Preface 

xiii  
PART ONE Sentential Logic 



1  (17) 


2  (2) 


4  (1) 


5  (2) 


7  (3) 


10  (2) 

The Nature of Symbolic Logic 


12  (1) 

The Scope of Symbolic Logic 


13  (5) 


15  (1) 


16  (1) 


16  (2) 

The Structure of Sentential Logic 


18  (12) 

Simple and Compound Sentences 


19  (4) 


23  (2) 

The Structure and Symbolism of Sentential Logic 


25  (5) 


27  (1) 


28  (1) 


28  (2) 


30  (17) 

Truth Tables for the Operators 


31  (8) 


39  (2) 

Truthfunctional Operators 


41  (1) 

Nontruthfunctional Operators 


42  (5) 


44  (1) 


44  (1) 


45  (2) 

Symbolizing English Sentences 


47  (19) 


48  (2) 

Truthfunctional and Nontruthfunctional Compounds 


50  (1) 

Symbolizing English Operators 


51  (8) 

Symbolizing Multiply Complex Sentences 


59  (7) 


63  (3) 

Truth Tables for Testing Validity 


66  (20) 

Constructing Base Columns for Truth Tables 


67  (4) 

The Truth Table Test for Validity 


71  (4) 


75  (6) 

Mechanical Decision Procedures 


81  (5) 


82  (1) 


82  (1) 


83  (3) 

Further Applications of the Truth Table Method 


86  (16) 

Tautologies, Contradictions, and Contingencies 


87  (3) 

Logical Implication and Logical Equivalence 


90  (3) 


93  (1) 

Statements and Statement Forms; Applying Truth Table Concepts 


94  (2) 

Four Kinds of Truth Table Problems and the Relations Between Them 


96  (6) 


97  (1) 


98  (1) 


98  (4) 

The Proof Method: Eight Basic Inference Rules 


102  (34) 

Form and Substitution Instance 


104  (3) 


107  (2) 

Eight Basic Inference Rules 


109  (8) 


117  (2) 

Constructing Simple Proofs 


119  (5) 

Constructing More Complex Proofs 


124  (12) 

Summary of Rules of Inference 


127  (1) 


128  (1) 


128  (8) 


136  (27) 

The Structure of Replacement Rules 


137  (1) 

The Ten Replacement Rules 


138  (9) 

Constructing Simple Proofs with Replacement Rules 


147  (3) 

Strategies for More Complex Proofs 


150  (13) 

Summary of Replacement Rules 


156  (1) 


156  (7) 

Conditional Proof and Indirect Proof 


163  (24) 


164  (4) 


168  (3) 

Discharging Assumptions; Restrictions on C.P. and I.P. 


171  (2) 


173  (3) 


176  (3) 


179  (1) 


180  (7) 

Summary of Rules of Conditional Proof and Indirect Proof 


181  (1) 


182  (1) 


183  (4) 
PART TWO Monadic Predicate Logic 



187  (10) 

Singular Sentences and Propositional Functions 


189  (3) 

Symbolizing Singular Sentences 


192  (5) 


195  (1) 


195  (2) 


197  (13) 

Universal and Existential Quantifiers 


198  (5) 

Free and Bound Variables; Scope of a Quantifier 


203  (1) 


203  (7) 


207  (1) 


208  (2) 


210  (24) 

The Four Categorical Propositions 


211  (3) 

Individuals, Sets, and Properties 


214  (1) 


215  (4) 

Symbolizing Categorical Propositions 


219  (2) 

Negated Categorical Propositions 


221  (2) 

Deriving C.Q.N. Rules from Q.N. Rules 


223  (1) 

Symbolizing English Categorical Sentences 


223  (11) 

Symmary of Categorical Propositions 


229  (1) 


230  (1) 


230  (4) 

Complex Subjects and Predicates 


234  (13) 

Complex Subjects and Predicates 


235  (4) 

Equivalent Symbolizations 


239  (8) 


244  (3) 

Quantifier Form and TruthFunctional Compounds of Quantifier Statements 


247  (8) 


248  (1) 

Truthfunctional Compounds and Quantifier Form 


249  (2) 

Symbolizing Truthfunctional Compounds 


251  (4) 


252  (1) 


252  (3) 

Proofs in Predicate Logic 


255  (25) 

Preliminary Statement of the Four Quantifier Rules 


256  (2) 

Instances of Quantified Formulas 


258  (1) 

The Rules of Universal Instantiation (U.I.) and Existential Generalization (E.G.) 


259  (1) 

The Rules of Existential Instantiation (E.I.) and Universal Generalization (U.G.); Flagging Restrictions 


260  (7) 

Constructing Proofs for ``Pure'' Quantifier Arguments 


267  (5) 

Constructing Proofs for Arguments Containing Truthfunctional Compounds 


272  (2) 

Constructing Proofs of Quantifier Theorems 


274  (6) 

Statement of the Quantifier Rules, with All Necessary Restrictions 


276  (1) 


277  (3) 

Invalidity in Quantifier Logic 


280  (15) 

The Natural Interpretation Method 


281  (3) 

Truth Conditions for Quantifier Statements 


284  (1) 

The Model Universe Method 


285  (10) 


292  (1) 


292  (3) 
PART THREE Relational Predicate Logic 


Symbolization in Relational Predicate Logic 


295  (28) 

Relational Predicates and Singular Sentences 


296  (3) 


299  (7) 


306  (3) 

Categorical Relational Statements; Complex Subjects and Predicates 


309  (4) 

Symbolizing English Sentences 


313  (10) 


317  (6) 

Proofs and Invalidity for Relational Predicate Logic 


323  (14) 

Proofs in Relational Predicate Logic 


324  (7) 

Invalidity in Relational Predicate Logic 


331  (6) 


335  (2) 

Identity and Definite Descriptions 


337  (16) 

Identity Statements and Their Negations 


338  (1) 

Exceptives and ``Only'' Statements 


339  (3) 


342  (1) 


343  (4) 


347  (6) 


348  (5) 

Proofs Involving Identity 


353  (12) 


353  (4) 

Proofs Containing Identity Statements 


357  (8) 

Summary of Identity Rules 


361  (1) 


361  (4) 
PART FOUR Extra Credit Units 


WellFormed Formulas for Sentential Logic 


365  (4) 


367  (2) 

Polish Notation for Sentential Logic 


369  (4) 


371  (2) 

Proof Trees for Sentential Logic 


373  (6) 


378  (1) 

Using Venn Diagrams to Prove Validity 


379  (6) 


383  (2) 

Stroke (nand) and Dagger (nor) Operators 


385  (4) 


388  (1) 

Proof Trees for Predicate Logic 


389  (12) 


399  (2) 
Answers to Starred Exercises 

401  (42) 
Index 

443  