TABLE OF CONTENTS
1. BASIC CONCEPTS OF LOGIC
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What is Logic? ..................................................................................................2
Inferences and Arguments ................................................................................2
Deductive Logic versus Inductive Logic ..........................................................5
Statements versus Propositions .........................................................................6
Form versus Content .........................................................................................7
Preliminary Definitions.....................................................................................9
Form and Content in Syllogistic Logic ...........................................................11
Demonstrating Invalidity Using the Method of Counterexamples .................13
Examples of Valid Arguments in Syllogistic Logic .......................................19
Exercises for Chapter 1 ...................................................................................22
Answers to Exercises for Chapter 1 ................................................................25
2. TRUTH-FUNCTIONAL CONNECTIVES
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Introduction.....................................................................................................28
Statement Connectives....................................................................................28
Truth-Functional Statement Connectives........................................................31
Conjunction.....................................................................................................33
Disjunction ......................................................................................................35
A Statement Connective that is not Truth-Functional.....................................37
Negation ..........................................................................................................38
The Conditional...............................................................................................39
The Non-Truth-Functional Version of If-Then...............................................40
The Truth-Functional Version of If-Then .......................................................41
The Biconditional............................................................................................43
Complex Formulas..........................................................................................44
Truth Tables for Complex Formulas...............................................................46
Exercises for Chapter 2 ...................................................................................54
Answers to Exercises for Chapter 2 ................................................................57
3. VALIDITY IN SENTENTIAL LOGIC
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Tautologies, Contradictions, and Contingent Formulas..................................62
Implication And Equivalence..........................................................................64
Validity in Sentential Logic ............................................................................66
Testing Arguments in Sentential Logic...........................................................67
The Relation Between Validity and Implication.............................................71
Exercises for Chapter 3 ...................................................................................74
Answers to Exercises for Chapter 3 ................................................................76
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Hardegree, Symbolic Logic
4. TRANSLATIONS IN SENTENTIAL LOGIC
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Introduction.....................................................................................................92
The Grammar of Sentential Logic; A Review ................................................93
Conjunctions ...................................................................................................94
Disguised Conjunctions ..................................................................................95
The Relational Use of ‘And’ ...........................................................................96
Connective-Uses of ‘And’ Different from Ampersand...................................98
Negations, Standard and Idiomatic ...............................................................100
Negations of Conjunctions............................................................................101
Disjunctions ..................................................................................................103
‘Neither...Nor’...............................................................................................104
Conditionals ..................................................................................................106
‘Even If’ ........................................................................................................107
‘Only If’ ........................................................................................................108
A Problem with the Truth-Functional If-Then..............................................110
‘If And Only If’.............................................................................................112
‘Unless’ .........................................................................................................113
The Strong Sense of ‘Unless’........................................................................114
Necessary Conditions....................................................................................116
Sufficient Conditions ....................................................................................117
Negations of Necessity and Sufficiency .......................................................118
Yet Another Problem with the Truth-Functional If-Then.............................120
Combinations of Necessity and Sufficiency .................................................121
‘Otherwise’....................................................................................................123
Paraphrasing Complex Statements................................................................125
Guidelines for Translating Complex Statements ..........................................133
Exercises for Chapter 4 .................................................................................134
Answers to Exercises for Chapter 4 ..............................................................138
5. DERIVATIONS IN SENTENTIAL LOGIC
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Introduction...................................................................................................142
The Basic Idea...............................................................................................143
Argument Forms and Substitution Instances ................................................145
Simple Inference Rules .................................................................................147
Simple Derivations........................................................................................151
The Official Inference Rules.........................................................................154
Inference Rules (Initial Set) ..........................................................................155
Inference Rules; Official Formulation ..........................................................156
Show-Lines and Show-Rules; Direct Derivation.........................................158
Examples of Direct Derivations ....................................................................161
Conditional Derivation..................................................................................164
Indirect Derivation (First Form)....................................................................169
Indirect Derivation (Second Form) ...............................................................174
Showing Disjunctions Using Indirect Derivation ........................................177
Further Rules.................................................................................................180
Showing Conjunctions and Biconditionals ...................................................181
The Wedge-Out Strategy ..............................................................................184
The Arrow-Out Strategy ...............................................................................187
Table of Contents
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Summary of the System Rules for System SL ..............................................189
Pictorial Summary of the Rules of System SL..............................................191
Pictorial Summary of Strategies....................................................................195
Exercises for Chapter 5 .................................................................................198
Answers to Exercises for Chapter 5 ..............................................................203
6. TRANSLATIONS IN MONADIC PREDICATE LOGIC
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Introduction...................................................................................................222
The Subject-Predicate Form of Atomic Statements.....................................223
Predicates ......................................................................................................224
Singular Terms..............................................................................................226
Atomic Formulas...........................................................................................228
Variables And Pronouns ...............................................................................230
Compound Formulas.....................................................................................232
Quantifiers.....................................................................................................232
Combining Quantifiers With Negation .........................................................236
Symbolizing The Statement Forms Of Syllogistic Logic .............................243
Summary of the Basic Quantifier Translation Patterns so far Examined .....248
Further Translations Involving Single Quantifiers .......................................251
Conjunctive Combinations of Predicates ......................................................255
Summary of Basic Translation Patterns from Sections 12 and 13................262
‘Only’ ............................................................................................................263
Ambiguities Involving ‘Only’.......................................................................267
‘The Only’.....................................................................................................269
Disjunctive Combinations of Predicates .......................................................272
Multiple Quantification in Monadic Predicate Logic ...................................277
‘Any’ and other Wide Scope Quantifiers......................................................282
Exercises for Chapter 6 .................................................................................289
Answers to Exercises for Chapter 6 ..............................................................295
7. TRANSLATIONS IN POLYADIC PREDICATE LOGIC
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Introduction...................................................................................................300
Simple Polyadic Quantification ....................................................................301
Negations of Simple Polyadic Quantifiers....................................................306
The Universe of Discourse............................................................................310
Quantifier Specification ................................................................................311
Complex Predicates.......................................................................................316
Three-Place Predicates ..................................................................................320
‘Any’ Revisited .............................................................................................322
Combinations of ‘No’ and ‘Any’ ..................................................................325
More Wide-Scope Quantifiers ......................................................................328
Exercises for Chapter 7 .................................................................................332
Answers to Exercises for Chapter 7 ..............................................................338
8. DERIVATIONS IN PREDICATE LOGIC
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Introduction...................................................................................................344
The Rules of Sentential Logic.......................................................................344
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Hardegree, Symbolic Logic
The Rules of Predicate Logic: An Overview ................................................347
Universal Out ................................................................................................349
Potential Errors in Applying Universal-Out .................................................351
Examples of Derivations using Universal-Out .............................................353
Existential In .................................................................................................355
Universal Derivation .....................................................................................359
Existential Out...............................................................................................367
How Existential-Out Differs from the other Rules .......................................374
Negation Quantifier Elimination Rules.........................................................376
Direct versus Indirect Derivation of Existentials .........................................382
Appendix 1: The Syntax of Predicate Logic ................................................390
Appendix 2: Summary of Rules for System PL (Predicate Logic) .............399
Exercises for Chapter 8 .................................................................................401
Answers to Exercises for Chapter 8 ..............................................................405
PICTORIAL SUMMARY OF RULES OF DERIVATION
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Rules for Sentential Logic.............................................................................417
Rules for Predicate Logic..............................................................................418