The Many Worlds of Logic

To the Instructor To the Student Acknowledgements Chapter 1 Fundamentals Recognizing Arguments Some Typical Conclusion Indicators Some Typical Premise Indicators Distinguishing Sentences and Statements Two General Categories of Argument: Deductive and Inductive Arguments Deductive and Inductive Indicator Words Evaluating Inductive Arguments: The Strong, the Weak, and the Inductively Sound Evaluating Deductive Arguments: The Valid, the Invalid, and the Dedictively Sound Deciding Whether an Argument is Valid or Invalid Consistency and Inconsistency Implication Logical Equivalence Necessity Ideals Appendix: Some Logical Puzzles Glossary PART I TRUTH-FUNCTIONAL LOGIC Chapter 2 Introductory Truth-Functional Logic Simple and Compound Sentences, Sentence Operators, and the Conjunction Negation Disjunction Truth-Functions and Truth-Functional Compound Sentences Conditional Sentences Biconditional Sentences Glossary Chapter 3 Translating English into Logical Symbols Symbolizing Sentences Containing More than One Operator Throwing the Tilde into the Mix From And to Or and Back Again--With a Few Nots Thrown In Some General Hints on Symbolizing Translating Conditionals and Biconditionals Symbolizing Necessary and Sufficient Conditions Glossary Chapter 4 Our New Language Gets a Name and a Formal Syntax The Language TL How to Calculate the Truth-Value of the Whole from the Values of the Parts Glossary Chapter 5 Truth-Table Analysis Constructing a Truth-Table for a Formula Eight-Row Tables How to Make Your Own Tautology Detector Using Just Paper and Pencil How to Make an Inexpensive Contradiction Detector for Home or Office The Contingency Detector: Don't Leave Home without It Testing an Argument for Validity Showing an Argument Invalid with a Partial Truth-Table Testing a Pair of Sentences for Equivalences Glossary Chapter 6 The Concept of Logical Form Sentence Forms Argument Forms The Disjunctive Syllogism Form The Modus Ponens Form The Modus Tollens Form The Hypothetical Syllogism Form Valid Argument Forms Invalid Argument Forms Concluding Comments Appendix: Contradictory and Tautological Sentence Forms Glossary Chapter 7 Truth-Functional Natural Deduction The Disjunctive Syllogism Rule The Modus Ponens Rule The Modus Tollens Rule The Hypothetical Syllogism Rule Proving that a Conclusion Validly Follows The System TD Proofs Glossary Chapter 8 Four More Inference Rules The Simplification Rule The Conjunction Rule The Addition Rule The Constructive Dilemma Rule Some Unsolicited Advice on Learning to Construct Proofs Proof Strategies Some Additional Suggestions Concerning Strategy Appendix: Some Common Deduction Errors Chapter 9 Indirect Proofs and Conditional Proofs The Indirect Proof Rule The Conditional Proof Rule Nested Proofs Proving Sentences Tautological The Law of Noncontradiction Glossary Chapter 10 Replacement Rules The Commutative Rule The Associative Rule The Double Negation Rule DeMorgan's Rule The Distribution Rule Five More Replacement Rules The Transposition Rule The Implication Rule The Exportation Rule The Tautology Rule The Equivalence Rule Are Replacement Rules Worth the Bother? Glossary Chapter 11 Indirect and Conditional Proofs with Replacement Rules Indirect Proofs with Replacement Rules Conditional Proof with Replacement Rules Proving Tautologies Glossary PART II TWO INFORMAL TOPICS Chapter 12 Definition The Purposes of Definition Five Types of Definition Two Types of Meaning Constructing a Definition: Techniques Rules for Intensional Definitions Glossary Chapter 13 Informal Fallacies Fallacies of No Evidence Fallacies of Little Evidence Fallacies of Language Glossary A Summary of the Fallacies PART III ARISTOTELIAN CATEGORICAL LOGIC Chapter 14 The Logic of Categorical Statements Categorical Sentences Quality and Quantity The Traditional Square of Opposition Translating English Sentences into Standard Categorical Forms Equivalence Rules for Aristotelian Logic