Logic For Dummies

by Zegarelli, Mark

• ISBN13: 9780471799412
• ISBN10: 0471799416
• Edition: 1st
• Format: Paperback
• Copyright: 2006-11-29
• Publisher: For Dummies

Summary

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.

Author Biography

Mark Zegarelli holds degrees in math and English from Rutgers University. He has written four books and numerous magazine articles on logic puzzles.

Table of Contents

Introduction	1	(1)
About This Book	1	(1)
Conventions Used in This Book	2	(1)
What You're Not to Read	3	(1)
Foolish Assumptions	3	(1)
How This Book Is Organized	3	(3)
Part I: Overview of Logic	4	(1)
Part II: Formal Sentential Logic (SL)	4	(1)
Part III: Proofs, Syntax, and Semantics in SL	4	(1)
Part IV: Quantifier Logic (QL)	5	(1)
Part V: Modern Developments in Logic	5	(1)
Part VI: The Part of Tens	5	(1)
Icons Used in This Book	6	(1)
Where to Go from Here	6	(1)
Part I: Overview of Logic	7	(42)
What Is This Thing Called Logic?	9	(10)
Getting a Logical Perspective	9	(4)
Bridging the gap from here to there	10	(1)
Understanding cause and effect	10	(2)
Everything and more	12	(1)
Existence itself	12	(1)
A few logical words	13	(1)
Building Logical Arguments	13	(2)
Generating premises	13	(1)
Bridging the gap with intermediate steps	14	(1)
Forming a conclusion	14	(1)
Deciding whether the argument is valid	15	(1)
Understanding enthymemes	15	(1)
Making Logical Conclusions Simple with the Laws of Thought	15	(2)
The law of identity	16	(1)
The law of the excluded middle	16	(1)
The law of non-contradiction	16	(1)
Combining Logic and Math	17	(2)
Math is good for understanding logic	17	(1)
Logic is good for understanding math	18	(1)
Logical Developments from Aristotle to the Computer	19	(14)
Classical Logic --- from Aristotle to the Enlightenment	20	(5)
Aristotle invents syllogistic logic	20	(3)
Euclid's axioms and theorems	23	(1)
Chrysippus and the Stoics	24	(1)
Logic takes a vacation	24	(1)
Modern Logic --- the 17th, 18th, and 19th Centuries	25	(4)
Leibniz and the Renaissance	25	(1)
Working up to formal logic	26	(3)
Logic in the 20th Century and Beyond	29	(4)
Non-classical logic	30	(1)
Godel's proof	30	(1)
The age of computers	31	(1)
Searching for the final frontier	32	(1)
Just for the Sake of Argument	33	(16)
Defining Logic	33	(4)
Examining argument structure	34	(2)
Looking for validation	36	(1)
Studying Examples of Arguments	37	(2)
Ice cream Sunday	37	(1)
Fifi's lament	38	(1)
Escape from New York	38	(1)
The case of the disgruntled employee	39	(1)
What Logic Isn't	39	(7)
Thinking versus logic	40	(1)
Reality --- what a concept!	41	(1)
The sound of soundness	42	(1)
Deduction and induction	43	(1)
Rhetorical questions	44	(2)
Whose Logic Is It, Anyway?	46	(3)
Pick a number (math)	46	(1)
Fly me to the moon (science)	47	(1)
Switch on or off (computer science)	47	(1)
Tell it to the judge (law)	48	(1)
Find the meaning of life (philosophy)	48	(1)
Part II: Formal Sentential Logic (SL)	49	(96)
Formal Affairs	51	(22)
Observing the Formalities of Sentential Logic	51	(2)
Statement constants	52	(1)
Statement variables	52	(1)
Truth value	53	(1)
The Five SL Operators	53	(10)
Feeling negative	54	(1)
Displaying a show of ands	55	(2)
Digging for or	57	(2)
Getting iffy	59	(2)
Getting even iffier	61	(2)
How SL Is Like Simple Arithmetic	63	(2)
