DOCSLIB.ORG

Appendix a Answers and Hints to Selected Exercises

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read 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. But this isn’t right. It makes the (false) claim that the string of letters ‘S1 and S2’ (a string that contains the variables ‘S1’ and ‘S2’) is an English sentence, whereas the intention of the original sentence was to say that strings like ‘Snow is white and grass is green’ and ‘Roses are red and violets are blue’ are English sentences. Really, what we want is something like this: If S1 is an English sentence and S2 is another English sentence, then the string consisting of S1, followed by ‘and’, followed by S2, is also an English sentence. Quine (1940, 36) invented a device for saying such things more concisely. In his notation, we could write instead: If S1 is an English sentence and S2 is another English sentence, then pS1 and S2q is also an English sentence. His “corner quotes”, ‘p’ and ‘q’, work like regular quotation marks, except when it comes to variables of the metalanguage such as ‘S1’ and ‘S2’. Expressions other than such variables simply refer to themselves within corner quotes, just as all expressions do within regular quotation marks. But metalanguage variables refer to their values—i.e., the linguistic expressions they stand for—rather than themselves, within Quine’s corner quotes. Thus, pS1 and S2q means the same as: the string consisting of S1, followed by ‘and’, followed by S2 Exercise 1.3 Let sentence S1 be ‘There exists an x such that x and x are identical’, and let S2 be ‘There exists an x such that there exists a y such that x and y are not identical’. Does S1 logically imply S2 according to the modal criterion? Well, that depends. It depends on what is possible. You might think that there could have existed only a single thing, in which case S1 would be true and S2 would be APPENDIX A. ANSWERS AND HINTS 337 false. If this is indeed possible, then S1 doesn’t logically imply S2 (given the modal criterion). But some people think that numbers exist necessarily, and in particular that it’s necessarily true that the numbers 0 and 1 exist and are not identical. If this is correct, then it wouldn’t be possible for S1 to be true while S2 is false (since it wouldn’t be possible for S2 to be false.) And so, S1 would logically imply S2, given the modal criterion. How about according to Quine’s criterion? Again, it depends—in this case on which expressions are logical expressions. If (as is commonly supposed) ‘there exists an x such that’, ‘there exists a y such that’, ‘not’, and ‘are identical’ are all logical expressions, then all expressions in S1 and S2 are logical expressions. So, since each sentence is in fact true, there’s no way to substitute nonlogical expressions to make S1 true and S2 false. So S1 logically implies S2 (according to Quine’s criterion). But suppose ‘are identical’ is not a logical expression. Then S1 would not logically imply S2, according to Quine’s criterion. For consider the result of substituting the predicate ‘are both existent’ for ‘are identical’. S1 then becomes true: ‘There exists an x such that x and x are both existent’, whereas S2 becomes false: ‘There exists an x such that there exists a y such that x and y are not both existent’. Exercise 1.4 Here is the de nition of the powerset of A: u : u A . The f ⊆ g powerset of 2,4,6 is ?, 2 , 4 , 6 , 2,4 , 2,6 , 4,6 , 2,4,6 . Notice that the powersetf of a setg alwaysf f containsg f g f g bothf theg f nullg setf andg f the setgg itself (look at the de nition of ‘subset’ to see why this is so.) Exercise 1.5 N and Z are equinumerous, because of the following function f : f (0) = 0, f (1) = 1, f (2) = 1, f (3) = 2, f (4) = 2, f (5) = 3, f (6) = 3,.... This function can be de ned− more rigorously as follows:− − ( n if n is even f n − 2 (for any n N) ( ) = n+1 2 if n is odd 2 Exercise 2.8 Hint: instead of trying to show directly that every wff without repetition of sentence letters has the feature of PL-invalidity, nd some feature F that is stronger than PL-invalidity (i.e., some feature F from which PL- invalidity follows), and show by induction that every wff without repeated sentence letters has this feature F ; and then, nally, conclude that every wff without repeated sentence letters is PL-invalid. APPENDIX A. ANSWERS AND HINTS 338 Exercise 2.10 Hint: call a sequent Γ φ valid iff Γ φ; prove by induction that every provable sequent is a valid sequent.) Exercise 3.7 We’re to show that there are no valid formulas in Kleene’s system. Consider the trivalent interpretation that assigns # to every sentence I letter. If there existed any Kleene-valid formula φ then KV (φ) would need to be 1, whereas we can show by induction that KV ( ) = # forI every wff . Base case: all the sentence letters are obviously # in I. Inductive step: assume that φ and are both # in . We need now to showI that φ , φ , and φ are all # in . But that’sI easy—just look at the truth tables^ for _, and . !# # is #, # # is I#, and # # is #. ^ _ ! ^ _ ! Exercise 3.8 Hint: use induction. Exercise 3.11 Hint: in each direction, prove the contrapositive. Exercise 3.8 might come in handy. Exercise 3.15 Hint: this isn’t hard, but it’s a bit tricky. It might help to note that every classical (bivalent) interpretation also counts as a trivalent interpretation, with itself as its only precisi cation. Exercise 3.16 We’re to argue that contraposition and reductio should fail, given a supervaluational semantics for (assuming the identi cation of truth with truth-in-all-sharpenings). Contraposition:4 as argued in the text, for all φ, φ logically implies “de nitely, φ”. So ‘Middling Mary is rich’ logically implies ‘Middling Mary is de nitely rich’. But ‘not: de nitely, Middling Mary is rich’ doesn’t logically imply ‘not: Middling Mary is rich’, since if Mary is a (de nite) borderline case of being rich, the rst is true on all sharpenings and hence is true, while the second is false under some sharpenings and so is not true. So to model these results, it should turn out under the supervaluationist semantics that P P but P 2 P. As for4 reductio,∼4 “Mary∼ is rich and Mary is not de nitely rich” cannot be true (on logical grounds), and so vacuously implies anything at all. (If it were true, then it would be true on all sharpenings; but then ‘Mary is rich’ would be true on all sharpenings; but then ‘Mary is not de nitely rich’ would be false.) So in particular, it logically implies both ‘Snow is white’ and ‘Snow is not white’ (say). But, contrary to reductio, ‘not: Mary is rich and Mary is not de nitely rich’ is not a logical truth, since it isn’t true. For there are sharpenings in which APPENDIX A. ANSWERS AND HINTS 339 both ‘Mary is rich’ and ‘Mary is not de nitely rich’ are true. Exercise 3.17 For the systems of Łukasiewicz, Kleene, and Priest, we are to nd intuitionistically provable sequents whose premises do not semantically imply their conclusions. Let’s begin with Kleene’s system. We showed in exercise 3.7 that there are no Kleene-valid wffs. Thus, ? 2K P P. But the following is an intuitionistically acceptable proof of the sequent ! P P: ? ) ! 1. P P RA (for conditional proof) ) 2. P P 1, I ? ) ! ! Next, ŁV ( (P P)) = # for any trivalent assignment in which P is #, I ∼ ∧∼ I so (P P) is Łukasiewicz-invalid. But ? (P P) is intuitionistically provable:∼ ∧∼ ) ∼ ∧∼ 1. P P P P RA (for reductio) ∧∼ ) ∧∼ 2.
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]
  • 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 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]