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CC3015 Logic

CC3015 Logic. Lecture 1 Introduction. Zeno of Elea 芝諾 490-425 BC. Logic 邏輯. For this course, logic, as a discipline, is the study of the principles of correct reasoning (or inference) 邏輯研究正確 推理 的原理. Reasoning 推理.

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CC3015 Logic

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  1. CC3015Logic Lecture 1 Introduction Zeno of Elea 芝諾 490-425 BC 12-13 Semester 2 Acknowledgment: the lecture ppts of this subject are prepared by Mr KC Wong

  2. Logic邏輯 • For this course, logic, as a discipline, is the study of the principles of correct reasoning (or inference) 邏輯研究正確推理的原理 12-13 Semester 2

  3. Reasoning推理 “The goal of reasoning is to find out, from the consideration of what we already know, something else which we do not know.” ---- C. S. Pierce Example of Reasoning: What day is today? I don’t know. But I know that yesterday was Sunday, and I know that if yesterday was Sunday then today is Monday. So, today is Monday. 12-13 Semester 2

  4. The Power of Reasoning Question: Are there two persons in Hong Kong who have exactly the same number of hairs on their heads? (Do not count those who are bald.) Two methods to solve this problem: 1. survey: count everyone’s hairs 2. purereasoning (i.e. logic) how? Which method is better? 12-13 Semester 2

  5. What is logic? • For this course, the object of study of logic is ARGUMENT (= written form of reasoning) 邏輯的研究對象是論證 • For this course, the goal of logic is to sort arguments (reasoning) into valid arguments (correct reasoning) and invalid arguments (incorrect reasoning) 邏輯的目的是區分有效的論證和沒有效的論證 • Example of Argument (in standard form): If yesterday was Sunday, then today is Monday. Yesterday was Sunday. ----------------------------------------------------------------- So, today is Monday. (valid or invalid?) 12-13 Semester 2

  6. What is Correct Reasoning / Valid Argument? • Is the following reasoning correct or not? / Is the following argument valid or not? Some animals are dogs. So, some animals are not dogs. Validity is about the form of the argument. Some A are D.  Some A are not D. Not valid. A counter-example is: Some men are humans. (True) So, some men are not humans. (False) 12-13 Semester 2

  7. What is Correct Reasoning / Valid Argument? • Is the following reasoning correct or not? / Is the following argument valid or not? Some cats are dogs. So, some dogs are cats. Validity is about the form of the argument. Some C are D.  Some D are C. Is this form valid or not? How can you show this? 12-13 Semester 2

  8. Validity 對確性/有效性 • The definition of validity: A (deductive) argument is valid if and only if it is logically impossible for all the premises to be true but the conclusion false 一個(演繹的)論證是對確的當且僅當前題全真時結論在邏輯上不可能假 對確 = “前真不能後假” = “前真一定後真” (Validity is a structural relationship between premises and conclusion.) 12-13 Semester 2

  9. Two kinds of logical validity • There are two kinds of logical validity: - formal validity (due to form) e.g., Some cats are dogs. So, some dogs are cats. - non-formal validity (due to meaning) e.g., Alan is  a father. So, Alan is a male. • In this course, we focus exclusively on formal validity 12-13 Semester 2

  10. Nature of Logic 邏輯的性質 • Consider the following argument: 1. Either Albert is an Abadab or he is a Glubphlab. 2. It is not true that Albert is an Abadab. --------------------------------------------------------------------- So, he is a Glubphlab. Question: Is this argument valid? 12-13 Semester 2

  11. Nature of Logic 邏輯的性質 • In fact, the validity of an argument (i.e., “前真不能後假”) is a property about its form (not its content): 1. Either ______ or _ _ _ _ _. 2. It is not true that ______. ---------------------------------------- So, _ _ _ _ _. Is this form valid or invalid? Can all the premises true but the conclusion false? 12-13 Semester 2

  12. Nature of Logic 邏輯的性質 Same argument form with different contents: 1. Eitherthere is Godorthere is Devil. 2. It is not true thatthere is Devil. ----------------------------------------------------- So, there is God. 1. Eitheryou are cleveroryou are stupid. 2. It is not true thatyou are stupid. --------------------------------------------------------- So, you are clever. 12-13 Semester 2

  13. Nature of Logic 邏輯的性質 Argument form in symbols: p or q Not p So, q (it’s called argument by elimination消去法; even dogs are capable of doing this reasoning.) 12-13 Semester 2

  14. Nature of Logic 邏輯的性質 Argument form in the symbols of our textbook p q ~p q 12-13 Semester 2

  15. Nature of Logic 邏輯的性質 Form 形式 Vs Content 內容 • Logic is about the form of, and not the content of, reasoning 邏輯關乎推理的形式，不關乎其內容 • Hence, logic is a science of forms 邏輯是一門關於形式的科學 • Hence, logic can be applied to any and every discipline which involves reasoning • Hence, logic has a wide range of applications 12-13 Semester 2