The ins and outs of values	63	(1)
There's no substitute for substitution	64	(1)
Parenthetical guidance suggested	65	(1)
Lost in Translation	65	(8)
The easy way --- translating from SL to English	66	(2)
The not-so-easy way --- translating from English to SL	68	(5)
The Value of Evaluation	73	(12)
Value Is the Bottom Line	74	(3)
Getting started with SL evaluation	75	(1)
Stacking up another method	76	(1)
Making a Statement	77	(5)
Identifying sub-statements	78	(1)
Scoping out a statement	79	(1)
The main attraction: Finding main operators	80	(2)
Eight Forms of SL Statements	82	(1)
Evaluation Revisited	83	(2)
Turning the Tables: Evaluating Statements with Truth Tables	85	(22)
Putting It All on the Table: The Joy of Brute Force	86	(1)
Baby's First Truth Table	87	(6)
Setting up a truth table	87	(2)
Filling in a truth table	89	(3)
Reading a truth table	92	(1)
Putting Truth Tables to Work	93	(7)
Taking on tautologies and contradictions	93	(1)
Judging semantic equivalence	94	(2)
Staying consistent	96	(2)
Arguing with validity	98	(2)
Putting the Pieces Together	100	(7)
Connecting tautologies and contradictions	101	(1)
Linking semantic equivalence with tautology	102	(1)
Linking inconsistency with contradiction	103	(2)
Linking validity with contradiction	105	(2)
Taking the Easy Way Out: Creating Quick Tables	107	(18)
Dumping the Truth Table for a New Friend: The Quick Table	108	(1)
Outlining the Quick Table Process	109	(4)
Making a strategic assumption	110	(1)
Filling in a quick table	110	(1)
Reading a quick table	111	(1)
Disproving the assumption	112	(1)
Planning Your Strategy	113	(3)
Tautology	113	(1)
Contradiction	114	(1)
Contingent statement	114	(1)
Semantic equivalence and inequivalence	114	(1)
Consistency and inconsistency	115	(1)
Validity and invalidity	115	(1)
Working Smarter (Not Harder) with Quick Tables	116	(9)
Recognizing the six easiest types of statements to work with	117	(2)
Working with the four not-so-easy statement types	119	(3)
Coping with the six difficult statement types	122	(3)
Truth Grows on Trees	125	(20)
Understanding How Truth Trees Work	125	(3)
Decomposing SL statements	126	(2)
Solving problems with truth trees	128	(1)
Showing Consistency or Inconsistency	128	(3)
Testing for Validity or Invalidity	131	(3)
Separating Tautologies, Contradictions, and Contingent Statements	134	(6)
Tautologies	134	(3)
Contradictions	137	(3)
Contingent Statements	140	(1)
Checking for Semantic Equivalence or Inequivalence	140	(5)
Part III: Proofs, Syntax, and Semantics in SL	145	(78)
What Have You Got to Prove?	147	(14)
Bridging the Premise-Conclusion Divide	148	(1)
Using Eight Implication Rules in SL	149	(12)
The → rules: Modus Ponens and Modus Tollens	150	(3)
The & rules: Conjunction and Simplification	153	(2)
The V rules: Addition and Disjunctive Syllogism	155	(3)
The Double → Rules: Hypothetical Syllogism and Constructive Dilemma	158	(3)
Equal Opportunities: Putting Equivalence Rules to Work	161	(14)
Distinguishing Implications and Equivalences	162	(1)
Thinking of equivalences as ambidextrous	162	(1)
Applying equivalences to part of the whole	162	(1)
Discovering the Ten Valid Equivalences	163	(12)
Double Negation (DN)	163	(1)
Contraposition (Contra)	164	(1)
Implication (Impl)	165	(1)
Exportation (Exp)	166	(1)
Commutation (Comm)	167	(1)
