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Homework notes from the clas

by: Tasfia Kamal

Homework notes from the clas

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Introduction to Logic
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Date Created: 08/26/16
Chapter 1 Exercise 1: (a) Determine whether each of the two arguments is sound or unsound respectively; and (b) Explain your answers by referring to the two constituent properties of soundness. [3.1] 1. Man is immortal. 2. Socrates is a man. 3. Therefore, Socrates is immortal. Answer: To be sound, an argument must meet two conditions. It must (a) be valid and (b) have all true premises. Since the above premise 1 is not true, that means this argument is not sound. Exercise 2: (a) Determine whether the argument given below is deductive or inductive; & (b) Justify your answer by referring to the three criteria on inference. [04.1] Amoco, Exxon, and Texaco are all listed on the New York Stock Exchange. All major oil companies must be listed on the NYSE. Answer: This is an inductive argument because it’s conclusion go beyond what is contained in its premises. [04.2] Anyone who is good at logic is a critical thinker. I am good at logic. So I am a critical thinker. Answer: This is a deductive argument because the conclusion is true based on all the presumably true premises that are given. Assignment 1: When does an argument become fallacious in a way that it "begs the question"? Answer: In logic, an argument is a form of reasoning where one person provides a set of reasons to prove a certain claim. Those reasons are called premises and the end claim or sentence is known as the conclusion. For example, 1. It's wrong to kill a human being. 2. Abortion take the life of (kills) a human being. 3. Therefore, Abortion is wrong. (Hauseman, Kahane & Tidman, 2). Here, sentence 1 & 2 are the premises which support sentence 3 also known as the conclusion. Often in arguments, not all the premises and conclusions are true. When the conclusion is assumed to be true by the premises and the premises used to prove the conclusion, it causes a fallacy within that argument. These types of fallacious arguments are often indicated by the phrase "begging the question" as it is simply based on assumption. For example, the statement, "America has the greatest liberty in the world because there are no countries with better liberty than USA" is based on one's assumption that the liberty of America is the greatest. Here, the person made an assumption about a controversial topic that not everyone would agree with or have different opinions about. Hence, if one's premises prove one's conclusion, but the premises are questionable/non-reliable/believed to be assumed, the argument becomes so fallacious that one is said to "beg the question". Tasfia Kamal July 26, 2016 Professor Kang Introduction to Logic Exercise 27 ... (For-Credit of 10 points) Do the three things with the argument given below: 1. Identify atomic sentences employed in the argument and assign the sentence constants (P, Q, R) to those atomic sentences (in their order of appearance in the argument). 2. Symbolize the whole argument by employing those sentence constants and the proper connectives. 3. Determine whether the following argument is valid or not by demonstrating how Truth Tables can be employed and interpreted: P1: If things are caused to exist, then the infinite regress of existence is not possible. P2: God is not the ultimate cause of existence, "only if" the infinite regress of existence is possible. P3: By the way, things are caused to exist. C: Therefore, God is the ultimate cause of existence. ANSWER: Here, in the first premise, “things are caused to exist” and “the infinite regress of existence is possible” (excluding “not” from the sentence) are the atomic sentences. We can assign “T” for “things are caused to exist” and “I” for “the infinite regress of existence is possible” as sentence constant. Moving to premise two, we have another atomic sentence which is “God is the ultimate cause of existence” (excluding “not” from the sentence) which can be assigned to “G” as sentence constant. Therefore, the atomic sentences derived from the above premises are: T: Things are caused to exist I: The infinite regress of existence is possible G: God is the ultimate cause of existence After identifying the atomic sentences, the next step is to symbolizing the whole argument. Thus, we can say, t1at P = T ⊃2~I, P = ~G 3 I & P = T; which gives us the conclusion that symbolizes C= G. With this, we conclude symbolizing the whole argument. Now, we can construct a Truth Table to see the validity of the argument. T I G T ⊃ ~I ~G ⊃ I T G T T T F T T F F T F T T T T T T F F T F F T T F F T F F F F T F F F F F Here, we see that only cell 4 is consist of all true premises which concludes to having true conclusion. Since, by definition, an argument with all true premises and true conclusion is called a valid argument, we can say that the above argument is valid. Assignment 3: extracted from textbook, Exercises 4-2 & 4-5 (You're encouraged to take on as many exercises as you can; Textbook has answers to even-numbered questions for your self-examination. Exercise 4-8 has many good ones too.) ==================================== [37-1] C: G ------------------- P1: ~M P2: N -> G P3: N v M 4. N 1,3 DS 5. G 2,4 MP (QED) ------------------- ==================================== [37-2] C: D ------------------- P1: ~G -> (A v B) P2: ~B P3: A -> D P4: ~G 5. (A v B) 1,4 MP 6. A 2, 5 DS 7. D 3,6 MP (QED) ------------------- ==================================== [37-3] C: ~B ------------------- P1: A -> (B -> C) P2: ~C P3: ~D -> A P4: C V ~D 5. ~D 2,4 DS 6. A 3, 5 MP 7. B -> C 1,6 MP 8. ~B 2,7 MT (QED) ------------------- ==================================== [37-4] C: D & E ------------------- P1: A -> (~B & C) P2: C -> D P3: E v B P4: A 5. ~B & C 1,4 MP 6. C 5 Simp 7. D 2,6 MP 8. ~ B 5 Simp 9. E 3,8 DS 10. D.E 7,9 Conj (QED) ------------------- ==================================== [37-5] C: ~F ------------------- P1: (F -> G) v H P2: ~G P3: ~H 4. (F -> G) 1,3 DS 5. ~F 2,5 MT (QED) ------------------- ==================================== [37-6] C: L ------------------- P1: ~A P2: (C v A) -> L P3: A v D P4: (D v U) -> C 5. D 1,3 DS 6. D v A 5 Add 7. D v U 4,6 HS 8. C 4,7 MP 9. C v A 8 Add 10. L 2,9 MP (QED) ------------------- ==================================== Assignment 4: Part 1: ---------------------------------------------------- [44-1] Exercise designed to appreciate comparative merit of CP: with the same argument below, [44-1.1] do the 1st proof without using CP; & [44-1.2] do the 2nd proof by using CP: 44-1.1: C: M -> R 1: ~M V N 2: ~R -> ~N 3: M -> N 1 Impl 4: N -> R 2 Contra 5: M -> R 3,4 HS ... QED 44-1.2: C: M -> R 1: ~M V N 2: ~R -> ~N -->3: M AP 4: N 1,3 DS 5: R 2,4 MT 6: M->R 3-5 CP ... QED ---------------------------------------------------- [44-2] Exercise designed to employ CP as part of the whole process: with the same argument below, [44-2.1] do the 1st proof by using CP with ~P as AP [44-2.2] do the 2nd proof by using CP with R as AP; & then Contra [44-2.1] C: ~P -> ~R 1: R -> (L & S) 2: (L V M) -> P --> 3: ~P AP | 4: ~(L V M) 2,3 MT | 5: ~L & ~M 4 DeM | 6: ~L 5 Simp | 7: ~L V ~S 6 Add | 8: ~(L & S) 7 DeM | 9: ~R 1,8 MT --------------------------------- 10: ~P -> ~R 3-9 CP .... QED [44-2.2] C: ~P -> ~R 1: R -> (L & S) 2: (L V M) -> P --> 3: R AP (to get P) | 4: L & S 1,3 MP | 5: L 4 Simp | 6: L V M 5 Add | 7: P 2,6 MP -------------------------------- 8: R -> P 3-7 CP 9: ~P -> ~R 8 Contra ... QED -------------------------------------------------------- [44-3] Exercise designed to see CP as a self-contained module: use CP to the get the 1st part of conjunction; use the conditional as a premise to get the 2nd part; & be careful not to take the 2nd part of conjunction from CP C: (H -> M) & ~F 1: ~H V ~F 2: ~M -> F 3: (~H V M) -> ~F --> 4: H AP | 5: ~F 1,4 DS | 6: M 2,5 MT 7: H -> M 4-6 CP 8: ~H V M 7 Impl 9: ~F 3,8 MP 10: (H->M) & ~F 7,9 Conj ... QED -------------------------------------------------------- [44-4] Exercise designed to employ CP twice to get an equivalence *. 1st proof for A -> B by CP *. 2nd proof for B -> A by CP *. combine them by Conj *. then use Equiv C: A <-> B 1: A -> ~C 2: ~B -> C 3: A V ~B 4: A AP 5: ~ C 1,4 MP 6: ~ ~ B 2,5 MT 7: B 6 DN 8: A -> B 4-7 CP 9: B AP 10: ~ ~ B 9 DN 11: A 3,10 DS 12: B -> A 9-11 CP 13: (A->B) & (B->A) 8,12 Conj 14: A <-> B 13 Equiv ========================================== Part 2: [45-1] C: ~A ------------------- 1: A -> (B & C) 2: ~B 3: A AP 4: B & C 1,3 MP 5: B 4 Simp 6: B & ~B 2,6 Conj 7: ~A 3-6 IP ========================================== [45-2] C: C --------------- 1: A -> ~B 2: B V C 3: A V C 4: ~C AP 5: A 3,4 DS 6: B 2,4 DS 7: ~A 1,6 MT 8: A & ~A 5,7 Conj 9: C 4-8 IP ========================================== [45-3] C: ~(A V C) ------------------- 1: A -> B 2: C -> D 3: (B V D) -> E 4: ~E 5: A V C AP 6: A 5 Simp 7: C 5 Simp 8: B 1-6 MP 9: D 2,7 MP 10: B V D 8,9 Add 11: E 3,10 MP 12: E & ~E 4,11 Conj 13: ~ (A V C) 5-12 IP ========================================== [45-4] C: ~(C & D) ---------------------- 1: ~A 2: (A V B) <-> C 3: ~B 4: C & D AP 5: C 4 Simp 6: [(A V B) -> C] & [C -> (A V B)] 2 Equiv 7: (A V B) -> C 6 Simp 8: A V B 5,7 MP 9: A 3,8 DS 10: A & ~A 1,9 Conj 11: ~ (C &D) 4-10 IP Shadowlands Review: Men have intellect and women have souls Are you trying to be offensive and just being stupid One of the things that stood out to me while watching the movie “Shadowland” is relating to the character of American Poet Joy Gresham. She seems to be an extremely outspoken and courageous woman in the world of men where she is never scared to speak of her mind. One of the first scenes, where Joy first gets to meet C. S. Lewis, the audience sees that she was not bothered by the fact that a room full of people were staring at her when she asked for Lewis. On the contrast, she seemed to be satisfied with her success of finding Lewis in a country she has never been, an author she has never met. I admire this kind of courage and absolutely can relate to her in this aspect. I believe having this kind of courage give us ladies a source of encouragement in conquering any difficulties risen between our goals or achievement. Later, in one of the scenes, the audience witness Joy standing up for the entire women population while confronted with an ignorant comment from one of the British authors. When one of the Oxford professor states, “Men have intellect and women have souls”, even I got extremely angry and wanted to offer my counter comment to that. In this scene, Joy politely answers, “Are you trying to be offensive and just being stupid?” to show that she is not taking anyone’s ignorant comments and will not let anyone humiliate entire women population. In this case, I believe I would have done the same; which is why I was very impressed by Joy’s answer and immediately fell in love with her character. After hearing Joy’s response, Lewis seemed to be very impressed as he shyly moved her away from them. Here, we can see that a shy character like Lewis seemed to be fond of an outspoken character like Joy simply because they are attracted by their opposite characteristics. Not only in movies, but also we see this in real life, where people are usually attracted towards opposite things or elements that they usually do not have. For instance, I am a very short person, hence I am attracted to tall boys; they seem very attractive to me whereas, they can seem to be unattractive to my tall friends. I always wanted to have a big house like my cousin, but when I tell her, she says it is not really a big deal to her and she would have been fine if she did not have a big house. People tend to not realize the important of a certain element while they have it or it exists but immediately realize it’s importance when it is gone. Another life lesson that this movie provides is that nothing is certain in life. You have something at one moment but not have it in the next second. This lesson I learned by viewing the end portion of the movie when the character Joy find out about her cancer; which reminded me the time when my mother had cancer. It was as if, one moment we were a happy and healthy family, but the next moment all of it was gone. This portion, if anything, teaches us to cherish every moment in life because no one knows when it will be taken away from us. Assignment 5: Symbolize & Prove the following argument: 1. All criminals are vicious. 