 9.3 Explosion, Double Negation, Contraposition................................. 9.4 Substitution of Equivalents.............................................................. 9.5 Reductio Ad Absurdum..................................................................... 9.6 Deductive Equivalence Between G՜ and L.................................... 9.7 Systems and Theories........................................................................ 79 79 81 83 85 87 88 89 10 Consistency, Soundness, Completeness................................................... 10.1 Consistency of L........................................................................ 10.2 Definitions of Soundness and Completeness................................... 10.3 Soundness of
L.................................................................................. 10.4 Completeness of L............................................................................ 10.5 Extension to G՜................................................................................. 91 91 92 93 93 96 11 Quantification............................................................................................... 11.1 Quantified Sentences.......................................... 11.2 A Brief Historical Survey................................................................. 11.3 Existential Import............................................................................. 11.4 Multiple Generality.......................................................................... 11.5 Definite Descriptions......................................................................... 99 99 101 103 104 106 5.2 5.3 5.4 5.5
Contents ix 12 The Symbols of Predicate Logic.............................................................. 109 12.1 Non-logical Expressions.................................................................. 109 12.2 Logical Constants and Auxiliary Symbols..................................... 110 12.3 Other Symbols................................................................................... Ill 12.4 Numerical Expressions..................................................................... 113 12.5 Multiple Generality and Scope Ambiguity..................................... 114 12.6 Existence............................................................................................ 115 13 The Language Lq....................................................................................... 13.1 Syntax................................................................................................. 13.2 Basic Semantic Notions.................................................................... 13.3 Satisfaction........................................................................................ 13.4 Truth.................................................................................... 13.5 Logical Consequence....................................................................... 13.6 Undecidability................................................................................... 119 119 121 122 123 126 127 14 The System Q.............................................................................................. 14.1 Axioms and Inference
Rule.............................................................. 14.2 Derivability in Q.............................................................................. 14.3 Generalization Theorem.................................................................. 14.4 Validity and Derivability......................................... 14.5 Deduction Theorem and Other Syntactic Results........................... 14.6 Alphabetic Variants.......................................................................... 131 131 132 133 133 134 135 15 Consistency, Soundness, Completeness.................................................. 15.1 Consistency of Q............................................................................... 15.2 Soundness of Q................................................................ 15.3 Completeness of Q............................................................................ 15.4 Compactness Theorem..................................................................... 15.5 Final Remarks................................................................................... 139 139 140 141 143 144 16 Undecidability and Related Results........................................................ 16.1 Undecidability of Q ......................................................................... 16.2 Gödel Numbering............................................................................. 16.3 Effective Enumerability of the Theorems of Q.............................. 16.4 A Further
Corollary.......................................................................... 16.5 Recursive Axiomatization and Decidability................................... 147 147 148 149 149 150 17 First-Order Logic........................................................................................ 153 17.1 First-Order Languages and Systems................................................ 153 17.2 First-Order Logic with Identity........................................................ 154 17.3 First-Order Theory.................. ;........................................................ 155 17.4 The Language of BasicArithmetic................................................... 156 17.5 Peano Arithmetic............................................................................... 158 18 Theories and Models.................................................................................. 18.1 Cardinality......................................................................................... 18.2 Löwenheim-SkolemTheorems............................................ 161 161 162
Contents x 18.3 18.4 18.5 Isomorphism.................................................................................... 164 Isomorphic Models of a Theory...................................................... 166 Categoricity........................................................................................ 167 19 Gödel’s Incompleteness Theorems............................................................ 19.1 Overview................................................................ 19.2 The Arithmetization of Syntax ................. 19.3 The Gödel Sentence.......................................................................... 19.4 First Incompleteness Theorem: Semantic Version......................... 19.5 First Incompleteness Theorem: Syntactic Version......................... 19.6 Second Incompleteness Theorem..................................................... 171 171 172 174 175 176 178 20 Rudiments ofModal Logic........................................................................ 20.1 Modal Operators................................................................................ 20.2 A Modaf Propositional Language................................................... 20.3 The System К................................................................................... 20.4 The Systems T, B, S4, S5................................................................. 20.5 A Modal Predicate Language.... ........... 20.6 Systems of Modal Predicate Logic.................................................. 20.7 Soundness and
Completeness........................................................... 181 181 182 184 187 191 193 195 Solutions............................................ 199 Bibliography....................................... 221 Index....................................................................................................................... 225
Springer Undergraduate Texts in Philosophy Andrea Iacona LOGIC: Lecture Notes for Philosophy, Mathematics, and ComputerScience This textbook is a logic manual which includes an elementary course and an advanced course. It covers more than most introductory logic textbooks, while maintaining a comfortable pace that students can follow. The technical exposition is clear, precise and follows a paced increase in complexity, allowing the reader to get comfortable with previous definitions and procedures before facing more difficult material. The book also presents an interesting overall balance between formal and philosophical discussion, making it suitable for both philosophy and more formal/science oriented students. This textbook is of great use to undergraduate philosophy students, graduate philosophy students, logic teachers, undergraduates and graduates in mathematics, computer science or related fields in which logic is required.
