Logic For Dummies

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