2. Some humans are criminals. C. Thus some humans are vicious. C: (E x (H &xV ) x 1. (x) (C x>V ) x 2. (E )x(H &xC ) x 3. C -a V a 1 UI 4. H &aC a 2 EI 5. H a 4 Simp. 6. C a 4 Simp. 7. V a 3,6 MP 8. H &aV a 5,7 Conj. 9. (E )x(H & x ) x 8 EG (QED) Valid Forms for Sentential Logic Valid Argument 1. Modus Ponens (MP): 5. Conjunction (Conj): Forms of Inference p ▯ q p p /∴ q q /∴ p ⋅ q 2. Modus Tollens (MT): 6. Hypothetical Syllogism (HS): p ▯ q p ▯ q ~ q /∴ ~ p q▯ r /∴ p ▯ r 3. Disjunctive Syllogism (DS): 7. Addition (Add): p ∨ q p /∴ p ∨ q ~ p /∴ q 8. Constructive Dilemma (CD): p ∨ q p ∨ q ~ q /∴ p p ▯ r 4. Simplification (Simp): q▯ s /∴ r ∨ s p ⋅ q /∴ p p ⋅ q /∴ q Valid Equivalence 9. Double Negation (DN): 14. Contraposition (Contra): Forms (Rule of p :: ~~ p ( ▯ q) :: (~ q ▯ ~ p) Replacement) 10. DeMorgan’s Theorem (DeM): 15. Implication (Impl): ~ p ⋅ q) :: (~ p ∨ ~ q) p ▯ q) :: (~ p ∨ q) ~ (p ∨ q) :: (~ p ⋅ ~ q) 16. Exportation (Exp): 11. Commutation (Comm): [p ⋅ q) ▯ r] :: [p ▯ (q ▯ r)] (p ∨ q) :: (q ∨ p) 17. Tautology (Taut): (p ⋅ q) :: (q ⋅ p) p :: (p ⋅ p) 12. Assocation (Assoc): p :: (p ∨ p) [p ∨ (q ∨ r)] :: [(p ∨ q) ∨ r] 18. Equivalence (Equiv): [ ⋅ (q ⋅ r)] :: [(p ⋅ q) ⋅ r] (p ▯ q) :: [(p ▯ q) ⋅ (q ▯ p)] 13. Distribution (Dist): (p ▯ q) :: [(p ⋅ q) ∨ (~ p ⋅ ~ q)] [p ⋅ (q ∨ r)] :: [(p ⋅ q) ∨ (p ⋅ r)] [p ∨ (q ⋅ r)] :: [(p ∨ q) ⋅ (p ∨ r)] Conditional and Conditional Proof Indirect Proof Indirect Proof → p AP /∴ q → ~ p AP /∴ p . . . . . . q q ⋅ ~ q p ▯ q CP p IP Rules for Predicate Logic Rule UI: (u)(. . . u . . .) /∴ (. . . w . . .) Provided: 1. (. . . w . . .) results from replacing each occur- rence of u free in (. . . u . . .) with a w that is either a constant or a variable free in (. . . w ...) (making no other changes). Rule EI: (∃u)(. . . u . . .) /∴ (. . . w . . .) Provided: 1. w is not a constant. 2. w does not occur free previously in the proof. 3. (. . .w . . .) results from replacing each occur- rence of u free in (. . . u . . .) with a w that is free in (. . . . . .) (making no other changes). Rule UG: (. . . u . . .) /∴ (w)(. . . w . . .) Provided: 1. u is not a constant. 2. u does not occur free previously in a line obtained by EI. 3. u does not occur free previously in an assumed premise that has not yet been discharged. 4. (. . . w . . .) results from replacing each occur- rence of u free in (. . . u . . .) with a w that is free in (. . . w . . .) (making no other changes) and there are no additional free occurrences of w already contained in (. . . w . . .). Rule EG: (. . . u . . .) /∴ (∃w)(. . . w . . .) Provided: 1. (. . . w . . .) results from replacing at least one occurrence of u, where u is a constant or a vari- able free in (. . . u . . .) with a w that is free in (... w . . .) (making no other changes) and there are no additional free occurrences of w already contained in (. . . w . . .). Rule QN: (u)(. . . u . . .) :: ~ (∃u) ~ (. . . u . . .) (∃u)(. . . u . . .) :: ~ (u) ~ (. . . u . . .) (u) ~ (. . . u . . .) :: ~ (∃u)(. . . u . . .) (∃u) ~ (. . . u . . .) :: ~ (u)(. . . u . . .) Rule ID: (. . . u . . .) (. . . u . . .) u = w /∴ (. . . w . . .) w = u /∴ (. . . w . . .) Rule IR: /∴ (x)(x = x) Logic and Philosophy A Modern Introduction Eleventh Edition Alan Hausman Hunter College, City University of NewYork Howard Kahane Late of University of Maryland Paul Tidman Mount Union College Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States Logic and Philosophy: © 2010, 2007 Wadsworth, Cengage Learning A Modern Introduction, Eleventh Edition ALL RIGHTS RESERVED. No part of this work covered by the copyright herein may be reproduced, transmitted, stored, Alan Hausman, Howard or used in any form or by any means graphic, electronic, or Kahane, Paul Tidman mechanical, including but not limited to photocopying, Publisher: Clark Baxter recording, scanning, digitizing, taping, Web distribution, Sr. Sponsoring Editor: information networks, or information storage and retrieval systems, except as permitted under Section 107 or 108 of the Joann Kozyrev 1976 United States Copyright Act, without the prior written Assistant Editor: permission of the publisher. Nathan Gamache Editorial Assistant: For product information and technology assistance, contact us at Michaela Henry Cengage Learning Customer & Sales Support1,-800-354-9706 Media Editor: For permission to use material from this text or product, Diane Akerman submit all requests online at Marketing Manager: Further permissions questions can be e-mailed to Mark Haynes Marketing Coordinator: Josh Hendrick Library of Congress Control Number: 2008940689 Content Project Manager: Student Edition: Alison Eigel Zade ISBN-13: 978-0-495-60158-6 Art Director: Faith Brosnan ISBN-10: 0-495-60158-6 Manufacturing Manager: Marcia Locke Wadsworth Senior Rights Account 20 Channel Center Manager—Text: Bob Kauser Boston, MA 02210 Production Service: USA Scratchgravel Publishing Services Cengage Learning products are represented in Canada by Cover Designer: RHDG/ Nelson Education, Ltd. Christopher Harris Cover image: David Muir/ Getty Images For your course and learning solutions, visit Compositor: Macmillan Publishing Solutions Purchase any of our products at your local college store or at our Cover Designer: preferred online store RHDG/Christopher Harris Printed in Canada 1 2 3 4 5 6 7 13 12 11 10 09 Contents Preface to the Eleventh Edition ix Preface to the Tenth Edition xi Chapter One: Introduction 1 1 The Elements of an Argument 1 2 Deduction and Induction 5 3 Deductive Argument Forms 7 4 Truth and Validity 8 5 Soundness 11 6 Consistency 12 7 Contexts of Discovery and Justification 14 