Logic for dummies

Logic concepts are more mainstream than you may realize. There’s logic every place you look and in almost everything you do, from deciding which shirt to buy to asking your boss for a raise, and even to watching television, where themes of such shows as CSI and Numbers incorporate a variety of logistical studies. Logic For Dummies explains a vast array of logical concepts and processes in easy-to-understand language that make everything clear to you, whether you’re a college student of a student of life. You’ll find out about:Formal LogicSyllogismsConstructing proofs and refutationsPropositional and predicate logicModal and fuzzy logicSymbolic logicDeductive and inductive reasoningLogic For Dummies tracks an introductory logic course at the college level. Concrete, real-world examples help you understand each concept you encounter, while fully worked out proofs and fun logic problems encourage you students to apply what you’ve learned.From the Back CoverFeatures real-world examples and worked-out proofsClarify your thinking and apply logic to everyday lifeLooking to learn logic, but feel lost? Relax! This friendly guide explains logic concepts in plain English, from proofs, predicate logic, and paradox to symbolic logic, semantic structures, and syllogisms. Step-by-step examples show you how to build and prove logical arguments and put equivalence rules to work. You even get tips on passing logic exams!Discover how toGain a logical perspectiveEvaluate statements with truth tablesConstruct proofs and refutationsProve arguments with quantifier logicMake logical conclusionsGrasp quantum and fuzzy logicAbout the AuthorMark Zegarelli is a professional writer with degrees in both English and Math from Rutgers University. He has earned his living for many years writing vast quantities of logic puzzles, a hefty chunk of software documentation, and the occasional book or film review. Along the way, he’s also paid a few bills doing housecleaning, decorative painting, and (for ten hours) retail sales. He likes writing best, though. About the Author......Page 4 Author’s Acknowledgments......Page 6 Contents at a Glance......Page 8 Table of Contents......Page 10 About This Book......Page 20 Conventions Used in This Book......Page 21 How This Book Is Organized......Page 22 Part III: Proofs, Syntax, and Semantics in SL......Page 23 Part VI: The Part of Tens......Page 24 Where to Go from Here......Page 25 Overview of Logic......Page 26 Getting a Logical Perspective......Page 28 Understanding cause and effect......Page 29 Existence itself......Page 31 Generating premises......Page 32 Forming a conclusion......Page 33 Making Logical Conclusions Simple with the Laws of Thought......Page 34 The law of non-contradiction......Page 35 Math is good for understanding logic......Page 36 Logic is good for understanding math......Page 37 Logical Developments from Aristotle to the Computer......Page 38 Aristotle invents syllogistic logic......Page 39 Euclid’s axioms and theorems......Page 42 Logic takes a vacation......Page 43 Leibniz and the Renaissance......Page 44 Working up to formal logic......Page 45 Logic in the 20th Century and Beyond......Page 48 Gödel’s proof......Page 49 The age of computers......Page 50 Searching for the final frontier......Page 51 Defining Logic......Page 52 Examining argument structure......Page 53 Looking for validation......Page 55 Ice cream Sunday......Page 56 Escape from New York......Page 57 What Logic Isn’t......Page 58 Thinking versus logic......Page 59 Reality — what a concept!......Page 60 The sound of soundness......Page 61 Deduction and induction......Page 62 Rhetorical questions......Page 63 Pick a number (math)......Page 65 Switch on or off (computer science)......Page 66 Find the meaning of life (philosophy)......Page 67 Formal Sentential Logic (SL)......Page 68 Observing the Formalities of Sentential Logic......Page 70 Statement variables......Page 71 The Five SL Operators......Page 72 Feeling negative......Page 73 Displaying a show of ands......Page 74 Digging for or......Page 76 Getting iffy......Page 78 Getting even iffier......Page 80 The ins and outs of values......Page 82 There’s no substitute for substitution......Page 83 Lost in Translation......Page 84 The easy way — translating from SL to English......Page 85 The not-so-easy way — translating from English to SL......Page 87 The Value of Evaluation......Page 92 Value Is the Bottom Line......Page 93 Getting started with SL evaluation......Page 94 Stacking up another method......Page 95 Making a Statement......Page 96 Identifying sub-statements......Page 97 Scoping out a statement......Page 98 The main attraction: Finding main operators......Page 99 Eight Forms of SL Statements......Page 101 Evaluation Revisited......Page 102 Turning the Tables: Evaluating Statements with Truth Tables......Page 104 Putting It All on the Table: The Joy of Brute Force......Page 105 Setting up a truth table......Page 106 Filling in a truth table......Page 108 Reading a truth table......Page 111 Taking on tautologies and contradictions......Page 112 Judging semantic equivalence......Page 113 Staying consistent......Page 115 Arguing with validity......Page 117 Putting the Pieces Together......Page 119 Connecting tautologies and contradictions......Page 120 Linking semantic equivalence with tautology......Page 121 Linking inconsistency with contradiction......Page 122 Linking validity with contradiction......Page 124 Taking the Easy Way Out: Creating Quick Tables......Page 126 Dumping the Truth Table for a New Friend: The Quick Table......Page 127 Outlining the Quick Table Process......Page 128 Filling in a quick table......Page 129 Reading a quick table......Page 130 Disproving the assumption......Page 131 Tautology......Page 132 Semantic equivalence and inequivalence......Page 133 Validity and invalidity......Page 134 Working Smarter (Not Harder) with Quick Tables......Page 135 Recognizing the six easiest types of statements to work with......Page 136 Working with the four not-so-easy statement types......Page 138 Coping with the six difficult statement types......Page 141 Understanding How Truth Trees Work......Page 144 Decomposing SL statements......Page 145 Showing