  16. Nature of Logic 邏輯的性質 • Reasoning  Argument  Validity  Form  Symbols Formal (Symbolic) Logic 形式邏輯 / 符號邏輯 • So, if logic is about correct reasoning, it is inevitablethat we need to use symbols; that is the nature of logic • Don’t be scared off by symbols • When you manipulate symbols, you need to understand the meaning behind them 12-13 Semester 2

  17. Soundness 真確 • Recall from CCT that an argument can be valid but still be a bad argument because it has one or more false premises • Good deductive argument = Sound argument = valid argument form + all premises true Note that although validity is not a sufficient condition for soundness, it is nevertheless a necessary condition. 12-13 Semester 2

  18. Soundness 真確 • In general, logic is not concerned with the truth of the premises of the argument (and hence its soundness), but its (formal) validity; E.g. in economics: If demand increases, then supply decreases. Supply decreases. ------------------------------------------------------------  Demand increases. Although logic cannot tell whether the premises are true or not, it can tell that this argument is not a good argument. 12-13 Semester 2

  19. How to memorize argument forms or logical rules? • One way to memorize abstract argument forms or logical rules is to put some interesting concretemeanings into the symbols; e.g., you will have to memorize this rule in Lecture 6: Either p or q. p q If p, then r. p r If q, then s. q s ---------------------- --------------  Either r or s  r s “Either you go by bus or take a taxi. If you go by bus, then you will be late. If you take a taxi, then you will die. So, you will be late or you will die.” 12-13 Semester 2

  20. Why study logic? • One practical use: Help you to tackle those questions in GRE/GMAT/LSAT/MBA entrance tests for further studies and those aptitude tests for government and big company jobs. 12-13 Semester 2

  21. Why this textbook? • Hausman, Kahane, and Tidman, Logic and Philosophy: A Modern Introduction, 12th edition, Wadsworth, 3 PolyU Library:http://library.polyu.edu.hk/search~S6?/tLogic+and+Philosophy/tlogic+and+philosophy/1%2C5%2C8%2CB/frameset&FF=tlogic+and+philosophy+a+modern+introduction&4%2C%2C4/indexsort=- CPCE Library:http://lib.cpce-polyu.edu.hk/search~/t?SEARCH=logic+and+philosophy • V. good • Cheap 12th edition ~ HK$220. • Humanistic orientation (logic and philosophy) 人文面向 (we aren’t teaching logic for computing, engineering, math, etc.) • There exists a Chinese translation (PolyU library: 邏輯與哲學 Call number: BC108.K312 1996) • Have plenty of good exercises and provide answers to all even-numbered questions at the back of the book • with a free software LogicCoach 12-13 Semester 2

  22. Contents of the course • We will cover three topics in this course: A. Syllogistic Logic 三段論邏輯 B. Sentential Logic 語句邏輯 C. Predicate Logic 謂詞邏輯 Note: different logics deal with different forms of arguments. • Teaching Plan 12-13 Semester 2

  23. A. Syllogistic logic 三段論邏輯 • Also called Categorical Logic 定言邏輯 \類稱邏輯 or Aristotelian Logic亞理士多德邏輯 • Deals with categoricalstatements and syllogisms A: All S are P. (e.g. All students are Chinese.) E: No S are P. (e.g. No students are Chinese.) I: Some S are P. (e.g. Some students are Chinese.) O: Some S are not P. (e.g. Some students are not Chinese.) 12-13 Semester 2

  24. A. Syllogistic logic 三段論邏輯 • A categorical syllogism is a three-line deductive argument in which allthree statements are standard-form categorical statements. E.g. Some doctors are professional football players. All cardiologists (心臟科醫生) are doctors. So, no cardiologists are professional football players. • The validity of categorical syllogisms can be effectively (= mechanically) 能行地 tested by Venn Diagrams 12-13 Semester 2

  25. A. Syllogistic logic 三段論邏輯 • Argument form: Some M are P. All S are M. So, No S are P. Valid or not? - Revise CC2002 CCT / CCN1004 CCT /CC2427 AOR lecture notes • We will learn, among other things, the Five Rules Method for determining validity in this logic, which uses the concept of “distribution” 普及 / 周延 (no need to draw diagrams; do it by heart; could be very fast) 12-13 Semester 2

  26. B. Sentential Logic 語句邏輯 • Also called “statement logic” or “propositional logic” 命題邏輯 • The logic of the connectives “and”並且 “or”或者 “not”並不是 “if–then”如果-那麼 “if and only if”當且僅當 12-13 Semester 2

  27. B. Sentential Logic 語句邏輯 • Example: Is the following argument valid or not? If you are clever, then you love logic and will take CC3015. Either you are stupid or you are clever. You are not stupid. So, you will take CC3015. Step 1. Symbolization 符號化 - to reveal the argument form Let C = “you are clever” L = “you love logic” S = “you are stupid” T = “you will take CC3015” 12-13 Semester 2

  28. B. Sentential Logic 語句邏輯 Argument form: In the symbols of our textbook: If C, then (L and T) C  (L • T) S or C C  S not S ~S So, T  T • We will learn two methods to determine validity in this logic, namely 1. the Truth-Table Method真值表法 2.the Natural Deduction Proof Method 自然演繹的証明方法 12-13 Semester 2