8 The Plan of This Book 14 Key Terms 16 Part One: Sentential Logic 17 Chapter Two: Symbolizing in Sentential Logic 19 1 Atomic and Compound Sentences 20 2 Truth-Functions 21 3 Conjunctions 21 4 Non–Truth-Functional Connectives 25 5 Variables and Constants 25 6 Negations 27 7 Parentheses and Brackets 28 8 Use and Mention 29 9 Disjunctions 30 10 “Not Both” and “Neither . . . Nor” 33 11 Material Conditionals 35 12 Material Biconditionals 38 13 “Only If” and “Unless” 40 14 Symbolizing Complex Sentences 41 15 Alternative Sentential Logic Symbols 48 Key Terms 50 v vi Contents Chapter Three: Truth Tables 53 1 Computing Truth-Values 53 2 Logical Form 58 3 Tautologies, Contradictions, and Contingent Sentences 63 4 Logical Equivalences 69 5 Truth Table Test of Validity 70 6 Truth Table Test of Consistency 73 7 Validity and Consistency 74 8 The Short Truth Table Test for Invalidity 76 9 The Short Truth Table Test for Consistency 80 10 A Method of Justification for the Truth Tables 81 Key Terms 85 Chapter Four: Proofs 86 1 Argument Forms 86 2 The Method of Proof: Modus Ponens and Modus Tollens 88 3 Disjunctive Syllogism and Hypothetical Syllogism 91 4 Simplification and Conjunction 93 5 Addition and Constructive Dilemma 94 6 Principles of Strategy 97 7 Double Negation and DeMorgan’s Theorem 103 8 Commutation, Association, and Distribution 106 9 Contraposition, Implication, and Exportation 107 10 Tautology and Equivalence 107 11 More Principles of Strategy 111 12 Common Errors in Problem Solving 115 Key Terms 122 Chapter Five: Conditional and Indirect Proofs 123 1 Conditional Proofs 123 2 Indirect Proofs 132 3 Strategy Hints for Using CP and IP 137 4 Zero-Premise Deductions 138 5 Proving Premises Inconsistent 139 6 Adding Valid Argument Forms 141 7 An Alternative to Conditional Proof? 142 8 The Completeness and Soundness of Sentential Logic 144 9 Introduction and Elimination Rules 146 Key Terms 149 Chapter Six: Sentential Logic Truth Trees 151 1 The Sentential Logic Truth Tree Method 151 2 The Truth Tree Rules 152 3 Details of Tree Construction 154 4 Normal Forms and Trees 160 5 Constructing Tree Rules for Any Function 161 Key Terms 163 Contents vii Part Two: Predicate Logic 165 Chapter Seven: Predicate Logic Symbolization 167 1 Individuals and Properties 167 2 Quantifiers and Free Variables 171 3 Universal Quantifiers 172 4 Existential Quantifiers 177 5 Basic Predicate Logic Symbolizations 178 6 The Square of Opposition 180 7 Common Pitfalls in Symbolizing with Quantifiers 180 8 Expansions 183 9 Symbolizing “Only,” “None but,” and “Unless” 186 Key Terms 190 Chapter Eight: Predicate Logic Semantics 191 1 Interpretations in Predicate Logic 191 2 Proving Invalidity 193 3 Using Expansions to Prove Invalidity 196 4 Consistency in Predicate Logic 197 5 Validity and Inconsistency in Predicate Logic 198 Key Terms 199 Chapter Nine: Predicate Logic Proofs 200 1 Proving Validity 200 2 The Four Quantifier Rules 202 3 The Five Main Restrictions 209 4 Precise Formulation of the Four Quantifier Rules 213 5 Mastering the Four Quantifier Rules 216 6 Quantifier Negation 220 Key Term 225 Chapter Ten: Relational Predicate Logic 226 1 Relational Predicates 226 2 Symbolizations Containing Overlapping Quantifiers 229 3 Expansions and Overlapping Quantifiers 229 4 Places and Times 234 5 Symbolizing “Someone,” “Somewhere,” “Sometime,” and So On 235 6 Invalidity and Consistency in Relational Predicate Logic 240 7 Relational Predicate Logic Proofs 241 8 Strategy for Relational Predicate Logic Proofs 248 9 Theorems and Inconsistency in Predicate Logic 250 10 Predicate Logic Metatheory 253 11 A Simpler Set of Quantifier Rules 254 viii Contents Chapter Eleven: Rationale Behind the Precise Formulation of the Four Quantifier Rules 257 1 Cases Involving the Five Major Restrictions 257 2 One-to-One Correspondence Matters 260 3 Accidentally Bound Variables and Miscellaneous Cases 264 4 Predicate Logic Proofs with Flagged Constants 269 Chapter Twelve: Predicate Logic Truth Trees 272 1 Introductory Remarks 272 2 General Features of the Method 273 3 Specific Examples of the Method 273 4 Some Advantages of the Trees 278 5 Example of an Invalid Argument with at Least One Open Path 279 6 Metatheoretic Results 280 7 Strategy and Accounting 283 Key Terms 285 Chapter Thirteen: Identity and Philosophical Problems of Symbolic Logic 286 1 Identity 286 2 Definite Descriptions 292 3 Properties of Relations 294 4 Higher-Order Logics 297 5 Limitations of Predicate Logic 299 6 Philosophical Problems 303 7 Logical Paradoxes 310 Key Terms 317 Chapter Fourteen: Syllogistic Logic 319 1 Categorical Propositions 319 2 Existential Import 322 3 The Square of Opposition 323 4 Conversion, Obversion, Contraposition 326 5 Syllogistic Logic—Not Assuming Existential Import 329 6 Venn Diagrams 332 7 Syllogisms 334 8 Determining Syllogism Validity 336 9 Venn Diagram Proofs of Validity or Invalidity 337 10 Five Rules for Determining Validity or Invalidity 342 11 Syllogistics Extended 345 12 Enthymemes 348 13 Sorites 349 14 Technical Restrictions and Limitations; Modern Logic and Syllogistic Logic Compared 351 Key Terms 355 Answers to Even-Numbered Exercise Items 358 Bibliography 419 Special Symbols 421 Index 423 Preface to the Eleventh Edition In this new edition of Logic and Philosophy, I have continued the task of ensuring accu- racy of print and thought and, as important, added some discussions of philosophical inter- est. In Chapter Five, I have inserted a section on an alternative to Conditional Proof that, I think, raises issues about the very nature of proofs and their relation to semantics. In Chapter Nine, I bring up issues related to inferences involving arbitrarily selected indi- viduals. In Chapter Twelve, I have added a discussion of alternatives to the flowchart method of constructing predicate trees, and also made some changes in the flowchart itself. The index has been extensively revised and reworked to make it much more con- ceptually accurate and, I hope, user friendly. Part Three of the former editions, which included such diverse