Introduction to Logic
Chapter 1 Introduction ......................................................................................... 1 History ....................................................................................................... 2 Types .......................................................................................................... 3 Syllogistic Logic ......................................................................................... 3 Propositional Logic .................................................................................... 4 Predicate Logic .......................................................................................... 4 Modal Logic ............................................................................................... 4 Informal Reasoning and Dialectic .............................................................. 5 Mathematical Logic ................................................................................... 6 Philosophical Logic .................................................................................... 7 Computational Logic .................................................................................. 7 Non-classical Logic ................................................................................... 8 Logical Form .............................................................................................. 8 Semantics ................................................................................................... 9 Inference .................................................................................................. 10 Logical Systems ........................................................................................ 11 Logic and Rationality ............................................................................... 12 Rival Conceptions .................................................................................... 12 Controversies ........................................................................................... 13 “Is Logic Empirical?” .............................................................................. 13 Implication: Strict or Material .................................................................. 13 Tolerating the Impossible ......................................................................... 14 Rejection of Logical Truth ....................................................................... 14 vi Chapter 2 Perceptions of Logic ......................................................................... 16 Informal Logic ......................................................................................... 16 History ..................................................................................................... 16 Proposed Definitions ................................................................................ 18 Criticisms ................................................................................................. 19 Relation to Critical Thinking ..................................................................... 19 Relation to Argumentation Theory ........................................................... 20 Mathematical Logic ................................................................................. 20 Subfields and Scope ................................................................................. 21 History ..................................................................................................... 21 Formal Logical Systems ........................................................................... 25 Set Theory ............................................................................................... 28 Model Theory ........................................................................................... 29 Recursion Theory .................................................................................... 30 Proof Theory and Constructive Mathematics ......................................... 31 Applications .............................................................................................. 31 Connections with Computer Science ....................................................... 31 Foundations of Mathematics .................................................................... 32 Logical Form ............................................................................................ 33 History ..................................................................................................... 33 Example of Argument Form .................................................................... 34 Importance of Argument Form ................................................................ 34 Logical Form in Modern Logic ................................................................ 35 Semantics of Logic .................................................................................. 35 Formal System ......................................................................................... 37 Background .............................................................................................. 37 Variants .................................................................................................... 38 Logic and Rationality ............................................................................... 39 Rival Conceptions of Logic ...................................................................... 40 Rival Conceptions of Logic ...................................................................... 40 Informal and Colloquial Definitions ......................................................... 41 Chapter 3 History of Logic ................................................................................. 42 Logic in the West ..................................................................................... 43 Prehistory of Logic .................................................................................. 43 Ancient Greece before Aristotle .............................................................. 43 Aristotle .................................................................................................... 45 Stoics ........................................................................................................ 46 Medieval Logic ........................................................................................ 48 Logic in the Middle East .......................................................................... 48 Logic in Medieval Europe ........................................................................ 49 Traditional Logic ...................................................................................... 50 The Textbook Tradition ............................................................................ 50 vii Logic in Hegel’s Philosophy ..................................................................... 51 Logic and Psychology .............................................................................. 52 Rise of Modern Logic .............................................................................. 52 Modern Logic ........................................................................................... 53 Embryonic Period .................................................................................... 54 Algebraic Period ...................................................................................... 54 Logicist Period ......................................................................................... 56 Metamathematical Period ........................................................................ 58 Logic after WWII .................................................................................... 60 Logic in the East ...................................................................................... 61 Logic in India ........................................................................................... 61 Logic in China .......................................................................................... 63 Chapter 4 Types of Logic ................................................................................... 64 Term Logic ............................................................................................... 64 Aristotle’s System .................................................................................... 64 Basics ....................................................................................................... 65 Term ......................................................................................................... 66 Proposition ............................................................................................... 67 Singular Terms ......................................................................................... 67 Influence on Philosophy ........................................................................... 68 Decline of Term Logic ............................................................................. 68 Revival ..................................................................................................... 69 Propositional Calculus .............................................................................. 69 Explanation ............................................................................................... 69 History ..................................................................................................... 70 Terminology .............................................................................................. 71 Basic Concepts ........................................................................................ 72 Soundness and Completeness of the Rules ............................................. 75 Graphical Calculi ...................................................................................... 77 Other Logical Calculi ............................................................................... 77 Solvers ..................................................................................................... 