Association (Assoc)	168	(1)
Distribution (Dist)	169	(1)
DeMorgan's Theorem (DeM)	170	(2)
Tautology (Taut)	172	(1)
Equivalence (Equiv)	172	(3)
Big Assumptions with Conditional and Indirect Proofs	175	(12)
Conditioning Your Premises with Conditional Proof	176	(5)
Understanding conditional proof	177	(1)
Tweaking the conclusion	178	(2)
Stacking assumptions	180	(1)
Thinking Indirectly: Proving Arguments with Indirect Proof	181	(3)
Introducing indirect proof	182	(1)
Proving short conclusions	183	(1)
Combining Conditional and Indirect Proofs	184	(3)
Putting It All Together: Strategic Moves to Polish Off Any Proof	187	(18)
Easy Proofs: Taking a Gut Approach	188	(3)
Look at the problem	188	(1)
Jot down the easy stuff	189	(1)
Know when to move on	190	(1)
Moderate Proofs: Knowing When to Use Conditional Proof	191	(4)
The three friendly forms: x → y, x V y, and ~(x & y)	191	(2)
The two slightly-less-friendly forms: x ↔ y and ~(x ↔ y)	193	(1)
The three unfriendly forms: x & y, ~(x v y), and ~(x → y)	194	(1)
Difficult Proofs: Knowing What to Do When the Going Gets Tough	195	(10)
Choose carefully between direct and indirect proof	195	(1)
Work backwards from the conclusion	196	(2)
Go deeper into SL statements	198	(4)
Break down long premises	202	(2)
Make a shrewd assumption	204	(1)
One for All and All for One	205	(8)
Making Do with the Five SL Operators	206	(2)
Downsizing --- A True Story	208	(5)
The tyranny of power	208	(1)
The blow of insurrection	209	(1)
The horns of dilemma	209	(1)
The (Sheffer's) stroke of genius	210	(2)
The moral of the story	212	(1)
Syntactical Maneuvers and Semantic Considerations	213	(10)
Are You WFF Us or Against Us?	214	(3)
Understanding WFFs (with a few strings attached)	215	(1)
Relaxing the rules	216	(1)
Separating WFFs from non-WFFs	216	(1)
Comparing SL to Boolean Algebra	217	(6)
Reading the signs	218	(2)
Doing the math	220	(1)
Understanding rings and things	221	(1)
Exploring syntax and semantics in Boolean algebra	221	(2)
Part IV: Quantifier Logic (QL)	223	(76)
Expressing Quantity with Quality: Introducing Quantifier Logic	225	(14)
Taking a Quick Look at Quantifier Logic	226	(5)
Using individual constants and property constants	226	(3)
Incorporating the SL operators	229	(1)
Understanding individual variables	230	(1)
Expressing Quantity with Two New Operators	231	(4)
Understanding the universal quantifier	231	(1)
Expressing existence	232	(1)
Creating context with the domain of discourse	233	(2)
Picking out Statements and Statement Forms	235	(4)
Determining the scope of a quantifier	236	(1)
Discovering bound variables and free variables	236	(1)
Knowing the difference between statements and statement forms	237	(2)
QL Translations	239	(12)
Translating the Four Basic Forms of Categorical Statements	239	(5)
``All'' and ``some''	240	(2)
``Not all'' and ``no''	242	(2)
Discovering Alternative Translations of Basic Forms	244	(3)
Translating ``all'' with E	245	(1)
Translating ``some'' with A	245	(1)
Translating ``not all'' with E	246	(1)
Translating ``no'' with A	246	(1)
Identifying Statements in Disguise	247	(4)
Recognizing ``all'' statements	247	(1)
Recognizing ``some'' statements	248	(1)
Recognizing ``not all'' statements	248	(1)
Recognizing ``no'' statements	249	(2)
Proving Arguments with QL	251	(24)
Applying SL Rules in QL	252	(4)