subjects as informal fal- lacies and modal logic, has been dropped from the text but is available online. I felt that the standard deductive logic sections of the book were the crucial ones for a one- or even two-semester course. In my own teaching of logic at Hunter College, deductive logic filled two semesters, and surveys of instructors who use this text showed pretty uniform agree- ment about this. My Hunter College colleagues Jim Freeman and Laura Keating have made very help- ful comments and suggestions; Professor Keating’s discussions with me were extensive, and I am most grateful to her for her contributions to this edition. Many of the additions and changes have been motivated by using the book in logic classes at Hunter College. My logic class in spring 2008 was excellent in this regard, and I want to thank the whole class for their contributions—especially Frank Boardman, Ben Herold,Arkady Etkin, and Philip Ross. Mr. Boardman, Mr. Herold, and Mr. Benyade Valencia were an invaluable help during preparation of the index. Several users of the book have also made very helpful suggestions: Ty Barnes, Green River College, and Martin Frické, University ofArizona, who constructed some very use- ful software for use with the book that he kindly sent me permission to reference: see Professor Richard Otte, Department of Philosophy, University of California, Santa Cruz, made a very helpful suggestion (which I adopted and adapted) concerning restrictions on the quantifier rules UG and EG. I am also grateful to Professor Michael Howard, The University of Maine, for a discussion concerning a possible short tabular test for sentential validity. I hope to include a discussion of this idea in the next edition (it came too late for this one). I also want to thank instructors who kindly answered ix x Preface to the Eleventh Edition inquiries from the publisher concerning their views of the book: James R. Goetsch, Jr., PhD, Eckerd College; Allan Hazlett, Fordham University; Michael Liston, University of Wisconsin—Milwaukee; Sulia A. Mason, University of South Carolina; Lucas Mather, Loyola Marymount University; Leemon McHenry, California State University, Northridge; Nathan M. Poage, Houston Community College Central; Larry J. Waggle, Illinois State University, Normal. Their comments were greatly appreciated and useful as I prepared this new edition. I have, throughout, tried to maintain the philosophical spirit that informed Howard Kahane’s original editions of this book. Given my long friendship with him and our innumerable philosophical discussions, I am in a unique position to do this. Kahane’s style of writing is one thing that first attracted many people to the work. My own style is different, and current readers may notice stylistic discontinuities. There is, however, a con- tinuity of idea. Kahane wanted this text to be a philosophical one; he wanted to raise issues about deductive logic. I have attempted to continue to serve that goal in this revision. Alan Hausman Hunter College City University of NewYork Preface to the Tenth Edition This new edition ofLogic and Philosophyhas been quite extensively redone. More specif- ically, the first fourteen chapters, all of which deal with deductive logic, have been care- fully re-edited.A great deal of effort has gone into ensuring the accuracy of the problems and exercises throughout the text. I have provided much new content, mainly of a more philosophical nature. In Chapters Six and Twelve, which deal with sentential and predicate trees respectively, there has been a complete rewrite. Chapter Six now stresses, beyond the construction of the trees, the the- oretical relationship to truth tables, demonstrated by the use of creative exercises. Chapter Twelve incorporates a flow chart for constructing trees, and a long section on metatheory. Much in the way of philosophical discussion has been added to topics such as existential import, the justification of truth tables, and the semantics of predicate logic, and exercises with a philosophical bent have been inserted after at least some of these discussions. The purpose of this book is to provide students with a clear, comprehensible introduc- tiontoacompletesystemforsentential(PartOne)andfirst-orderpredicatelogic(PartTwo). Includedaswellareadiscussionoftherelationshipbetweentraditionalsyllogisticlogicand its modern expression, and a wealth of material in Part Three of a supplementary nature. This text is designed for those who desire a comprehensive introduction to logic that is both rigorous and student friendly. In Parts One and Two, numerous exercise sets take the student from sentential logic through first-order predicate logic with identity. The rules are carefully motivated and compared to other systems of rules for sentential and predi- cate logic. Part Three includes a solid range of additional material, including chapters devoted to informal logic, inductive logic, and modal, epistemic, and deontic logics. Thus the text is designed to provide instructors with maximum flexibility in course design. This great variety of material enables instructors to choose topics of interest to them and best suited to the needs of their students. Through all editions, the goal has been to make symbolic logic understandable for the typical student. Several features of the book contribute to this goal. Help for the Mathematically Anxious or Averse The symbols in a symbolic logic text and the rules for their manipulation are mathemati- cal in character; the discussions of both are philosophical. Logic and Philosophy stresses the relationships between the mathematical and the philosophical in ways that are xi xii Preface to the Tenth Edition designed to engage student interest and at the same time to provide explanations of why the symbols work as they do. I know from sad firsthand experience how often both expla- nation and interest are missing in mathematics courses and texts, and it is such texts that most beginning students in logic have as their model of mathematical thinking. The result is all