Consistency or Inconsistency......Page 147 Testing for Validity or Invalidity......Page 150 Tautologies......Page 153 Contradictions......Page 156 Checking for Semantic Equivalence or Inequivalence......Page 159 Proofs, Syntax, and Semantics in SL......Page 164 What Have You Got to Prove?......Page 166 Bridging the Premise-Conclusion Divide......Page 167 Using Eight Implication Rules in SL......Page 168 The......Page 169 The & rules: Conjunction and Simplification......Page 172 The......Page 174 Rules: Hypothetical Syllogism and Constructive Dilemma......Page 177 Equal Opportunities: Putting Equivalence Rules to Work......Page 180 Applying equivalences to part of the whole......Page 181 Double Negation (DN)......Page 182 Contraposition (Contra)......Page 183 Implication (Impl)......Page 184 Exportation (Exp)......Page 185 Commutation (Comm)......Page 186 Association (Assoc)......Page 187 Distribution (Dist)......Page 188 DeMorgan’s Theorem (DeM)......Page 189 Equivalence (Equiv)......Page 191 Big Assumptions with Conditional and Indirect Proofs......Page 194 Conditioning Your Premises with Conditional Proof......Page 195 Understanding conditional proof......Page 196 Tweaking the conclusion......Page 197 Stacking assumptions......Page 199 Thinking Indirectly: Proving Arguments with Indirect Proof......Page 200 Introducing indirect proof......Page 201 Proving short conclusions......Page 202 Combining Conditional and Indirect Proofs......Page 203 Putting It All Together: Strategic Moves to Polish Off Any Proof......Page 206 Look at the problem......Page 207 Jot down the easy stuff......Page 208 Know when to move on......Page 209 The three friendly forms: x......Page 210 The two slightly-less-friendly forms: x......Page 212 The three unfriendly forms: x & y, ~(x......Page 213 Choose carefully between direct and indirect proof......Page 214 Work backwards from the conclusion......Page 215 Go deeper into SL statements......Page 217 Break down long premises......Page 221 Make a shrewd assumption......Page 223 One for All and All for One......Page 224 Making Do with the Five SL Operators......Page 225 The tyranny of power......Page 227 The horns of dilemma......Page 228 The (Sheffer’s) stroke of genius......Page 229 The moral of the story......Page 231 Syntactical Maneuvers and Semantic Considerations......Page 232 Are You WFF Us or Against Us?......Page 233 Understanding WFFs (with a few strings attached)......Page 234 Separating WFFs from non-WFFs......Page 235 Comparing SL to Boolean Algebra......Page 236 Reading the signs......Page 237 Doing the math......Page 239 Exploring syntax and semantics in Boolean algebra......Page 240 Quantifier Logic (QL)......Page 242 Expressing Quantity with Quality: Introducing Quantifier Logic......Page 244 Using individual constants and property constants......Page 245 Incorporating the SL operators......Page 248 Understanding individual variables......Page 249 Understanding the universal quantifier......Page 250 Expressing existence......Page 251 Creating context with the domain of discourse......Page 252 Picking out Statements and Statement Forms......Page 254 Discovering bound variables and free variables......Page 255 Knowing the difference between statements and statement forms......Page 256 Translating the Four Basic Forms of Categorical Statements......Page 258 Discovering Alternative Translations of Basic Forms......Page 263 Translating “some” with......Page 264 Translating “no” with......Page 265 Recognizing “all” statements......Page 266 Recognizing “not all” statements......Page 267 Recognizing “no” statements......Page 268 Proving Arguments with QL......Page 270 Comparing similar SL and QL statements......Page 271 Transferring the eight implication rules from SL into QL......Page 272 Employing the ten SL equivalence rules in QL......Page 274 Introducing QN......Page 275 Using QN in proofs......Page 276 Easy rule #1: Universal Instantiation (UI)......Page 279 Easy rule #2: Existential Generalization (EG)......Page 281 Not-so-easy rule #1: Existential Instantiation (EI)......Page 284 Not-so-easy rule #2: Universal Generalization (UG)......Page 289 Good Relations and Positive Identities......Page 294 Defining and using relations......Page 295 Connecting relational expressions......Page 296 Making use of quantifiers with relations......Page 297 Working with multiple quantifiers......Page 298 Writing proofs with relations......Page 299 Identifying with Identities......Page 302 Understanding identities......Page 303 Writing proofs with identities......Page 304 Using the decomposition rules from SL......Page 306 Adding UI, EI, and QN......Page 308 Using UI more than once......Page 310 Non-Terminating Trees......Page 314 Modern Developments in Logic......Page 318 Computer Logic......Page 320 Turing and his UTM......Page 321 The Modern Age of Computers......Page 323 Hardware and logic gates......Page 324 Software and computer languages......Page 326 Sporting Propositions: Non-Classical Logic......Page 328 Three-valued logic......Page 329 Multi-valued logic......Page 330 Fuzzy logic......Page 332 Getting into a New Modality......Page 334 Taking Logic to a Higher Order......Page 336 Moving Beyond Consistency......Page 337 Introducing quantum logic......Page 339 Playing the shell game......Page 340 Grounding Logic in Set Theory......Page 342 Setting things up......Page 343 Trouble in paradox: Recognizing the problem with set theory......Page 344 Developing a solution in the Principia Mathematica......Page 345 Discovering the Axiomatic System for SL......Page 346 Proving Consistency and Completeness......Page 347 Formalizing logic and mathematics with the Hilbert Program......Page 348 Gödel’s Incompleteness Theorem......Page 349 How he did it......Page 350 Pondering the Meaning of It All......Page 351 The Part of Tens......Page 352 Ten Quotes about Logic......Page 354 Gottfried Leibniz (1646–1716)......Page 356 Georg Cantor (1845–1918)......Page 357 David Hilbert (1862–1943)......Page 358 Alan Turing (1912–1954)......Page 359 Start by Glancing over the Whole Exam......Page 360 If You REALLY Get Stuck, Move On......Page 361 Admit Your Mistakes......Page 362 Stay Until the Bitter End......Page 363 Index......Page 364

