DOCSLIB.ORG

PHIL12A Section Answers, 23 February 2011

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
PHIL12A Section Answers, 23 February 2011 PHIL12A Section answers, 23 February 2011 Julian Jonker 1 How much do you know? 1. The following questions are adapted form exercises 5.1-5.6. Decide whether each pattern of inference is valid. If it is, show that it is using truth tables. If it is not, give example sentences that show how the conclusion can be false though the premises are true. (a) From P and :Q, infer P ^ Q. This is invalid, as the following sentences exemplify: 1 P = Logic is fun. True 2 :Q = Logic is not easy. True 3 P ^ Q = Logic is fun and easy. False (b) From :P _:Q and :P , infer :Q. This is invalid, as the following sentences exemplify: 1 :P _:Q = Either soft drinks are unhealthy or water is unhealthy. True 2 :P = Soft drinks are unhealthy. True 3 :Q = Water is unhealthy. False (c) (Ex 5.6) From P ^ Q and :P , infer R. This is valid, as a truth table will show. You need to show that every row of the truth table which makes both P ^ Q and :P true also makes R true. However, because P ^ Q and :P are contradictory, there is no row of the truth table which makes both of them true (try it!). So the definition for TT validity is (vacuously) fulfilled. 1 2. Write informal proofs for the following arguments, using proof by cases. Be as explicit as possible about each step in your proof. (a) Suppose you know that (Cube(a)^Tet(b))_(Cube(c)^Tet(b)). Show that Tet(b) follows. There are two cases to consider. First, suppose that (Cube(a)^Tet(b)) is true. Since both of the conjuncts of a true conjunction are true, we can conclude that Tet(b) is true. Second, consider that (Cube(c)^Tet(b)) is true. Again, both the conjuncts of a true conjunction are true, so we conclude that Tet(b) is true. Since Tet(b) follows in each case, we conclude that Tet(b) is true. (b) Suppose you know that (Cube(a)^Small(b))_(Cube(c)^Small(c)). Show that Small(b)_Small(c) follows. There are two cases to consider. First, suppose that Cube(a)^Small(b) is true. Since both con- juncts of a conjunction are true, we know that Small(b) is true. But then we can conclude that Small(b)_Small(c) is true, since a disjunction is true whenever one of the disjuncts is true. Second, suppose that Cube(c)^Small(c) is true. Since both conjuncts of a conjunction are true, we know that Small(c) is true. But then we can conclude that Small(b)_Small(c) is true, since a disjunction is true whenever one of the disjuncts is true. It follows that Small(b)_Small(c) is true, since it is true in every case. (c) (Ex 5.7) 1 Home(max) _ Home(claire) 2 :Home(max) _ Happy(carl) 3 :Home(claire _ Happy(carl) 4 Happy(carl) We want to show that if we make the premises of the argument true, the conclusion must be true. In holding premise 1 true, there are two cases we can consider: either Max is home or Claire is home. First, suppose that Max is home. By the second premise, Carl is happy, since at least one of the disjuncts of the disjunction must be true. This also makes the third premise true. Second, suppose that Claire is home. By the third premise, Carl is happy, since at least one of the disjuncts of the disjunction must be true. This also makes the second premise true. It follows that Carl is happy, since we can conclude this in every possible case. 2 3. Write informal proofs to show that the following arguments are valid. (a) (Ex 5.16) 1 Max or Claire is at home but either Scruffy or Carl is unhappy. 2 Either Max is not home or Carl is happy. 3 Either Claire is not home or Scruffy is unhappy. 