78 First-order Logic ...................................................................................... 78 Syntax ...................................................................................................... 80 Deductive Systems .................................................................................. 83 Metalogical Properties ............................................................................. 85 Limitations ................................................................................................ 87 Automated Theorem Proving and Formal Methods ................................ 88 Modal Logic ............................................................................................. 89 Development of Modal Logic .................................................................. 90 Alethic Logic ........................................................................................... 91 Epistemic Logic ....................................................................................... 92 Temporal Logic ........................................................................................ 93 Doxastic Logic ......................................................................................... 94 Other Modal Logics ................................................................................. 95 The Ontology of Possibility ...................................................................... 95 Further Applications ................................................................................. 95 Controversies ........................................................................................... 96 Logic and Dialectic .................................................................................. 96 Defeasibility ............................................................................................. 96 Dialog Games ........................................................................................... 96 Logic in Computer Science ...................................................................... 96 Theoretical Foundations and Analysis ..................................................... 97 Computers to Assist Logicians ................................................................ 97 Logic Applications for Computers ........................................................... 97 Non-classical Logic ................................................................................. 99 Examples of Non-classical Logics ........................................................... 99 Classification of Non-classical Logics According to Specific Authors .... 99 Chapter 5 Critical Thinking and Informal Logic ............................................ 101 History ................................................................................................... 101 Definitions .............................................................................................. 101 Logic and Rationality ............................................................................. 102 Inductive versus Deductive Thinking .................................................... 103 Critical Thinking and Rationality ............................................................ 103 Functions ................................................................................................ 103 Procedure ............................................................................................... 104 Habits or Traits of Mind......................................................................... 104 Research ................................................................................................ 105 Education ............................................................................................... 105 Efficacy.................................................................................................. 106 Importance in Academia ........................................................................ 107 Critical Thinking in Computer-Mediated Communication ...................... 108 Philosophical Analysis ............................................................................ 109 Method of Analysis ................................................................................ 109 Controversy ............................................................................................. 110 Ambiguity ................................................................................................ 111 Linguistic Forms ..................................................................................... 111 Music ...................................................................................................... 114 Visual Art ................................................................................................ 114 Constructed Language ............................................................................ 115 Computer Science ................................................................................... 115 Argument ................................................................................................ 115 Formal and Informal ............................................................................... 116 Standard Types ....................................................................................... 116 viii Deductive ................................................................................................ 117 Inductive ................................................................................................. 119 Defeasible Arguments and Argumentation Schemes ............................. 119 By Analogy .............................................................................................120 Other Kinds .............................................................................................120 Fallacies and Non-arguments .................................................................122 Belief .......................................................................................................122 Knowledge and Epistemology .................................................................123 As a Psychological Phenomenon ............................................................123 Epistemological Belief Compared to Religious Belief ............................125 Formation ................................................................................................126 Justified True Belief ................................................................................127 Modification ............................................................................................128 Partial ......................................................................................................128 Prediction ................................................................................................128 Religion ...................................................................................................129 Systems ...................................................................................................134 Bias .........................................................................................................135 Cognitive Biases .....................................................................................136 Conflicts of Interest ................................................................................139 Statistical Biases .....................................................................................141 Contextual Biases ...................................................................................141 Prejudices ................................................................................................143 Credibility ................................................................................................144 Journalistic Credibility .............................................................................144 Academic Credibility ..............................................................................145 Scientific Credibility ................................................................................146 Medical Credibility ..................................................................................146 Street Credibility .....................................................................................147 Business Leadership Credibility ..............................................................148 Two-phase Model of Credibility ..............................................................148 Evidence .................................................................................................148 Burden of Proof ......................................................................................149 Evidence in Science ................................................................................150 Evidence in Law .....................................................................................150 Types of Evidence ..................................................................................151 Explanation ..............................................................................................159 Explanatory Power .................................................................................161 Fact .........................................................................................................162 Fallacy .....................................................................................................165 Enquiry ....................................................................................................171 Opinion ....................................................................................................173 ix Premise .................................................................................................. 175 Propaganda ............................................................................................ 