# Plato s Allegory of the Cave

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1 Logic

2 Plato s Allegory of the Cave

3 The First Word of the Day is Troglodyte From the Greek word for cave (trōglē). The Troglodytae (Τρωγλοδῦται) or Troglodyti (literally cave goers ) are those who live in caves.

4 The Second Word of the Day is Allegory From the Greek allēgoria from allos (other) + agoria (speaking). An allegory is a story, poem, or picture that can be interpreted to reveal a hidden meaning behind the literal meaning.

5 One representation of Plato s Cave

6 Overview: Logic Intro (1) Sentences and Statements and Propositions (2) Simple and Complex Statements (3) Necessary and Sufficient Conditions (4) An argument as a reason to believe. (5) Arguments vs Explanations (6) Enthymemes (7) Practice Exercises

7 What is Logic? The study of good arguments or good reasoning. The study of how we ought to reason (not how we do reason). A normative science (as opposed to a descriptive science). Logic was first formally studied by Aristotle ( BCE). Sample Argument: All Athenians are Mortal. Aristotle is an Athenian. Therefore: Aristotle is Mortal.

8 Logic and Language

9 Sentences and Statements A sentence is a string of words that follows the rules of grammar. A statement is a sentence with a truth-value (a sentence that is either true or false). Statements are functionally informative, and typically have the form of assertions.

10 Statements and Propositions A statement is a sentence with a truth-value. A proposition is the meaning of the statement.

11 Sentences and Statements Sentence 1 I am sitting. [spoken by Steve Naragon at noon on June 3, 2016] 2 He is sitting. [spoken by another of Steve Naragon at noon on June 3, 2016] 3 I was sitting at noon. [spoken by Steve Naragon later that day] 4 I am sitting. [spoken by Alice Miller at noon on June 3, 2016] Statement Steve Naragon was sitting at noon on June 3, Steve Naragon was sitting at noon on June 3, Steve Naragon was sitting at noon on June 3, Alice Miller was sitting at noon on June 3, 2016.

12 Simple and Complex Statements

13 Some Complex Statements Conditional: If P, then Q If you mow my yard this afternoon, then I ll pay you \$20. P Q Conjunction: P and Q Ed and Bob are Republicans. = Ed is a Republican and Bob is a Republican P Q Disjunction: P or Q Either that s a hedgehog climbing up your trousers or it s a porcupine. = A hedgehog is climbing up your trousers or a porcupine is climbing up your trousers. P Q

14 Necessary and Sufficient Conditions

15 NC and SC are Everywhere Conditional Statements: If P, then Q If Fido is a dog, then Fido is a mammal. P: Fido is a dog. Q: Fido is a mammal. Categorical Statements: All P are Q All dogs are mammals. P: dogs Q: mammals P is a sufficient condition of Q Q is a necessary condition of P empty dogs mammals P is sufficient for Q = P being true is enough for Q to be true. Q is necessary for P = P cannot be true unless Q is true.

16 SG Practice: Necessary and Sufficient Having a son is a of being a parent. Sufficient Condition Having four sides is a of being a square. Necessary Condition Being bald is a of being a professor. Neither necessary nor sufficient Being an unmarried man is a of being a bachelor. Both necessary and sufficient

17 SG Practice: Necessary and Sufficient Taking the exam is a of passing the exam. Necessary Condition Drinking that entire bottle of whisky at one sitting is a of becoming inebriated. Sufficient Condition Lighting a match is a of starting a fire. Sufficient Condition Being a mammal is a of being an otter. Necessary Condition

18 Identifying Arguments

20 Logic is about arguments. and An argument is a set of statements that support the truth of a further statement.

21 A Few Logic Terms Sentence: A string of words following the rules of grammar. Statement: A sentence with a truth-value. Proposition (or propositional content ): The meaning of the statement. Truth-Value: The property of being true or of being false. Argument: An attempt to prove the truth of some statement (the conclusion) by appealing to other statements already believed to be true (the premises). Inference: The relationship that ties the premises to the conclusion, suggesting that the conclusion is true because the premises are true. Fallacy: An argument that appears to support the conclusion, more than it actually does.

22 Sample Argument (1) If human beings consist entirely of matter, then none of their actions are free. (2) But human beings do consist entirely of matter. (3) Therefore none of their actions are free. Premise Premise Conclusion The premises, if true, give us a reason to believe that the conclusion is true.

23 Sample Argument (1) If human beings consist entirely of matter, then none of their actions are free. (2) But human beings do consist entirely of matter. (3) Therefore none of their actions are free. Premise Premise Conclusion The premises and conclusion do not include indicator -words and other particles.

24 In Summary Argument = premises + conclusion + inference Premises/conclusion are all statements, each expressing some proposition. A fallacy is an argument whose inference is weaker than it appears.