4 Scruffy is unhappy Premise 1 is a conjunction, and when we hold it to be true we hold both conjuncts to be true. So we accept that Max or Claire is at home, since this is one of the conjuncts. Now we use a proof by cases to show that Scruffy is unhappy. There are two cases to consider. In the first case, we suppose that Max is at home. Then by the second premise we know that Carl is happy. But not that by premise 1, it is true that either Scruffy or Carl is unhappy. This is a disjunction, so one of the disjuncts must be true. We conclude that in this case Scruffy is unhappy. In the second case, we suppose that Claire is at home. The third premise is a disjunction, so one of the disjuncts must be true. Therefore Scruffy is unhappy in this case. It follows that Scruffy is unhappy, since this is true in every case. (b) (Ex 5.17) 1 Cube(a) _ Tet(a) _ Large(a) 2 :Cube(a) _ a=b _ Large(a) 3 :Large(a) _ a=c 4 :(c=c ^ Tet(a)) 5 a=b _ a=c In order to hold premise 1 true, at least one of the disjuncts must be true. Let’s consider each disjunct: there are three cases. In each case we wish to show that the conclusion, a=b_a=c, holds. Case 1: a is a cube. By premise 2 it follows that a and b are identical, or a is large, or both. Now we show by cases that a=b_a=c holds. Suppose first that a and b are identical. Then the disjunction a=b_a=c is true since the first disjunct is true. Suppose next that a is large. Then by premise 3 we have that a and c are identical. But then it follows that the disjunction a=b_a=c is true, since the second disjunct is true. It follows that a=b_a=c is true in case 1. Case 2: a is a tetrahedron. By premise 4 we have that either c is not identical to c or a is not a tetrahedron. In either case, we have a contradiction! But anything follows from a contradiction: including a=b_a=c. 3 Case 3: a is large. By premise 3 we have that a and c are identical. But then we know that a=b_a=c is true, since the second disjunct is true. It follows that a=b_a=c is true, since it follows in each case. Note that the our proof contained proofs by cases embedded within a proof by cases. The structure of this would have been much easier to follow if we had uses a formal proof! 4. Construct formal proofs for the following arguments. (a) (Ex 6.4) 1 (A ^ B) _ C 2 C _ B Proof: 1 (A ^ B) _ C 2 (A ^ B) 3 B ^Elim: 2 4 C _ B _Intro: 3 5 C 6 C _ B _Intro: 5 7 C _ B _Elim: 1, 2-4, 5-6 4 (b) (Ex 6.2) 1 P _ (Q ^ R) 2 (P _ Q) ^ (P _ R) Proof: 1 P _ (Q ^ R) 2 P 3 P _ Q _Intro: 2 4 P _ R _Intro: 2 5 (P _ Q) ^ (P _ R) ^Intro: 3,4 6 Q ^ R 7 Q ^Elim: 6 8 P _ Q _Intro: 7 9 R ^Elim: 6 10 P _ R _Intro: 9 11 (P _ Q) ^ (P _ R) ^Intro: 8, 10 12 (P _ Q) ^ (P _ R) _Elim: 1, 2-5, 6-11 5 (c) For the following proof, you should recall that you can nest subproofs within subproofs. 1 (P _ Q) ^ (P _ R) 2 P _ (Q ^ R) (I will use Reit steps in some of these proofs to clarify things, even though not strictly necessary.) Proof: 1 (P _ Q) ^ (P _ R) 2 P _ Q ^Elim: 1 3 P 4 P _ (Q ^ R) _Intro: 3 5 Q 6 P _ R ^Elim: 1 7 P 8 P _ (Q ^ R) _Intro: 7 9 R 10 Q Reit: 5 11 Q ^ R ^Intro: 9, 10 12 P _ (Q ^ R) _Intro: 11 13 P _ (Q ^ R) _Elim: 6, 7-8, 9-12 14 P _ (Q ^ R) _Elim: 2, 3-4, 5-13 6 (d) (Ex 6.8) 1 P ^ (Q _ ::R) 2 ::P ^ (Q _ R) 1 P ^ (Q _ ::R) 2 P ^Elim: 1 3 :P 4 P Reit: 2 5 ??Intro: 3,4 6 ::P :Intro: 3-5 7 Q _ ::R ^Elim: 1 8 Q 9 Q _ R _Intro: 8 10 ::R 11 R :Elim: 10 12 Q _ R _Intro: 11 13 Q _ R _Elim: 7, 8-9, 10-12 14 ::P ^ (Q _ R) ^Intro: 6,13 7 (e) 1 :(P _ Q) 2 :P ^ :Q Proof: 1 :(P _ Q) 2 P 3 P _ Q _Intro: 2 4 :(P _ Q) Reit: 1 5 ??Intro: 3,4 6 :P :Intro: 2-5 7 Q 8 P _ Q _Intro: 7 9 :(P _ Q) Reit: 1 10 ??Intro: 8,9 11 :Q :Intro: 10 12 :P ^ :Q ^Intro: 6,11 8 (f) (Ex 6.19) 1 A _ B 2 :B _ C 3 A _ C Here’s one proof: 1 A _ B 2 :B _ C 3 A 4 A _ C _Intro: 3 5 B 6 :B 7 ??Intro: 5,6 8 A _ C ?Elim: 7 9 C 10 A _ C _Intro: 9 11 A _ C _Elim: 2, 6-8, 9-10 12 A _ C _Elim: 1, 3-4, 5-11 (Note that you might have found other proofs for some of these exercises.
