Preface 

xi  


1  (18) 

The Elements of an Argument 


1  (4) 


5  (2) 


7  (1) 


8  (3) 


11  (1) 


12  (1) 

Consistency and Validity Compared 


13  (2) 

Contexts of Discovery and Justification 


15  (1) 


15  (4) 


16  (3) 

Part One: Sentential Logic 


19  (144) 

Symbolizing in Sentential Logic 


21  (34) 

Atomic and Compound Sentences 


22  (1) 


23  (1) 


23  (4) 

NonTruthFunctional Connectives 


27  (1) 


27  (2) 


29  (1) 


30  (1) 


31  (1) 


32  (3) 

``Not Both'' and ``Neither . . . Nor'' 


35  (2) 


37  (3) 


40  (2) 

``Only If'' and ``Unless'' 


42  (1) 

Symbolizing Complex Sentences 


43  (7) 

Alternative Sentential Logic Symbols 


50  (5) 


52  (3) 


55  (33) 


55  (5) 


60  (5) 

Tautologies, Contradictions, and Contingent Sentences 


65  (6) 


71  (1) 

Truth Table Test of Validity 


72  (3) 

Truth Table Test of Consistency 


75  (1) 


76  (2) 

The Short Truth Table Test for Invalidity 


78  (4) 

The Short Truth Table Test for Consistency 


82  (1) 

A Method of Justification for the Truth Tables 


83  (5) 


87  (1) 


88  (37) 


88  (2) 

The Method of Proof: Modus Ponens and Modus Tollens 


90  (3) 

Disjunctive Syllogism and Hypothetical Syllogism 


93  (2) 

Simplification and Conjunction 


95  (1) 

Addition and Constructive Dilemma 


96  (3) 


99  (6) 

Double Negation and DeMorgan's Theorem 


105  (3) 

Commutation, Association, and Distribution 


108  (1) 

Contraposition, Implication, and Exportation 


109  (1) 

Tautology and Equivalence 


109  (4) 

More Principles of Strategy 


113  (4) 

Common Errors in Problem Solving 


117  (8) 


124  (1) 

Conditional and Indirect Proofs 


125  (25) 


125  (9) 


134  (5) 

Strategy Hints for Using CP and IP 


139  (1) 


140  (1) 

Proving Premises Inconsistent 


141  (2) 

Adding Valid Argument Forms 


143  (1) 

The Completeness and Soundness of Sentential Logic 


144  (1) 

Introduction and Elimination Rules 


145  (5) 


149  (1) 

Sentential Logic Truth Trees 


150  (13) 

The Sentential Logic Truth Tree Method 


150  (1) 


151  (2) 

Details of Tree Construction 


153  (6) 


159  (1) 

Constructing Tree Rules for Any Function 


160  (3) 


162  (1) 

Part Two: Predicate Logic 


163  (192) 

Predicate Logic Symbolization 


165  (24) 

Individuals and Properties 


165  (4) 

Quantifiers and Free Variables 


169  (1) 


170  (5) 


175  (1) 

Basic Predicate Logic Symbolizations 


176  (2) 


178  (1) 

Common Pitfalls in Symbolizing with Quantifiers 


178  (3) 


181  (3) 

Symbolizing ``Only,'' ``None but,'' and ``Unless'' 


184  (5) 


187  (2) 

Predicate Logic Semantics 


189  (9) 

Interpretations in Predicate Logic 


189  (2) 


191  (3) 

Using Expansions to Prove Invalidity 


194  (1) 

Consistency in Predicate Logic 


195  (1) 

Validity and Inconsistency in Predicate Logic 


196  (2) 


197  (1) 


198  (24) 


198  (1) 

The Four Quantifier Rules 


199  (6) 

The Five Main Restrictions 


205  (5) 

Precise Formulation of the Four Quantifier Rules 


210  (2) 

Mastering the Four Quantifier Rules 


212  (4) 


216  (6) 


221  (1) 

Relational Predicate Logic 


222  (32) 


222  (3) 

Symbolizations Containing Overlapping Quantifiers 


225  (1) 

Expansions and Overlapping Quantifiers 


225  (5) 


230  (1) 

Symbolizing ``Someone,'' ``Somewhere,'' ``Sometime,'' and So On 


231  (5) 

Invalidity and Consistency in Relational Predicate Logic 


236  (1) 

Relational Predicate Logic Proofs 


237  (7) 

Strategy for Relational Predicate Logic Proofs 


244  (3) 

Theorems and Inconsistency in Predicate Logic 


247  (3) 

Predicate Logic Metatheory 


250  (1) 

A Simpler Set of Quantifier Rules 


250  (4) 

Rationale Behind the Precise Formulation of the Four Quantifier Rules 


254  (15) 

Cases Involving the Five Major Restrictions 


254  (3) 

OnetoOne Correspondence Matters 


257  (4) 

Accidentally Bound Variables and Miscellaneous Cases 


261  (4) 

Predicate Logic Proofs with Flagged Constants 


265  (4) 

Predicate Logic Truth Trees 


269  (14) 


269  (1) 

General Features of the Method 


270  (1) 

Specific Examples of the Method 


270  (5) 

Some Advantages of the Trees 


275  (1) 

Example of an Invalid Argument with at Least One Open Path 


275  (1) 


276  (4) 


280  (3) 


282  (1) 

Identity and Philosophical Problems of Symbolic Logic 


283  (33) 


283  (6) 


289  (2) 


291  (3) 


294  (2) 

Limitations of Predicate Logic 


296  (4) 


300  (7) 


307  (9) 


314  (2) 


316  (39) 


316  (3) 


319  (1) 


320  (3) 

Conversion, Obversion, Contraposition 


323  (3) 

Syllogistic LogicNot Assuming Existential Import 


326  (3) 


329  (2) 


331  (2) 

Determining Syllogism Validity 


333  (1) 

Venn Diagram Proofs of Validity or Invalidity 


334  (5) 

Five Rules for Determining Validity or Invalidity 


339  (3) 


342  (3) 


345  (1) 


346  (2) 

Technical Restrictions and Limitations; Modern Logic and Syllogistic Logic Compared 


348  (7) 


352  (3) 

Part Three: Other Systems of Logic 


355  (86) 


357  (19) 


357  (2) 


359  (17) 


375  (1) 


376  (35) 

A Mistaken View of Induction and Deduction 


376  (2) 

Kinds of Inductive Arguments 


378  (10) 


388  (3) 


391  (7) 


398  (1) 


399  (2) 


401  (2) 

Induction Is UnjustifiedThe Old Riddle of Induction 


403  (3) 

Not All Instances of Theories Confirm ThemThe New Riddle of Induction 


406  (5) 


409  (2) 


411  (14) 

The Nature of an Axiom System 


411  (1) 

Interpreted and Uninterpreted Systems 


412  (1) 

Properties of Axiom Systems 


413  (3) 

Outline of an Axiom System for Sentential Logic 


416  (5) 

Axiom Systems for Predicate Logic 


421  (1) 

Other Kinds of Axiom Systems 


422  (1) 

Objections to Axiom Systems 


423  (2) 


423  (2) 


425  (16) 


425  (1) 


426  (2) 


428  (1) 


429  (1) 


430  (1) 


430  (1) 


431  (2) 

Epistemic Logic: The Logic of Knowledge and Belief 


433  (2) 


435  (2) 


437  (1) 

Problems with Deontic Systems 


438  (3) 


440  (1) 
Answers to EvenNumbered Exercise Items 

441  (64) 
Bibliography 

505  (4) 
Special Symbols 

509  (2) 
Index 

511  