Elements of Deductive Inference : an Introduction to Symbolic Logic

The text covers elementary logic, from statement logic through relational logic with identity and function symbols. The authors acquaint students with formal techniques at a level appropriate for undergraduates, but extends far enough and deep enough into the subject that it is suitable for a brief first-year graduate course. The text covers full and brief truth tables, and presents the method of truth (consistency) trees and natural deduction for the whole of elementary logic. The text's organization allows instructors to cover just statement logic, or statement logic combined with various extensions into predicate logic: monadic logic with or without identity, or the preceding plus relational logic with or without identity and with or without function symbols. At each stage, the instructor may elect to pursue truth trees and/or natural deduction. A final chapter provides a perspective for further study and applications of logic. The text may be used with or without the accompanying software.

Editor's Preface. Acknowledgments. I. INTRODUCTION. Logic and Argument. Deduction and Induction. Statements, Propositions, and Context. Use and Mention. II. STATEMENT LOGIC I: A NEW LANGUAGE. Introduction. Truth-Functionally Compound Statements. Symbolizing Simple and Compound Statements. Symbolizing More Complex Statements and Arguments. Spelling It Out Formally. III. STATEMENT LOGIC II: SEMANTIC METHODS. Introduction. Truth Tables. Formalized Semantics for SL. Truth-Functional Validity and Tautologousness. Further Semantic Properties and Relationships. Truth-Functional Consistency. The Material Conditional Revisited. Brief Truth Tables. Truth Trees. Using Truth Trees to Test for Other Semantic Properties. The Adequacy of the Tree Method. IV. STATEMENT LOGIC III: SYNTACTIC METHODS. Introduction. Whole Line Inference Rules for DSL. Replacement Rules for DSL. Conditional Proof and Reduction ad Absurdum. Proof Strategy. Proving Tautologousness and Other Semantic Properties. The Adequacy of the Natural Deduction System DSL. Additional Inference Rules. A Second Look at the Truth Table for the Material Conditional. V. PREDICATE LOGIC I: SYNTAX AND SEMANTICS. Introduction. Informal Introduction to the Language of Predicate Logic. Syntax for L. Formal Semantics I: Interpretations. Formal Semantics II: Truth Under an Interpretation. Symbolizing English I: Monadic Logic and Categorical Forms. Symbolizing English II: Polyadic Logic and Nested Quantifiers. Semantic Properties and Relationships for L. Classifying Logical Relations. VI. PREDICATE LOGIC II: SEMANTIC METHODS. Introduction. Truth Trees. Reading Interpretations From Finished Open Paths. The Problem of Infinite Trees. The Adequacy of the Tree Method for Lm and L. Soundness, Completeness, and Undecidability. VII. MONADIC PREDICATE LOGIC III: SYNTACTIC METHODS. Introduction. The Rules UI, EG, and Q. The Rules UG, R, PA-EI and EI. The Adequacy of DL. VIII. EXTENSIONS TO L: IDENTITY, RELATIONS AND FUNCTIONS. Introduction. Syntax and Semantics for L=. Symbolization in L= I: 'At Least', 'At Most', and 'Exactly'. Symbolization in L= II: Identity and Polyadic Predicates. Truth Trees for L=. Natural Deduction in L=. Syntax and Semantics for L*. Symbolization in L*. Truth Trees in L*. Natural Deduction in L*. IX. SOME APPLICATIONS AND LIMITATIONS OF L*. Introduction. Definite Descriptions and Ontological Commitment. Axiom Systems for Arithmetic. The Incompleteness of Arithmetic. Applications of Axiomatic Theories to the Philosophy of Science. Higher Order Logic. Modal Logic. Strict and Counterfactual Conditionals. General Intensional Logic. Deontic Logic. Free Logic. Many-Valued Logic. Appendix One: Further Reading in Logic. Appendix Two: Answer to Selected Exercises. Index.