Logic concepts are more mainstream than you may realize. There’s logic every place you look and in almost everything you do, from deciding which shirt to buy to asking your boss for a raise, and even to watching television, where themes of such shows as CSI and Numbers incorporate a variety of logistical studies. Logic For Dummies explains a vast array of logical concepts and processes in easy-to-understand language that make everything clear to you, whether you’re a college student of a student of life. You’ll find out about:

• Formal Logic
• Syllogisms
• Constructing proofs and refutations
• Propositional and predicate logic
• Modal and fuzzy logic
• Symbolic logic
• Deductive and inductive reasoning

Logic For Dummies tracks an introductory logic course at the college level. Concrete, real-world examples help you understand each concept you encounter, while fully worked out proofs and fun logic problems encourage you students to apply what you’ve learned.

A straightforward guide to logic concepts Logic concepts are more mainstream than you may realize. There's logic every place you look and in almost everything you do, from deciding which shirt to buy to asking your boss for a raise, and even to watching television, where themes of such shows as CSI and Numbers incorporate a variety of logistical studies. Logic For Dummies explains a vast array of logical concepts and processes in easy-to-understand language that make everything clear to you, whether you're a college student of a student of life. You'll find out about: Formal Logic Syllogisms Constructing proofs and refutations Propositional and predicate logic Modal and fuzzy logic Symbolic logic Deductive and inductive reasoning L ogic For Dummies tracks an introductory logic course at the college level. Concrete, real-world examples help you understand each concept you encounter, while fully worked out proofs and fun logic problems encourage you students to apply what you've learned.