UGED1111C LOGIC: CATEGORICAL SYLLOGISMS (I) 2021-22, Summer Sapphires, WONG Sin Ting Soundness 2 Sound argument ◼ A deductive argument that is valid and has all true premises. 1. 所有人都有觸手。 2. 所有有觸手的都是殺手。 (F) (F) 1. All buskers perform in public places. 2. Some buskers are singers. (T) (T) 3. 所有人都是殺手。 (F) 3. Some singers perform in public places. (T) unsound sound Soundness 3 Valid Argument Invalid Argument True Premises & True Conclusion ✓ ✓ True Premises & False Conclusion ✓ False Premises & True Conclusion ✓ ✓ False Premises & False Conclusion ✓ ✓ Sound Argument Arguments, Truth and Validity 4 Recap: Evaluating Arguments 5 Regardless of the type of argument, there are two steps of evaluation: ◼ ◼ How do its premises support its conclusion? Are all premises true? How about the inductive arguments? Inductive Standard 6 Two key concepts: ◼ Strength (Strong/weak) How do its premises support its conclusion? ◼ Cogency Are all premises true? Strength 7 Strong argument ◼ If all its premises are true, it is probable that its conclusion is true. Weak argument ◼ Its conclusion does not follow probably from its premises. Important Remark 8 Unlike validity, the strength of an inductive argument is not an all-or-nothing concept. It is a matter of degree. 30 1. Around 97 percent of CUHK students use a smartphone. 2. Mia is a CUHK student. 3. Mia uses a smartphone. Cogency 9 Cogent argument ◼ An inductive argument that is strong and has all true premises. 1. 大部份人從7層樓或以上的高度墮下都會死。 2. 梁振英於2018年8月24日從8樓墮下。 (T) (F) 3. 梁振英已經死了。 (F) uncogent Summary 10 Sound Valid Deductive Argument Unsound Invalid (all are unsound) Argument Cogent Strong Uncogent Inductive Argument Weak (all are uncogent) Agenda 11 Categorical Proposition Translation Venn Diagrams Transformation Procedures ◼ Conversion, Obversion and Contraposition Categorical Logic 12 This logical system was developed by Aristotle more than 2000 years ago. It is the major logical system before the 20th Century. Categorical Proposition 13 A categorical proposition is proposition that relates two classes, or categories. ◼ ◼ The classes in question are denoted respectively by the term subject term and the predicate term. On top of that, every categorical proposition contains a quantifier and a copula (which links the terms). Categorical Proposition 14 Quantifier: All Subject term: CUHK students Copula: are Predicate term: clever students Categorical Proposition 15 There are only 4 kinds of categorical propositions: (A) All S are P. (E) No S are P. (I) Some S are P. (O) Some S are not P. Aside: The labeling comes from the Latin words Affirmo and Nego, “A” and “I” from the former and “E” and “O” from the latter. Quality of Categorical Proposition 16 Every categorical proposition is either affirmative or negative. ◼ ◼ “All S are P” and “Some S are P” are categorical propositions affirming the class membership between S and P. “No S are P” or “Some S are not P” are categorical propositions denying the class membership between S and P. Quantity of Categorical Proposition 17 Every categorical proposition is either universal or particular. ◼ If a proposition concerns every member of the subject term, it is a universal categorical proposition. ◼ ◼ “All S are P” and “No S are P”. If a proposition concerns some members of the subject term, it is a particular categorical proposition. ◼ “Some S are P” and “Some S are not P”. Quality and Quantity 18 Affirmative Negative