Comparing similar SL and QL statements	252	(1)
Transferring the eight implication rules from SL into QL	253	(2)
Employing the ten SL equivalence rules in QL	255	(1)
Transforming Statements with Quantifier Negation (QN)	256	(3)
Introducing QN	256	(1)
Using QN in proofs	257	(2)
Exploring the Four Quantifier Rules	259	(16)
Easy rule #1: Universal Instantiation (UI)	260	(2)
Easy rule #2: Existential Generalization (EG)	262	(3)
Not-so-easy rule #1: Existential Instantiation (EI)	265	(5)
Not-so-easy rule #2: Universal Generalization (UG)	270	(5)
Good Relations and Positive Identities	275	(12)
Relating to Relations	276	(7)
Defining and using relations	276	(1)
Connecting relational expressions	277	(1)
Making use of quantifiers with relations	278	(1)
Working with multiple quantifiers	279	(1)
Writing proofs with relations	280	(3)
Identifying with Identities	283	(4)
Understanding identities	284	(1)
Writing proofs with identities	285	(2)
Planting a Quantity of Trees	287	(12)
Applying Your Truth Tree Knowledge to QL	287	(8)
Using the decomposition rules from SL	287	(2)
Adding UI, El, and QN	289	(2)
Using UI more than once	291	(4)
Non-Terminating Trees	295	(4)
Part V: Modern Developments in Logic	299	(34)
Computer Logic	301	(8)
The Early Versions of Computers	302	(2)
Babbage designs the first computers	302	(1)
Turing and his UTM	302	(2)
The Modern Age of Computers	304	(5)
Hardware and logic gates	305	(2)
Software and computer languages	307	(2)
Sporting Propositions: Non-Classical Logic	309	(14)
Opening up to the Possibility	310	(5)
Three-valued logic	310	(1)
Multi-valued logic	311	(2)
Fuzzy logic	313	(2)
Getting into a New Modality	315	(2)
Taking Logic to a Higher Order	317	(1)
Moving Beyond Consistency	318	(2)
Making a Quantum Leap	320	(3)
Introducing quantum logic	320	(1)
Playing the shell game	321	(2)
Paradox and Axiomatic Systems	323	(10)
Grounding Logic in Set Theory	323	(4)
Setting things up	324	(1)
Trouble in paradox: Recognizing the problem with set theory	325	(1)
Developing a solution in the Principia Mathematica	326	(1)
Discovering the Axiomatic System for SL	327	(1)
Proving Consistency and Completeness	328	(2)
Consistency and completeness of SL and QL	329	(1)
Formalizing logic and mathematics with the Hilbert Program	329	(1)
Godel's Incompleteness Theorem	330	(2)
The importance of Godel's theorem	331	(1)
How he did it	331	(1)
Pondering the Meaning of It All	332	(1)
Part VI: The Part of Tens	333	(12)
Ten Quotes about Logic	335	(2)
Ten Big Names in Logic	337	(4)
Aristotle (384--322 BC)	337	(1)
Gottfried Leibniz (1646--1716)	337	(1)
George Boole (1815--1864)	338	(1)
Lewis Carroll (1832--1898)	338	(1)
Georg Cantor (1845--1918)	338	(1)
Gottlob Frege (1848--1925)	339	(1)
Bertrand Russell (1872--1970)	339	(1)
David Hilbert (1862--1943)	339	(1)
Kurt Godel (1906--1978)	340	(1)
Alan Turing (1912--1954)	340	(1)
Ten Tips for Passing a Logic Exam	341	(4)
Breathe	341	(1)
Start by Glancing over the Whole Exam	341	(1)
Warm up with an Easy Problem First	342	(1)
Fill in Truth Tables Column by Column	342	(1)
If You Get Stuck, Jot Down Everything	342	(1)
If You Really Get Stuck, Move On	342	(1)
If Time Is Short, Finish the Tedious Stuff	343	(1)
Check Your Work	343	(1)
Admit Your Mistakes	343	(1)
Stay Until the Bitter End	344	(1)
Index	345