175 Types ...................................................................................................... 177 Techniques ............................................................................................. 183 Models ................................................................................................... 183 Children .................................................................................................. 185 Prudence ................................................................................................ 186 As the “Mother” of all Virtues .............................................................. 186 Versus Imprudence, Cunning and False Prudence ................................ 187 Integral Parts ......................................................................................... 188 Prudential Judgement ............................................................................. 188 In Rhetoric ............................................................................................. 188 In Accounting ......................................................................................... 190 Reason ................................................................................................... 190 Philosophical History ............................................................................. 191 Reason Compared to Related Concepts ................................................ 195 Traditional Problems Raised Concerning Reason .................................. 199 Reason in Particular Fields of Study...................................................... 203 Relevance .............................................................................................. 206 Definition ................................................................................................ 206 Epistemology .......................................................................................... 207 Relevance Logic .................................................................................... 207 Application ............................................................................................. 207 Rhetoric .................................................................................................. 209 Uses ....................................................................................................... 209 History ................................................................................................... 215 Canons ................................................................................................... 219 Modern ................................................................................................... 226 French .................................................................................................... 232 Animal Rhetoric ..................................................................................... 235 Rigour ..................................................................................................... 236 Intellectual Rigour .................................................................................. 236 Mathematics .......................................................................................... 237 Vagueness .............................................................................................. 239 Importance ............................................................................................. 239 Approaches ............................................................................................ 240 Legal Principle ....................................................................................... 241 Chapter 6 Theories of Deduction .................................................................... 242 Deductive Reasoning ............................................................................. 242 Simple Example ..................................................................................... 242 Law of Detachment ............................................................................... 243 Law of Syllogism ................................................................................... 243 x Law of Contrapositive ............................................................................244 Validity and Soundness ...........................................................................244 Logicism ..................................................................................................245 Origin of the Name “Logicism” ..............................................................246 Intent, or Goal, of Logiism ......................................................................246 Epistemology Behind Logicism ...............................................................248 The Logistic Construction of the Natural Numbers _ ............................251 Neo-logicism ...........................................................................................256 Constructivism (Mathematics) ................................................................256 Constructive Mathematics ......................................................................257 The Place of Constructivism in Mathematics .........................................258 Mathematicians who have Made Major Contributions to Constructivism ........................................................................................259 Dialetheism .............................................................................................259 Motivations ..............................................................................................260 Advantages .............................................................................................261 Criticisms ................................................................................................261 Examples of True Contradictions that Dialetheists Accept ....................261 Fictionalism .............................................................................................262 Finitism ....................................................................................................262 History ....................................................................................................262 Main Idea ................................................................................................263 Classical Finitism vs. Strict Finitism ........................................................263 Views Regarding Infinite Mathematical Objects ....................................263 Other Related Philosophies of Mathematics ..........................................264 Formalism (Philosophy of Mathematics) ................................................264 Deductivism ............................................................................................265 Hilbert’s Formalism .................................................................................265 Axiomatic Systems .................................................................................267 Principia Mathematical ...........................................................................267 Criticisms of Formalism ..........................................................................267 Intuitionism ..............................................................................................268 Truth and Proof .......................................................................................268 Intuitionism and Infinity ...........................................................................269 History of Intuitionism.............................................................................270 Logical Atomism .....................................................................................271 Origin ......................................................................................................271 Principles .................................................................................................271 Influence and Decline .............................................................................274 Nominalism .............................................................................................274 The Problem of Universals .....................................................................275 Varieties of Nominalism ..........................................................................276 History ....................................................................................................278 xi Criticisms ............................................................................................... 279 Platonic Realism .................................................................................... 279 Universals .............................................................................................. 280 Particulars .............................................................................................. 281 Criticism ................................................................................................. 281 Pragmatism ............................................................................................ 282 Origins .................................................................................................... 283 Core Tenets ............................................................................................ 284 In other Fields of Philosophy ................................................................. 288 Analytical, Neo-classical, and Neo-pragmatism .................................... 293 Legacy and Contemporary Relevance .................................................. 294 Criticisms ............................................................................................... 296 Philosophical Realism ............................................................................ 296 History ................................................................................................... 297 Realism in Contemporary Analytic Philosophy ...................................... 298