Load more

Recommended publications
  • Logic Is Awesome. PROOF THAT LOGIC IS AWESOME
    PROOF THAT LOGIC IS AWESOME Call the following sentence P: “If this conditional is true, then Logic is Awesome”. I will now prove that P is in fact true. P is a conditional. I will assume its antecedent and then show that its consequent would follow. So we assume that “this conditional is true”. “this conditional” refers to sentence P so I now have by assumption that P is true. But since P is true and we also have its antecedent, then its consequent follows by modus ponens so given my assumption of “this conditional is true” then Logic is Awesome so by conditional proof (→Intro), I have now proved P. But then the antecedent “this conditional is true” is in fact true since I have just proved that P is true. Therefore by modus ponens, it follows that: Logic is Awesome. Monday, May 5, 2014 SUMMARY OF THE COURSE Monday, 5 May Monday, May 5, 2014 WHAT IS LOGIC? Monday, May 5, 2014 WHAT IS LOGIC? Reasoning is ubiquitous in every aspect of life. Arguably, the foundation of reasoning is consistency. Consistency is intimately related to the concept of logical consequence. Monday, May 5, 2014 WHAT IS LOGIC? Reasoning is ubiquitous in every aspect of life. Arguably, the foundation of reasoning is consistency. Consistency is intimately related to the concept of logical consequence. We say that some proposition P is a logical consequence of a set of propositions S if it is impossible for every proposition in S to be true and P to be false. Monday, May 5, 2014 WHAT IS LOGIC? Reasoning is ubiquitous in every aspect of life.
    [Show full text]
  • The Logic of Boolean Connectives
    Introduction & Review Truth Tables Logical Truths & Tautologies Equivalence Consequence Introduction & Review Truth Tables Logical Truths & Tautologies Equivalence Consequence Announcements For 09.15.11 1 HW1 was due on Tuesday The Logic of Boolean Connectives • If you didn't turn it in, it's late 2 HW2 & HW3 are due next Tuesday (09.20) Truth Tables, Tautologies & Logical Truths • Electronic HW must be submitted before class • Written HW must be handed in at beginning of class William Starr • Otherwise, it's late 3 Optional sections have been scheduled • Wednesday 1:25-2:10, Uris 307 09.15.11 • Thursday 1:25-2:10, Uris G22 William Starr j Phil 2621: Minds & Machines j Cornell University 1/45 William Starr j Phil 2621: Minds & Machines j Cornell University 2/45 Introduction & Review Truth Tables Logical Truths & Tautologies Equivalence Consequence Introduction & Review Truth Tables Logical Truths & Tautologies Equivalence Consequence Outline Introduction Truth Functions • Last class, we learned the meaning of ^; _; : in terms 1 Introduction & Review of truth functions • We also saw that truth functions allowed us to do 2 Truth Tables something useful: • Figure out the truth value of a complex sentence from 3 Logical Truths & Tautologies the truth values of its atomic parts, and vice versa • For example, we know that :(Cube(a) _ Cube(b)) is true in a world where neither a nor b are cubes, since: 4 Equivalence • If :(Cube(a) _ Cube(b)) is true, then Cube(a) _ Cube(b) is false 5 Consequence • If Cube(a) _ Cube(b) is false, then Cube(a) and
    [Show full text]
  • Boolean Connectives and Formal Proofs
    Formal Proofs in Fitch The Boolean Connectives Logik f¨urInformatiker Logic for computer scientists Boolean Connectives and Formal Proofs Till Mossakowski WiSe 2009/10 Till Mossakowski Logic Formal Proofs in Fitch The Boolean Connectives Formal proofs in Fitch P Q R S1 Justification 1 ::: ::: Sn Justification n S Justification n+1 Till Mossakowski Logic Formal proofs / 55 Notice that on the right of every step below the Fitch bar, we give a justification of the step. In our deductive system, a justification indicates justification which rule allows us to make the step, and which earlier steps (if any) the rule is applied to. In giving an actual formal proof, we will number the steps, so we can refer to them in justifying later steps. We already gave one example of a formal proof in the system , back on F page 48. For another example, here is a formalization of our informal proof of the symmetry of identity. 1. a = b 2. a = a = Intro 3. b = a = Elim: 2, 1 In the right hand margin of this proof you find a justification for each step Formal Proofs in Fitch belowThe the Boolean Fitch Connectives bar. These are applications of rules we are about to introduce. Fitch rule: IdentityThe numbers introduction at the right of step 3 show that this step follows from steps 2 and 1 by means of the rule cited. The first rule we use in the above proof is Identity Introduction. This = Intro rule allows you to introduce, for any name (or complex term) n in use in the proof, the assertion n = n.