25 Identifying Arguments When individuals voluntarily abandon property, they forfeit any expectation of privacy in it that they might have had. Therefore, a warrantless search or seizure of abandoned property is not unreasonable under the Fourth Amendment. (Judge Stephanie Kulp Seymour, United States v. Jones)

26 Identifying Arguments (1) When individuals voluntarily abandon property, they forfeit any expectation of privacy in it that they might have had. Therefore, (2) a warrantless search or seizure of abandoned property is not unreasonable under the Fourth Amendment. (Judge Stephanie Kulp Seymour, United States v. Jones) (1) gives us reason to believe (2).

27 Identifying Arguments If Socrates is a student of nature, then he is an atheist. He is a student of nature. Therefore, he is an atheist. (Plato, Apology)

28 Identifying Arguments (1) If Socrates is a student of nature, then he is an atheist. (2) He is a student of nature. Therefore, (3) he is an atheist. (Plato, Apology) (1) and (2) give us reason to believe (3).

29 Indicator Words Conclusion Indicators therefore, wherefore, thus, consequently, hence, accordingly, entails that, for this reason, so, it follows that, as a result, suggests that, proves that, indicates that, is likely that, Premise Indicators since, because, given that, in that, as indicated by, for, owing to, inasmuch as, may be inferred from, in view of

30 Explanations (as backward arguments)

31 Arguments vs Explanations (1) John broke the window because he tripped. (2) John broke the window because he had forgotten his key and needed to get in. (3) John broke the window because he was the only person in the house. (Fisher, Logic of Real Arguments, p. 18)

32 Arguments vs Explanations (1) Explanandum: (John broke the window) Why did he do that? Because Explanans: (he tripped). (2) Explanandum: (John broke the window) Why did he do that? Because Explanans: (he forgot his key). (3) Conclusion: (John broke the window) Why should I believe that? Because Premise: (he was the only person in the house).

33 Enthymemes

34 Enthymemes #1 An enthymeme is any deductive argument that is missing either a premise or a conclusion, or sometimes both. For example: We cannot trust this man, for he has lied in the past.

35 Enthymemes #1 An enthymeme is any deductive argument that is missing either a premise or a conclusion, or sometimes both. For example: We cannot trust this man, for he has lied in the past. The complete argument would read: Those who have lied in the past cannot be trusted. This man has lied. Therefore, we cannot trust this man

36 Enthymemes #2 If you spend that much time playing video games, then you will not do well in college.

37 Enthymemes #2 If you spend that much time playing video games, then you will not do well in college. The complete argument could read: If you spend that much time playing video games, then you will not do well in college. You are spending that much time playing video games. Therefore, you will not do well in college.

38 Enthymemes #2 If you spend that much time playing video games, then you will not do well in college. The complete argument could read: If you spend that much time playing video games, then you will not do well in college. You are spending that much time playing video games. Therefore, you will not do well in college. Or, more hopefully, it might read: If you spend that much time playing video games, then you will not do well in college. You want to do well in college. Therefore, you ll spend less time playing video games.

39 Logic Review (1) Sentences and Statements and Propositions (2) Simple and Complex Statements (3) Necessary and Sufficient Conditions (4) Premises give a reason to believe the conclusion. (5) Arguments vs Explanations (6) Enthymemes

40 Practice

41 What s the Conclusion? (1/3) That bicycle must belong to Mary, since it s either John s or Mary s, and it s way too big for John.

42 What s the Conclusion? (1/3) (1) That bicycle must belong to Mary, since (2) it s either John s or Mary s, and (3) it s way too big for John. (2) and (3) give us reason to believe (1).

43 What s the Conclusion? (2/3) Belief by U.S. citizens in global warming dropped by 20%, even though the scientific data still unequivocally support the claim that global temperatures are rising. This suggests that public opinion is being swayed by something other than the facts.

44 What s the Conclusion? (2/3) (1) Believe by U.S. citizens in global warming dropped by 20%, even though (2) the scientific data still unequivocally support the claim that global temperatures are rising. This suggests that (3) public opinion is being swayed by something other than the facts. (1) and (2) give us reason to believe (3).

45 What s the Conclusion? (3/3) If Ed has black hair, then Ed is Italian, and Ed does indeed have black hair, so he must be Italian.

46 What s the Conclusion? (3/3) (1) If Ed has black hair, then Ed is Italian. (2) Ed does have black hair, so (3) Ed is Italian. (1) and (2) give us reason to believe (3).

47 What s Missing? (1/2) He s clearly not home, since his light is not on.

48 What s Missing? (1/2) (1) He s clearly not home, since (2) his light is not on. [and (3) If his light is not on, then he s not home] (2) and the missing premise (3) give us reason to believe (1).

49 What s Missing? (2/2) If I want pizza, then I ll order pizza, but I don t want pizza.

50 What s Missing? (2/2) (1) If I want pizza, then I ll order pizza, but (2) I don t want pizza. [Therefore (3) I won t order pizza.] (1) and (2) give us reason to believe the omitted conclusion (3).

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