too often fear and loathing in mathematics courses. Students can overcome math anxiety once they see that logic is intelligible in a sense they can understand. Many students who fear mathematics do not fear the learning of a language. Throughout, the learning of logical systems is treated like the learning of a foreign lan- guage, the happy difference being the absolute rigor of logic’s grammatical rules (some- what analogous to the once popular idea of a universal language to be spoken, Esperanto, which had completely regular grammatical rules). I have found in my own classes that this approach is helpful to many. “Walk-Through” Sections For crucial exercises there are “how-to” type sections where the student is carefully walked through a moderately difficult sample problem step by step. Coverage of Basic Concepts Beginning with the first chapter, attention is paid to such topics as logical form and the relationship between consistency and validity. The fundamental concept of a semantic interpretation is used to provide a unified explanation of such basic concepts as validity, consistency, logical equivalence, and logical implication in both sentential and predicate logic. The changes in the Tenth Edition are built on this foundation. Some of these changes were of course motivated by errors in the previous edition. Obviously, logic texts cannot afford such mistakes. Every effort has been made to make the text and exercises accurate. Just as important, it is my intent in this and future revisions to restore much of the origi- nal philosophical purpose that Howard Kahane had when he wrote the book. Since I was a close friend and was intimately connected to the writing of the first edition, I feel uniquely qualified to undertake this task. Here are some of the highlights of these changes: • Philosophical discussion and justification of basic concepts has markedly increased in many sections. For example, a section on the justification of the truth tables has been substantially revised. There is a discussion of the freeing of variables by existential and universal instantiation that does not occur in most logic texts, and there is an expanded look at the issues surrounding existential import. • I have also added suggestions for philosophical discussions below some of the exer- cises. I must add that I do not envision these discussions as directed at philosophy majors alone. Many of them arise quite naturally from the material itself—questions, for example, that those fearful of math are so often afraid to ask but, if answered, replace anxiety with understanding. I, for example, was a mathematics major as an undergraduate; each year my puzzlement over what was going on in my math classes got worse and worse. When as a graduate student I took a symbolic logic class from Professor Gustav Bergmann, he carefully explained what a function was, and years of mystery dissolved overnight. The concepts of logic are far from cut and dried, and they need not ever be dry. Preface to the Tenth Edition xiii • Chapter Six has been rewritten. Exercises on how to construct the tree rules them- selves show the relationship of the rules to truth tables. • Chapter Twelve has been carefully revised so that the nature and structure of the con- struction of predicate trees is now much clearer. Metatheoretic results have been included as well. Acknowledgments I wish to express my thanks to the many teachers who sent in comments and suggestions about the Ninth Edition. Thanks especially to the following for their helpful comments: Nathan Andersen, Eckerd College; David Gibson, Pepperdine University; Robert Lane, University of West Georgia; Larry J. Waggle, Illinois State University; Nancy Brown, De Anza College;Alan Gorfin, Western New England College; Larry J. Kaye, University of Massachusetts, Boston; and Roderick M. Stewart, Austin College. Christopher Hom, University of California, Santa Cruz, checked Ninth Edition errata sheets. Jon Wulff, Bellevue Community College; Peter Horban, Simon Fraser University; Matthew Phillips, Rutgers University; Jeffrey Buechner, Rutgers University; Reina Hayaki, University of Nebraska, Lincoln; and Lisa Warenski, Union College, all helped with detecting and correcting errors. It is my hope that the Tenth will be a worthy successor to the editions edited by Paul Tidman, who took over from Howard Kahane after he fell ill. Howard Kahane’sLogic and Contemporary Rhetoric, a direct response to student cries for relevance in the late 1960s, was a groundbreaking work that has set a standard for all subsequent informal logic texts. Logic and Philosophy, although not revolutionary in that way, was unique in its informal pedagogical style. Both works typified the intense com- mitment Kahane had to his students and his work. He lived the life of an intellectually honest man. His death in 2001 deprived the world of a fine philosopher, a wonderful human being, and a dear friend. Alan Hausman Hunter College City University of NewYork This page intentionally left blank Logic and Philosophy This page intentionally left blank Chapter One Introduction 1 The Elements of an Argument Consider the following simple example of reasoning: Identical twins often have different IQ test scores.Yet such twins inherit the same genes. So environment must play some part in determining IQ. Logicians call this kind of reasoning an argument. (But they don’t have in mind shouting or fighting. Rather, their concern is arguing for or presenting reasons for a conclusion.) In this case, the argument consists of three statements: 1. Identical twins often have different IQ test scores. 2. Identical twins inherit the same genes. 