    [Show full text]
  • Glossary of Logical Terms
    Glossary Antecedent: The antecedent of a conditional is its first component clause (the if clause). In P → Q, P is the antecedent and Q is the consequent. Antisymmetric: a binary relation R is antisymmetric iff no two things ever bear R to one another, i.e., R satisfies the condition that ∀x∀y [(R(x, y) ∧ R(y, x)) → x = y]. Argument: “Argument” is used in two different senses in logic. 1. Arguments as pieces of reasoning: an argument is a sequence of statements in which one (the conclusion) is supposed to follow from or be supported by the others (the premises). 2. Arguments in the mathematical sense: an argument is an individual symbol (variable or constant) taken by a predicate in an atomic wff. In the atomic wff LeftOf(x, a), x and a are the arguments of the binary predicate LeftOf. Aristotelian forms (A, E, I, O): The four main sentence forms treated in Aristotle’s logic: the A form (universal affirmative) All P’s are Q’s, the E form (universal negative) No P’s are Q’s, the I form (particular affirmative) Some P’s are Q’s, and the O form (particular negative) Some P’s are not Q’s. Arity: The arity of a predicate indicates the number of individual constants (names) it takes to combine with the predicate to form a complete sentence. A predicate with an arity of one is called unary. A predicate with an arity of two is called binary. It’s possible for a predicate to have any arity, so we can talk about 6-ary or even 113-ary predicates.
    [Show full text]
  • Logik Für Informatiker Formal Proofs for Propositional Logic
    The rule system of Fitch (natural deduction) Logik f¨urInformatiker Formal proofs for propositional logic Till Mossakowski, Lutz Schr¨oder WiSe 2011/12 Till Mossakowski, Lutz Schr¨oder Logic The rule system of Fitch (natural deduction) The truth table method A sentence is a tautology if and only if it evaluates to TRUE in all rows of its complete truth table. Truth tables can be constructed with the program Boole. Till Mossakowski, Lutz Schr¨oder Logic The rule system of Fitch (natural deduction) Tautological equivalence and consequence Two sentences P and Q are tautologically equivalent, if they evaluate to the same truth value in all valuations (rows of the truth table). Q is a tautological consequence of P1;:::; Pn if and only if every row that assigns TRUE to each of P1;::: Pn also assigns TRUE to Q. If Q is a tautological consequence of P1;::: Pn, then Q is also a logical consequence of P1;:::; Pn. Some logical consequences are not tautological ones. Till Mossakowski, Lutz Schr¨oder Logic The rule system of Fitch (natural deduction) Tautological equivalence and consequence Two sentences P and Q are tautologically equivalent, if they evaluate to the same truth value in all valuations (rows of the truth table). Q is a tautological consequence of P1;:::; Pn if and only if every row that assigns TRUE to each of P1;::: Pn also assigns TRUE to Q. If Q is a tautological consequence of P1;::: Pn, then Q is also a logical consequence of P1;:::; Pn. Some logical consequences are not tautological ones. Till Mossakowski, Lutz Schr¨oder Logic The rule system of Fitch (natural deduction) Tautological equivalence and consequence Two sentences P and Q are tautologically equivalent, if they evaluate to the same truth value in all valuations (rows of the truth table).