  29. B. Sentential Logic 語句邏輯 • Truth-Table Method: the argument is valid because there are no前真後假cases C L S T C  (L • T) (C  S) ~S T 0 0 0 0 1 0 1 0 0 0 0 1 1 0 1 1 0 0 1 0 1 1 0 0 0 0 1 1 1 1 0 1 0 1 0 0 1 0 1 0 0 1 0 1 1 0 1 1 1 = true 0 1 1 0 1 1 0 0 0 1 1 1 1 1 0 1 0 = false 1 0 0 0 0 1 1 0 1 0 0 1 0 1 1 1 1 0 1 0 0 1 0 0 1 0 1 1 0 1 0 1 1 1 0 0 0 1 1 0 1 1 0 1 1 1 1 1 1 1 1 0 0 1 0 0 1 1 1 1 1 1 0 1 12-13 Semester 2

  30. B. Sentential Logic 語句邏輯 • Short truth-table method: Assume all premises true but the conclusion false; see if there is any contradiction C  (L • T) 1 C  S 1 ~S 1  T 0 If so, the argument is valid; otherwise, invalid. 12-13 Semester 2

  31. B. Sentential Logic 語句邏輯 • Natural deduction proof: 1. C (L • T) Premise 2. C  S Premise 3. ~S Premise 4. C 2,3 DS 5. L • T 1,4 MP 6. T 5 Simp 12-13 Semester 2

  32. C. Predicate Logic 謂詞邏輯 • Also called Quantification Logic 量化邏輯 • Quantifers量詞: for all 所有, for some 有 (there exists; at least one) • Analyzes the internal structure of an atomic statement in terms of individuals個體 and predicates謂詞 • E.g. symbolize “Mary loves John” as Lmj Lxy means “x loves y” m = Mary j = john • L is a predicate; x and y are individual variables; m and j are individuals 12-13 Semester 2

  33. C. Predicate Logic 謂詞邏輯 Question: Does “Everyone loves someone (or other)” 每個人都鍾意人 logically imply 邏輯地推出 “There exists someone who is loved by everyone” ? 所有人都鍾意同一人 12-13 Semester 2

  34. C. Predicate Logic 謂詞邏輯 • Assume the Domain of Discourse論域 is persons • Let Lxy = x loves y- L is a predicate; x, y are individual variables • “Everybody loves y” as for every person x, x loves person y In the symbols of our textbook: (x)Lxy - (x) is the universal quantifier • “Someone loves y” as for some person x, x loves person y In the symbols of our textbook:(x)Lxy - (x) is the existential quantifier 12-13 Semester 2

  35. C. Predicate Logic 謂詞邏輯 No, it doesn’t! Argument form: (x)(y)Lxy  Everyone loves someone (or other) --------------------  (y)(x)Lxy  There exists someone who is loved by everyone This form is invalid. A counter-example 反例which has the same argument form: Everyone has a mother, but there is no such a mother who is the mother of everyone! Let Lxy = y is the mother of x 12-13 Semester 2

  36. C. Predicate Logic 謂詞邏輯 Is the following argument valid or invalid? “All dogs are animals. Therefore, all heads of dogs are heads of animals.” 12-13 Semester 2

  37. C. Predicate Logic 謂詞邏輯 Let Dx = x is a dog Ax = x is an animal Hxy = x is the head of y Argument form: (x)(Dx  Ax) / (x)[y(Dy • Hxy)  y(Ay • Hxy)] 12-13 Semester 2

  38. C. Predicate Logic 謂詞邏輯 Natural deduction proof: 1. (x)(Dx  Ax) / (x)[y(Dy • Hxy)  y(Ay • Hxy)] 2. y(Dy • Hxy) AP 3. Dy • Hxy 2 EI y is an unknown 4. Dy 3 Simp 5. Hxy 3 Simp 6. Dy  Ay 1UI 7. Ay 4, 6 MP 8. Ay • Hxy 5, 7 Conj 9. y(Ay • Hxy) 8 EG 10. y(Dy • Hxy)  y(Ay • Hxy) 2-9 CP 11. (x)[y(Dy • Hxy)  y(Ay • Hxy)] 10 UG The argument is valid. 12-13 Semester 2

  39. C. Predicate Logic 謂詞邏輯 • We will learn two methods to determine validity in this logic, namely 1. Method of Counter-Example 反例法 2. Natural Deduction Proof Method 自然演繹的証明方法 12-13 Semester 2

  40. How to study this subject • Learn by doing plentyof exercises • Understand and memorize the definitions of key terms such as validity, and the logical rules • Follow closely; don’t skip classes • Ask questions 12-13 Semester 2

  41. Moodle e-learning System • We use Moodle • You need to download lecture notes and tutorial exercise sheets from there yourself • Do the tutorial exercises before coming to class 12-13 Semester 2

  42. Critical Thinking Web at HKU • 思方網 at HKU: http://philosophy.hku.hk/think/logic/index.php 12-13 Semester 2

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