3. So environment must play some part in determining IQ. The first two statements in this argumenreasons for accepting the third. In logic terms, they are said to be premises of the argument, and the third statement is called the argument’s conclusion. An argument can be defined as a series of statements, one of which is the conclusion (the thing argued for) and the others are the premises (reasons for accepting the conclusion). In everyday life, few of us bother to explicitly label premises or conclusions. We usu- ally don’t even bother to distinguish one argument from another. But good writing pro- vides clues that signal the presence of an argument. Such words as “because,” “since,” and “for” usually indicate that what follows is a premise. And words such as “therefore,” “hence,” “consequently,” and “so” usually signal a conclusion. Similarly, such expressions as “It has been observed that . . . ,” “In support of this . . . ,” and “The relevant data . . .” generally introduce premises, whereas expressions such as “It follows that . . . ,” “The result is . . . ,” “The point of all this is . . . ,” and “The implication is . . .” usually signal conclusions. Here is a simple example: Since it’s wrong to kill a human being, it follows that abortion is wrong, because abortion takes the life of (kills) a human being. 1 2 Introduction In this example, the words “since” and “because” signal premises offered in support of the conclusion signaled by the phrase “it follows that.” Put into textbook form, the argument reads, 1. It’s wrong to kill a human being. 2. Abortion takes the life of (kills) a human being. ∴ 3. Abortion is wrong. The symbol “ ∴” represents the word “therefore” and indicates that what follows is a conclusion. This particular argument has two premises, but an argument may have any number of premises (even only one!) and may be surrounded by or embedded in other arguments. Not just any group of sentences makes an argument. The sentences in an argument must express statements—that is, say something that is either true or false. Many sen- tences are used for other purposes: to ask questions, to issue commands, or to give vent to emotions. In ordinary contexts none of the following express statements: Open the door. (command) Who’s the boss here? (question) Thank goodness! (expression of emotion) Of course, sometimes nondeclarative sentences are indeed used to make statements. “Who’s the boss here?” can be used to make a statement, particularly if the boss is talk- ing. In this case the boss is not really asking a question at all, but rather is saying, “ I am the boss here,” thus declaring a fact under the guise of asking a question. But even if every sentence in a group of sentences expresses a statement, the result is not necessarily an argument. The statements must be related to one another in the appro- priate way. Something must be argued for (the conclusion), and there must be reasons for accepting the conclusion. Thus, mere bald assertions are not arguments, anecdotes gener- ally are not arguments, nor are most other forms of exposition or explanation. It’s impor- tant to understand the difference between rhetoric that is primarily expository or explana- tory and rhetoric that is basically argumentative.A passage that contains only exposition gives us no reason to accept the “facts” in it other than the authority of the writer or speaker, whereas passages that contain arguments give reasons for some of their claims (conclusions) and call for a different sort of evaluation than merely an evaluation of the authority of the writer. Examples Two of the following groups of statements constitute arguments, and two do not. These examples also illustrate that although words such as “therefore” and “because” usually signal the presence of an argument, this is not always the case. 1. I believe in God because that is how I was raised. (This is biography, not an argu- ment. “Because” is used here to indicate the cause of the speaker’s belief—to give an explanation—not to signal a premise.) Introduction 3 2. I believe in God because life has meaning. If there is no God, life would be mean- ingless. (This is an argument. The speaker is advancing a reason to believe that God exists.) Here is the argument put into textbook form: 1. Life has meaning. 2. If there were no God, life would be meaningless. ∴ 3. God exists. (Notice that in this case the word “because” does signal that a premise is to follow.) 3. Biff was obviously afraid of making a commitment to a long-term relationship. Therefore, Susie was not surprised when they eventually broke up. (This is not an argument. This is an explanation of why Susie was not surprised.) 4. We’ll get a tax break if we marry before the end of the year. Therefore, I think we should move our wedding date up and not wait until January. (This is an argument.) 1. We’ll get a tax break if we marry before the end of the year. ∴ 2. We should move our wedding date up and not wait until January. Exercise 1-1 Here are fifteen passages (the first six are from student papers and exams, modestly edited). Determine which contain arguments and which do not. Label the premises and conclusions of those that do, and explain your answers. Paraphrase if that makes things clearer. (Even-numbered items in most exercise sets are answered in a section at the back of the book.) 1. I don’t like big-time college football. I don’t like pro football on TV either. In fact, I don’t like sports, period. 2. My summer vacation was spent working in Las Vegas. I worked as a waitress at the Desert Inn and made tons of money. But I guess I got addicted to the slots and didn’t save too much. Next summer my friend Hal and I are going to work in Reno if we can find jobs there. 3. Well, I have a special reason for believing in big-time college football. After all, I wouldn’t have come here if Ohio State’s football team hadn’t gone to the Rose Bowl, because that’s how I heard about this place to begin with. 4. At the present rate of consumption, the oil will be used up in 20 to 25 years. And we’re sure not going to reduce consumption in the near future. So we’d better start developing solar power, windmills, and other “alternative energy sources” pretty soon. 5. The abortion issue is blown all out of proportion. How come we don’t hear nearly as much about the evils of birth control pills?After all, a lot more potential people are “killed” by the pill than by abortion. 