Universal All S are P (A) No S are P (E) Particular Some S are P (I) Some S are not P (O) Remarks 19 “Some” = “at least one” ◼ There are only 4 kinds of categorical propositions. ◼ “Some S are P” means “there exists at least one S that is also P.” “All S are not P” and “No S are not P” are not standard form. Subject and predicate terms must be plural nouns/noun phrases as long as they are countable. Exercise (1) 20 Which of the following categorical propositions is in standard form? All students who go to school by bus are LIHKG users. 沒有人是孤獨的。 (A) Some birds fly south during the winter. (E) 所有男人都是禽獸。 (I) Only CUHK students are lunatics. (O) 阿德是執葉的人。 All animals are not objects. 有些熊貓不是黑白色的動物。 All S are P. No S are P. Some S are P. Some S are not P. Translation 21 Subject and predicate terms must be both plural nouns/noun phrases ◼ ◼ 「沒有人是孤獨的。」 →「沒有人是孤獨的人。」 “All politicians are hypocritical.” → “All politicians are hypocrites/hypocritical people.” Translation 22 Copula must be “are” or “are not” “Some birds fly south during the winter.” → “Some birds are animals that fly south during the winter.” ◼ ◼ 「有些人喜歡盲目追隨潮流。」 →「有些人是喜歡盲目追隨潮流的人。」 Translation 23 Every categorical proposition must begin with standard quantifiers ◼ 「阿德是執葉的人。」 →「所有與阿德等同的人都是執葉的人。 」 “Oscar Wilde is a famous English writer.” → “All people identical to Oscar Wilde are famous English writers.” ◼ Translation 24 Adverbs “You should never read Hegel’s writings.” → “No times are times when you should read Hegel’s writings”. ◼ “Whenever Beerus wants to sleep he sleeps.” → “All times Beerus wants to sleeps are times he sleeps.” ◼ Translation 25 Unexpressed Quantifiers ◼ ◼ * Context-dependent * “Sapphires are blue gems.” → “All sapphires are blue gems.” 「獅子住在動物園。」 → 「有些獅子是住在動物園的動物。」 Translation 26 Definite/indefinite articles * Context-dependent * ◼ “A cat is a living creature.” → “All cats are living creatures.” ◼ “A man is killed in the accident.” → “Some men are people killed in the accident.” Translation 27 Exclusive propositions ◼ ◼ “Only CUHK students are lunatics.” → “All lunatics are CUHK students”. 「只有偉大的人才能撰寫歷史。」 → 「所有能撰寫歷史的人都是偉大的人。」 Translation 28 Conditional propositions ◼ “If a man is rich, he is attractive.” → “All rich men are attractive men.” “Unless a boy misbehaves, he will be treated decently.” → “All boys who do not misbehave are boys who will be treated decently.” ◼ Venn Diagrams 29 Venn Diagrams were invented by John Venn. It is a great tool for representing the relationships between classes, or categories. Venn Diagrams 30 All S are P S No S are P P S = Emptiness P Venn Diagrams 31 Some S are P S Some S are not P P S = Existence P Singular Propositions 32 「比路斯是破壞神。」 所有與比路斯等同的人都是破壞神。 比路斯等同的人 破壞神 有些與比路斯等同的人是破壞神。 比路斯等同的人 破壞神 If vs Only if 33 If (只要) ◼ 只要是億萬富豪，就能娶女神。 所有億萬富豪都是能娶女神的人。 億萬富豪 能娶女神的人 所有能娶女神的人都是億萬富豪。 億萬富豪 能娶女神的人 If vs Only if 34 Only if (只有) ◼ 只有是億萬富豪，才能娶女神。 所有億萬富豪都是能娶女神的人。 億萬富豪 能娶女神的人 所有能娶女神的人都是億萬富豪。 億萬富豪 能娶女神的人 Exercise (2) 35 Translate the following proposition to standardized categorical proposition. References 36 ➢ ➢ ➢ Copi, I., Cohen, C., & McMahon, K. (2011). Introduction to Logic (14th ed. / Irving M. Copi, Carl Cohen, Kenneth McMahon ed.). Upper Saddle River, NJ: Pearson Education. Hurley, P. (2015). A Concise Introduction to Logic (12th ed.). Australia ; Stamford, Ct.: Cengage Learning. Lau, J. (2011). An Introduction to Critical Thinking and Creativity : Think More, Think Better. Hoboken, N.J.: Wiley.