    [Show full text]
  • The Logic of Quantifiers
    Introduction FO Validity FO Consequence Introduction FO Validity FO Consequence Announcements 03.31 The Logic of Quantifiers Logical Truth & Consequence in Full Fol 1 The Midterm has been returned If you haven't gotten yours back, see me after class William Starr 03.31.09 William Starr | The Logic of Quantifiers (Phil 201.02) | Rutgers University 1/34 William Starr | The Logic of Quantifiers (Phil 201.02) | Rutgers University 2/34 Introduction FO Validity FO Consequence Introduction FO Validity FO Consequence Outline Overview The Big Picture Now that we've added 8 and 9, we have introduced every connective of fol: 1 Introduction 8 9 $ ! _ ^ : = For six of these symbols we've studied: 2 FO Validity 1 It's semantics: truth-tables, satisfaction, game rules 2 How to translate English sentences using it 3 FO Consequence 3 It's role in logic: which sentences containing it are logical truths and which arguments containing it are valid 4 It's role in proofs: which inference steps and methods of proof it supports and how these can be formalized For 8 and 9, we've only done the first two Today, we'll get started on the third! William Starr | The Logic of Quantifiers (Phil 201.02) | Rutgers University 3/34 William Starr | The Logic of Quantifiers (Phil 201.02) | Rutgers University 5/34 Introduction FO Validity FO Consequence Introduction FO Validity FO Consequence Overview The Logical Concepts Today Logical Truth & Logical Consequence Logical Truth A is a logical truth iff it is impossible for A to be false given the meaning of the logical vocabulary it
    [Show full text]
  • Teaching Philosophy
    Teaching Philosophy Jonathan Simon At the introductory level, I believe the philosophy professor's task is to show the student how to simultaneously regard the world with more won- der and more critically than before. These two things go together, since the more critically one engages the more one sees the riddles, puzzles and open questions everywhere. I generally prefer a `bottom-up' approach, teaching Socratically or conversationally where possible, beginning with issues more likely to independently pique interest and stimulate conversation (trolley problems, the existence of God, etc), then gradually developing rigor, con- necting the more concrete to the more abstract as the course goes on. This is in part because in my experience it is what works, but in part because I hold the substantive philosophical position that the more abstract questions (of, say, meta-ethics or metaphysics) are ultimately inseparable from the more concrete questions which lead to them. I have taught absolute introductory courses like Central Problems in Philosophy (Fall 2009), and Intro to Philos- ophy (Summer 2010) in this manner, and also advanced introductory courses like Metaphysics (2010) and Minds and Machines (2009). 1 Teaching Statement Jonathan Simon November 2016 I have also taught Logic (twice so far, in Fall 2010 and Spring 2011). Logic is a special case since the fascinating questions at the core of the discipline are often passed over in favor of a more skills driven approach. I attempt to balance the desiderata that students learn all of the requisite skills, with the desiderata that they attain some perspective on why these skills matter and on the philosophical questions that drive the discipline.
    [Show full text]
  • Basic Concepts in Modal Logic1
    Basic Concepts in Modal Logic1 Edward N. Zalta Center for the Study of Language and Information Stanford University Table of Contents Preface Chapter 1 { Introduction x1: A Brief History of Modal Logic x2: Kripke's Formulation of Modal Logic Chapter 2 { The Language Chapter 3 { Semantics and Model Theory x1: Models, Truth, and Validity x2: Tautologies Are Valid x2: Tautologies Are Valid (Alternative) x3: Validities and Invalidities x4: Validity With Respect to a Class of Models x5: Validity and Invalidity With Repect to a Class x6: Preserving Validity and Truth Chapter 4 { Logic and Proof Theory x1: Rules of Inference x2: Modal Logics and Theoremhood x3: Deducibility x4: Consistent and Maximal-Consistent Sets of Formulas x5: Normal Logics x6: Normal Logics and Maximal-Consistent Sets Chapter 5 {Soundness and Completeness x1: Soundness x2: Completeness Chapter 6 { Quantified Modal Logic x1: Language, Semantics, and Logic x2: Kripke's Semantical Considerations on Modal Logic x3: Modal Logic and a Distinguished Actual World 1Copyright c 1995, by Edward N. Zalta. All rights reserved. 1 Preface These notes were composed while teaching a class at Stanford and study- ing the work of Brian Chellas (Modal Logic: An Introduction, Cambridge: Cambridge University Press, 1980), Robert Goldblatt (Logics of Time and Computation, Stanford: CSLI, 1987), George Hughes and Max Cresswell (An Introduction to Modal Logic, London: Methuen, 1968; A Compan- ion to Modal Logic, London: Methuen, 1984), and E. J. Lemmon (An Introduction to Modal Logic, Oxford: Blackwell, 1977). The Chellas text influenced me the most, though the order of presentation is inspired more by Goldblatt.2 My goal was to write a text for dedicated undergraduates with no previous experience in modal logic.