6. I’ve often wondered how they make lead pencils. Of course, they don’t use lead, they use graphite. But I mean, How do they get the graphite into the wood? That’s my 4 Introduction problem. The only thing I can think of is maybe they cut the lead into long round strips and then cut holes in the wood and slip the lead in. 7. Punishment, when speedy and specific, may suppress undesirable behavior, but it cannot teach or encourage desirable alternatives. Therefore, it is crucial to use posi- tive techniques to model and reinforce appropriate behavior that the person can use in place of the unacceptable response that has to be suppressed. —Walter and Harriet Mischel, Essentials of Psychology 8. There was no European language that Ruth could not speak at least a little bit. She passed the time in the concentration camp, waiting for death, by getting other pris- oners to teach her languages she did not know. Thus did she become fluent in Romany, the tongue of the gypsies. —Kurt Vonnegut, Jailbird 9. The death of my brother was another instance in which I realized the inadequacy of the superstition of progress in regard to life.A good, intelligent, serious man, he was still young when he fell ill. He suffered for over a year and died an agonizing death without ever understanding why he lived and understanding even less why he was dying. No theories could provide any answers to these questions, either for him or for me, during his slow and painful death. —Leo Tolstoy, Confession 10. To be sustained under the EighthAmendment, the death penalty must “comport with the basic concept of human dignity at the core of theAmendment”; the objective in imposing it must be “consistent with our respect for the dignity of other men.” Under these standards, the taking of life “because the wrongdoer deserves it” surely must fail, for such a punishment has as its very basis the total denial of the wrongdoer’s dignity and worth. —Justice Thurgood Marshall, Dissenting Opinion in Gregg v. Georgia 11. The electoral college should be abolished. Everyone’s vote should count the same. With the electoral college the votes of those who live in small states count for more. That’s how Bush won the election, even though Gore got more votes. So I think we should do away with it. 12. Every event must have a cause. Since an infinite series of causes is impossible, there must be a first uncaused cause of everything: God. 13. If God were all good, he would want his creatures to always be happy. If God were all powerful, he would be able to accomplish anything he wants. Therefore, God must be lacking in either power or goodness or both. 14. It is now some years since I detected how many were the false beliefs that I had from my earliest youth admitted as true, and how doubtful was everything I had since con- structed on this basis: and from that time I was convinced that I must once for all seriously undertake to rid myself of all the opinions which I had formerly accepted, and commence to build anew from the foundation, if I wanted to establish any firm and permanent structure in the sciences. —René Descartes, Meditations 15. Ifyoucandiscoverabetterwayoflifethanofficeholdingforyourfuturerulers,awell- governed city becomes a possibility. For only in such a state will those rule who are truly rich, not in gold, but in the wealth that makes happiness—a good and wise life. —Plato, The Republic Introduction 5 2 Deduction and Induction Deduction and induction are commonly thought to be the cornerstones of good reasoning. The fundamental logical property of adeductively valid argument is this: If all its prem- ises are true, then its conclusion must be true. In other words, an argument is valid just in case it is impossible for all its premises to be true and yet its conclusion be false.The truth of the premises of a valid argument guarantees the truth of its conclusion. To determine whether or not an argument is valid, one must ask whether there are any possible circumstances under which the premises could all be true and yet the conclusion be false. If not, the argument is valid. If it is possible for the premises to be true and the conclusion false, the argument is invalid.Aninvalid argument is simply an argument that is not valid. The question naturally arises as to why it is impossible for the conclusion of a valid argument to be false if all its premises are true. Why do its premises, if true, “guarantee” the truth of its conclusion? Unfortunately, there is no simple answer to this question. A clear answer for some types of deductive arguments is given in this book. It is revealing to notice that in a typical case the information contained in the conclusion of a deductively valid argument is already “contained” in its premises. We tend not to notice this fact because it is usually contained in the premises implicitly (along with other information not contained in the conclusion). Indeed, cases in which the conclusion is explicitly men- tioned in a premise tend to be rather trivial. Examples Here is an example of a deductively valid argument whose conclusion is implicitly contained in its premises: 1. All wars are started by miscalculation. 2. The Iraq conflict was a war. ∴ 3. The Iraq conflict was started by miscalculation. Having said in the first premise that all wars are started by miscalculation and in the second that the Iraq conflict was a war, we have implicitly said that the Iraq conflict was started by miscalculation. And this is what is asserted by the argument’s conclusion. Here is another example: 1. If Bonny has had her appendix taken out, then she doesn’t have to worry about getting appendicitis. 2. She has had her appendix taken out. ∴ 3. She doesn’t have to worry about getting appendicitis. The first premise states that if she has had her appendix out, then she doesn’t have to worryaboutappendicitis,andthesecond,thatshehasinfacthadherappendixout,which implicitly asserts the conclusion that she doesn’t have to worry about appendicitis. 6


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