Dropping the Assumption of Existential Import The Modern Square of Opposition Glossary Chapter 15 Categorical Syllogisms Logical Form Venn Diagrams Testing a Categorical Syllogism for Validity with Venn Diagrams Diagramming Aristotelian Categorical Syllogisms Diagramming from the Boolean Standpoint The Sorites Testing a Sorites with Venn Diagrams Enthememes Refutation by Logical Analogy Appendix: Rules for Evaluating Categorical Syllogisms PART IV MODERN QUANTIFICATIONAL LOGIC Chapter 16 Quantificational Logic I: The Language QL Two Types of Sentences General Sentences A Syntax for our New Language The Vocabulary of QL Symbolizing General Sentences Categorical Sentences The Old "Quantifier Switch" Trick Switching Quantifiers on Categoricals Symbolizing Complicated General Sentences Denying Existence The Only Way to Go What is a Cat-Dog? Glossary Chapter 17 The Language of Quantificational Logic II: Relations Sentences with a Quantifier-Dyadic Predicate Combo Any and Every Reflexive Sentences Sentences with Overlapping Quantifiers "What Are You Talking About?" The Universe of Discourse Dean Martin, Universal Love, and a Summary of Logic Relations To Be or Not To Be: The Logic of Identity The Identity Sign Appendix: Properties of Relations Glossary Chapter 18 Proofs with Monadic Predicates The Universal Instantiation Rule Existential Generalization Existential Instantiation Memories of Geometry Class: Universal Generalization One New Replacement Rule: Quantifier Exchange Naming Our System Glossary Chapter 19 Interpretations, Invalidity, and Semantics Interpretations of Multiply Quantified Sentences Using Interpretations to Show Invalidity Semantics The Monadic Predicate Test Glossary Chapter 20 Conditional and Indirect Quantifier Proofs Adding Truth-Functional Replacement Rules to the Mix Putting QD on a Diet: A Reduced Set of Quantifier Rules Proving Logical Truths Chapter 21 Proofs with Overlapping Quantifiers Properties of Relations Chapter 22 Proofs with Identity Properties of the Identity Relation Glossary PART V MODAL LOGIC Chapter 23 Introductory Modal Logic To Shave or Not to Shave: That Is the Question Five Modal Properties Possible Truths, Possible Falsehoods, Contingencies Necessary Truths Necessary Falsehoods Putting Statements into Symbols Translating English Sentences into Modal Symbols A Name and Syntax for our Modal Language The Vocabulary for ML The Grammar for ML Linking Modal Operators "It Ain't Necessarily So," Or, Trading a Diamond for a Box and a Box for a Diamond Modal Operators Need Scope, Too Modal Relations Scopes of the Dyadic Modal Operators Symbolizing with Dyadic Operators Modal Operators Are Not Truth-Functional Appendix: There's Nothing New under the Sun Glossary Chapter 24 Modal Logic: Methods of Proof Five Modal Principles Six Inference Rules The Possibility to Necessity Rule The Necessitation Rule Four Modal Replacement Rules Validity in S5 Proving Theorems of S5 Another Inference Rule: The Tautology Necessitation Rule Appendix 1: Putting an S5 Formula on a Diet: S5 Reduction Appendix 2: The Modal Fallacy Glossary PART VI INDUCTION Chapter 25 Inductive Reasoning Analogical Reasoning Evaluating Analogical Arguments Analogies as Models Enumerative Induction Statistical Inductive Generalization Inference to the Best Explanation What Makes One Explanation Better than Another? Glossary Chapter 26 Scientific Reasoning Scientific Reasoning Comments on the Steps Confirming and Disconfirming Scientific Hypotheses The Confirmation of a Scientific Hypothesis The Disconfirmation of a Scientific Hypothesis The Fact of the Cross What Makes One Hypothesis Better than Another? Case Studies Cause and Effect and Mill's Method Cause and Effect Mill's Method of Agreement Mill's Method of Difference The Joint Method of Agreement and Difference Mill's Method of Residues Mill's Method of Concomitant Variation Glossary Appendices 1: Truth-Trees 2: Truth Answers to Selected Exercises Index