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| adam_txt |
Contents 1 Basic Notions. 1.1 What Is Logic?. 1.2 Arguments and Their Formulation. 1.3 Complex Reasoning. 1.4 Truth and Falsity. 1.5 Bivalence. 1 1 3 4 6 8 2 Validity. 2.1 Some Set-Theoretical Notions. 2.2 True Premises. 2.3 Validity as Necessary Truth Preservation. 2.4 Other Logical Properties and Relations. 2.5 Important Facts About Validity. 2.6 Validity Is Not Everything. 11 11 12 13 15 16 20 3 Formality. 3.1 Formal Validity. 3.2 Formal
Invalidity. 3.3 Formal Language. 3.4 Formal System. 3.5 Object Language and Metalanguage. 3.6 Further Set-Theoretical Notions. 25 25 28 29 30 31 32 4 The Symbols of Propositional Logic. 4.1 Sentence Letters. 4.2 Sentential Connectives. 4.3 Brackets. 4.4 Expressive Completeness. 4.5 Truth-Functionality and Substitutivity . 4.6 Formalization in a PropositionalLanguage. 35 35 36 38 39 41 41 5 The Language L. 5.1 Formation Rules. 45 45 vii
viii Contents Syntactic Trees. Scope. Interpretation. Truth Tables . 46 47 48 49 б Logical Consequence in L. 6.1 Definition of Logical Consequence.-. 6.2 Other Logical Properties and Relations . 6.3 Important Facts About Logical Consequence. 6.4 Logical Consequence as a Test for Validity. 6.5 Effective Computability. 53 53 54 55 56 57 7 The System G,. 7.1 Derivation. 7.2 Rules for . 7.3 Rules for D . 7.4 Rules for л . 7.5 Rules for v . 61 61 62 64 65 68 8
Derivability in G. 8.1 Derivability and Related Notions. 8.2 Important Facts About Derivability. 8.3 Some Tips. 8.4 Derived Rules. 8.5 Other Natural Deduction Systems. 71 71 72 73 75 76 9 The System L. 9.1 Axioms and Inference Rule. 9.2 Deduction Theorem. 9.3 Explosion, Double Negation, Contraposition. 9.4 Substitution of Equivalents. 9.5 Reductio Ad Absurdum. 9.6 Deductive Equivalence Between G՜ and L. 9.7 Systems and Theories. 79 79 81 83 85 87 88 89 10 Consistency, Soundness, Completeness. 10.1 Consistency of L. 10.2 Definitions of Soundness and Completeness. 10.3 Soundness of
L. 10.4 Completeness of L. 10.5 Extension to G՜. 91 91 92 93 93 96 11 Quantification. 11.1 Quantified Sentences. 11.2 A Brief Historical Survey. 11.3 Existential Import. 11.4 Multiple Generality. 11.5 Definite Descriptions. 99 99 101 103 104 106 5.2 5.3 5.4 5.5
Contents ix 12 The Symbols of Predicate Logic. 109 12.1 Non-logical Expressions. 109 12.2 Logical Constants and Auxiliary Symbols. 110 12.3 Other Symbols. Ill 12.4 Numerical Expressions. 113 12.5 Multiple Generality and Scope Ambiguity. 114 12.6 Existence. 115 13 The Language Lq. 13.1 Syntax. 13.2 Basic Semantic Notions. 13.3 Satisfaction. 13.4 Truth. 13.5 Logical Consequence. 13.6 Undecidability. 119 119 121 122 123 126 127 14 The System Q. 14.1 Axioms and Inference
Rule. 14.2 Derivability in Q. 14.3 Generalization Theorem. 14.4 Validity and Derivability. 14.5 Deduction Theorem and Other Syntactic Results. 14.6 Alphabetic Variants. 131 131 132 133 133 134 135 15 Consistency, Soundness, Completeness. 15.1 Consistency of Q. 15.2 Soundness of Q. 15.3 Completeness of Q. 15.4 Compactness Theorem. 15.5 Final Remarks. 139 139 140 141 143 144 16 Undecidability and Related Results. 16.1 Undecidability of Q . 16.2 Gödel Numbering. 16.3 Effective Enumerability of the Theorems of Q. 16.4 A Further
Corollary. 16.5 Recursive Axiomatization and Decidability. 147 147 148 149 149 150 17 First-Order Logic. 153 17.1 First-Order Languages and Systems. 153 17.2 First-Order Logic with Identity. 154 17.3 First-Order Theory. ;. 155 17.4 The Language of BasicArithmetic. 156 17.5 Peano Arithmetic. 158 18 Theories and Models. 18.1 Cardinality. 18.2 Löwenheim-SkolemTheorems. 161 161 162
