Philosophy 1100

1. Philosophy 1100 Title: Critical Reasoning Instructor: Paul Dickey E-mail Address: pdickey2@mccneb.edu Today: Exercises 8-1 & 8-2 Exercise 8-4 (odd numbered problems) Exercise 8-11, problems #1-5 Next week: Exercise 8-11, problems #6-10 not done in class tonight Read Chapter 9, pages 295-311, pp. 317-330 Exercise 9-1, all problems *** Please Note: Argumentative Essay will NOT be due Next week. 1

2. Chapter EightDeductive Arguments:Categorical Logic

3. Four Basic Kinds of Claims in Categorical Logic (Standard Forms) A: All _________ are _________. (Ex. All Presbyterians are Christians. E: No ________ are _________. (Ex. No Muslims are Christians. ___________________________________ I: Some ________ are _________. (Ex. Some Arabs are Christians. O: Some ________ are not _________. (Ex. Some Muslims are not Sunnis.

4. Three Categorical Operations • Conversion – The converse of a claim is the claim with the subject and predicate switched, e.g. • The converse of “No Norwegians are Swedes” is “No Swedes are Norwegians.” • Obversion – The obverse of a claim is to switch the claim between affirmative and negative (A -> E, E -> A, I -> O, and O -> I and replace the predicate term with the complementary (or contradictory) term, e.g. • The obverse of “All Presbyterians are Christians” is “No Presbyterians are non-Christians.” • Contrapositive – The contrapositive of a claim is the cliam with the subject and predicate switched and replacing both terms with complementary terms (or contradictory terms), e.g. • The contrapositive of “Some citizens are not voters” is “Some non-voters are not noncitiizens.

5. OK, So where is the beef? • By understanding these concepts, you can apply the • three rules of validity for deductive arguments: • Conversion – The converses of all E- and I- claims, but not A- and O- claims are equivalent to the original claim. • Obversion – The obverses of all four types of claims are equivalent to their original claims. • Contrapositive – The contrapositives of all A- and O- claims, but not E- and I- claims are equivalent to the original claim.

6. Categorical Syllogisms • A syllogism is a deductive argument that has two premises -- and, of course, one conclusion (claim). • A categorical syllogism is a syllogism in which: • each of these three statements is a standard form, and • there are three terms which occur twice, once each in two of the statements.

7. Three Terms of a Categorical Syllogism • For example, the following is a categorical syllogism: • (Premise 1) No Muppets are Patriots. • (Premise 2) Some Muppets are puppets that support themselves financially. • (Conclusion) Some puppets that support themselves financially are not Patriots.. • The three terms of a categorical syllogism are: • 1) the major term (P) – the predicate term of the conclusion (e.g. Patriots). • 2) the minor term (S) – the subject term of the conclusion (e.g. Self-supporting Puppets) • 3) the middle term (M) – the term that occurs in both premises but not in the conclusion (e.g. Muppets).

8. USING VENN DIAGRAMS TO TEST ARGUMENT VALIDITY • Identify the classes referenced in the argument (if there are more than three, something is wrong). • When identifying subject and predicate classes in the different claims, be on the watch for statements of “not” and for classes that are in common. • Make sure that you don’t have separate classes for a term and it’s complement. • 2. Assign letters to each classes as variables. • 3. Given the passage containing the argument, rewrite the argument in standard form using the variables. M = “xxxx “ S = “ yyyy“ P = “ zzzz“ No M are P. Some M are S. ____________________ Therefore, Some S are not P.

9. Draw a Venn Diagram of three intersecting circles. • Look at the conclusion of the argument and identify the subject and predicate classes. • Therefore, Some S are not P. • Label the left circle of the Venn diagram with the name of the subject class found in the conclusion. (10 A.M.) • Label the right circle of the Venn diagram with the name of the predicate class found in the conclusion. • Label the bottom circle of the Venn diagram with the middle term.

10. No M are P. Some M are S. • Diagram each premise according the standard Venn diagrams for each standard type of categorical claim (A,E, I, and O). • If the premises contain both universal (A & E-claims) and particular statements (I & O-claims), ALWAYS diagram the universal statement first (shading). • When diagramming particular statements, be sure to put the X on the line between two areas when necessary. • 10. Evaluate the Venn diagram to whether the drawing of the conclusion "Some S are not P" has already been drawn. If so, the argument is VALID. Otherwise it is INVALID.

11. Power of Logic Exercises: http://www.poweroflogic.com/cgi/Venn/venn.cgi?exercise=6.3B ANOTHER GOOD SOURCE: http://www.philosophypages.com/lg/e08a.htm

12. Class Workshop: • Exercise 8-11, #1-5

13. Using the Rules Method To Test Validity Background – ***If a claim refers to all members of the class, the term is said to be distributed. Table of Distributed Terms: A-claim: All S are P E-claim: No S are P I-Claim: Some S are P O-Claim: Some S are not P The bold, italic, underlined term is distributed. Otherwise, the term is not distributed.

14. Some Dogs are Not Poodles. Why is this a statement about all poodles? Say a boxer is a dog which is not a poodle. Thus, the statement above says that “all poodles are not boxers” and thus “poodles” is distributed.