    [Show full text]
  • Language, Proof and Logic
    Language, Proof and Logic Language, Proof and Logic Second Edition Dave Barker-Plummer, Jon Barwise and John Etchemendy in collaboration with Albert Liu, Michael Murray and Emma Pease Copyright © 1999, 2000, 2002, 2003, 2007, 2008, 2011 CSLI Publications Center for the Study of Language and Information Leland Stanford Junior University First Edition 1999 Second Edition 2011 Printed in the United States 15 14 13 12 11 1 2 3 4 5 Library of Congress Cataloging-in-Publication Data Barker-Plummer, Dave. Language, proof, and logic. { 2nd ed. / Dave Barker-Plummer, Jon Barwise, and John Etchemendy in collaboration with Albert Liu, Michael Murray, and Emma Pease. p. cm. { Rev. ed. of: Language, proof, and logic / Jon Barwise & John Etchemendy. Includes index. ISBN 978-1-57586-632-1 (pbk. : alk. paper) 1. Logic. I. Barwise, Jon. II. Etchemendy, John, 1952- III. Barwise, Jon. Language, proof, and logic. IV. Title. BC71.B25 2011 160{dc23 2011019703 CIP 1 The acid-free paper used in this book meets the minimum requirements of the American National Standard for Information Sciences|Permanence of Paper for Printed Library Materials, ANSI Z39.48-1984. Acknowledgements Our primary debt of gratitude goes to our main collaborators on this project: Gerry Allwein and Albert Liu. They have worked with us in designing the entire package, developing and implementing the software, and teaching from and refining the text. Without their intelligence, dedication, and hard work, LPL would neither exist nor have most of its other good properties. In addition to the five of us, many people have contributed directly and in- directly to the creation of the package.
    [Show full text]
  • Appendix a Answers and Hints to Selected Exercises
    Appendix A Answers and Hints to Selected Exercises Exercise 1.1a “‘P P’ is a logical truth” is a sentence of the metalanguage, and (I would say) is false.∨∼ ‘P P’ contains the meaningless letter ‘P’, so it isn’t a logical truth. Rather, it represents∨∼ logical truths (assuming the law of the excluded middle is correct! See chapter 3.) Exercise 1.1b ‘(P Q) (Q P)’ is a sentence of the object language. Since it contains meaningless_ expressions! _ (‘P’, ‘Q’), it isn’t true. (Not that it’s false!) Exercise 1.1c This is a bit of a trick question. “‘Frank and Joe are brothers’ logically implies ‘Frank and Joe are siblings’” is a sentence of English, which is talking about further sentences of English. So English is functioning here both as the object language and as the metalanguage. As for whether the sentence is true, I would say no, since the implication is not “formal”. Exercise 1.2a ‘Attorney and lawyer are synonyms’ confuses use and mention; inserting quotation marks thus xes the problem: ‘Attorney’ and ‘lawyer’ are synonyms. Exercise 1.2b How can we insert quotation marks to remove the use-mention confusion in ‘If S1 is an English sentence and S2 is another English sentence, then the string S1 and S2 is also an English sentence’? This is again a bit of a trick question. You might think to do it this way: 335 APPENDIX A. ANSWERS AND HINTS 336 If S1 is an English sentence and S2 is another English sentence, then the string ‘S1 and S2’ is also an English sentence.
    [Show full text]