Contents x 18.3 18.4 18.5 Isomorphism. 164 Isomorphic Models of a Theory. 166 Categoricity. 167 19 Gödel’s Incompleteness Theorems. 19.1 Overview. 19.2 The Arithmetization of Syntax . 19.3 The Gödel Sentence. 19.4 First Incompleteness Theorem: Semantic Version. 19.5 First Incompleteness Theorem: Syntactic Version. 19.6 Second Incompleteness Theorem. 171 171 172 174 175 176 178 20 Rudiments ofModal Logic. 20.1 Modal Operators. 20.2 A Modaf Propositional Language. 20.3 The System К. 20.4 The Systems T, B, S4, S5. 20.5 A Modal Predicate Language. . 20.6 Systems of Modal Predicate Logic. 20.7 Soundness and
Completeness. 181 181 182 184 187 191 193 195 Solutions. 199 Bibliography. 221 Index. 225
Springer Undergraduate Texts in Philosophy Andrea Iacona LOGIC: Lecture Notes for Philosophy, Mathematics, and ComputerScience This textbook is a logic manual which includes an elementary course and an advanced course. It covers more than most introductory logic textbooks, while maintaining a comfortable pace that students can follow. The technical exposition is clear, precise and follows a paced increase in complexity, allowing the reader to get comfortable with previous definitions and procedures before facing more difficult material. The book also presents an interesting overall balance between formal and philosophical discussion, making it suitable for both philosophy and more formal/science oriented students. This textbook is of great use to undergraduate philosophy students, graduate philosophy students, logic teachers, undergraduates and graduates in mathematics, computer science or related fields in which logic is required. |
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| spelling | Iacona, Andrea Verfasser (DE-588)1204290180 aut LOGIC: lecture notes for philosophy, mathematics, and computer science Andrea Iacona Cham, Switzerland Springer [2021] x, 227 Seiten txt rdacontent n rdamedia nc rdacarrier Springer undergraduate texts in philosophy 2569-8737 Philosophy, general Philosophy of Language Philosophy Language and languages—Philosophy Logik (DE-588)4036202-4 gnd rswk-swf Logik (DE-588)4036202-4 s DE-604 Erscheint auch als Online-Ausgabe 978-3-030-64811-4 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=033767703&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=033767703&sequence=000003&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
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| title | LOGIC: lecture notes for philosophy, mathematics, and computer science |
| title_auth | LOGIC: lecture notes for philosophy, mathematics, and computer science |
| title_exact_search | LOGIC: lecture notes for philosophy, mathematics, and computer science |
| title_exact_search_txtP | LOGIC: lecture notes for philosophy, mathematics, and computer science |
| title_full | LOGIC: lecture notes for philosophy, mathematics, and computer science Andrea Iacona |
| title_fullStr | LOGIC: lecture notes for philosophy, mathematics, and computer science Andrea Iacona |
| title_full_unstemmed | LOGIC: lecture notes for philosophy, mathematics, and computer science Andrea Iacona |
| title_short | LOGIC: lecture notes for philosophy, mathematics, and computer science |
| title_sort | logic lecture notes for philosophy mathematics and computer science |
| topic | Philosophy, general Philosophy of Language Philosophy Language and languages—Philosophy Logik (DE-588)4036202-4 gnd |
| topic_facet | Philosophy, general Philosophy of Language Philosophy Language and languages—Philosophy Logik |
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