15. The Rules of the Syllogism • A syllogism is valid if and only if all three of the following conditions are met: • The number of negative claims in the premises and the conclusion must be the same. (Remember: these are the E- and the O- claims) • At least one premise must distribute the middle term. • Any term that is distributed in the conclusion must be distributed in its premises.

16. Class Workshop: • Exercise 8-13, 8-14, • & 8-15, 8-16

17. You must perform all of the following • on the given argument: • Translate the premises and conclusion to standard logical forms and put the argument into a syllogistic form. • Identify the type of logical form for each statement. • For each statement, give an equivalent statement and name the operation that you used to do so. • Identify the minor, major, and middle terms of the syllogism. • Draw the appropriate Venn Diagram for the premises. • Identify all distributed terms of the argument and the number of negative claims in the premises and conclusion. • What, if any, rules of validity are broken by the argument? • State if the argument is valid or invalid.

18. Everything that Pete won at the carnival must be junk. I know that Pete won everything that Bob won, and all the stuff Bob won is junk. • Translate the premises and conclusion to standard logical forms and put the argument into a syllogistic form. • Identify the type of logical form for each statement. • Define terms – • P: Pete’s winnings at the carnival • J: Thing that are junk • B: Bob’s winnings at the carnival • A-claim – All B is P • A-claim - All B is J • A-claim – All P is J

19. Everything that Pete won at the carnival must be junk. I know that Pete won everything that Bob won, and all the stuff Bob won is junk. • For each statement, give an equivalent statement and name the operation that you used to do so. • Identify the minor, major, and middle terms of the syllogism. • A-claim – All B is P • Contrapositive is equivalent – All non-P are non-B. • A-claim - All B is J • Obverse is equivalent – No B is non-J. • A-claim – All P is J • Obverse is equivalent – No P is non-J. • Minor term is P; Major term is J; and Middle term is B.

20. Everything that Pete won at the carnival must be junk. I know that Pete won everything that Bob won, and all the stuff Bob won is junk. • Draw the appropriate Venn Diagram for the premises.

21. Everything that Pete won at the carnival must be junk. I know that Pete won everything that Bob won, and all the stuff Bob won is junk. • Identify all distributed terms of the argument and the number of negative claims in the premises and conclusion. • What, if any, rules of validity are broken by the argument? • State if the argument is valid or invalid. • All B is P • All B is J • All P is J • Since A-claims distribute their subject terms, B is • Distributed in the premises and P is distributed in the • conclusion. There are no negative claims in either the • premises or the conclusion. • Since P is distributed in the conclusion, but not in • either premise rule 3 is broken. Thus, the argument is invalid.

22. The Game • You must perform all of the following • on the given argument: • Translate the premises and conclusion to standard logical forms and put the argument into a syllogistic form. • Identify the type of logical form for each statement. • For each statement, give an equivalent statement and name the operation that you used to do so. • Identify the minor, major, and middle terms of the syllogism. • Draw the appropriate Venn Diagram for the premises. • Identify all distributed terms of the argument and the number of negative claims in the premises and conclusion. • What, if any, rules of validity are broken by the argument? • State if the argument is valid or invalid. • Exercises 8-19, p. 290, Problems #8 & #19.

23. Philosophy 1100 Chapter Nine Deductive Arguments:Truth-Functional Logic

24. Truth Functional Logic • Truth Functional logic is important because it gives us a consistent tool to determine whether certain statements are true or false based on the truth or falsity of other statements. • A sentence is truth-functional if whether it is true or not depends entirely on whether or not partial sentences are true or false. • For example, the sentence "Apples are fruits and carrots are vegetables" is truth-functional since it is true just in case each of its sub-sentences "apples are fruits" and "carrots are vegetables" is true, and it is false otherwise. • Note that not all sentences of a natural language, such as English, are truth-functional, e.g. Mary knows that the Green Bay Packers won the Super Bowl.

25. Truth Functional Logic: The Basics • Please note that while studying Categorical Logic, we used uppercase letters (or variables) to represent classes about which we made claims. • In truth-functional logic, we use uppercase letters (variables) to stand for claims themselves. • In truth-functional logic, any given claim P is true or false. • Thus, the simplest truth table form is: • P • _ • T • F

26. Truth Functional Logic: The Basics • Perhaps the simplest truth table operation is negation: • P ~P • T F • F T

27. Truth Functional Logic: The Basics • Now, to add a second claim, to account for all truth-functional possibilities our representation must state: • P Q • T T • T F • F T • F F • And the operation of conjunction is represented by: • P Q P & Q • T T T • T F F • F T F • F F F

28. Truth Functional Logic: The Basics • The operation of disjunction is represented by: • P Q P V Q • T T T • T F T • F T T • F F F • The operation of the conditional is represented by: • P Q P -> Q • T T T • T F F • F T T • F F T

29. Now, using these basic principles, we can construct truth tables for more complex statements. Consider the claim: If Paula goes to work, then Quincy and Rogers will get a day off. • We represent the claims like this: • P = Paula goes to work • Q = Quincy gets a day off • R = Rogers gets a day off, and • We symbolize the complex claim as P -> (Q & R) • The truth table looks like this: • P Q R Q & R P -> (Q & R) • T T T T T • T T F F F • T F T F F • T F F F F • F T T T T • F T F F T • F F T F T • F F F F T

30. Class Workshop: Exercises 9-1