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Philosophy 1100. Title: Critical Reasoning Instructor: Paul Dickey E-mail Address: pdickey2@mccneb.edu. Today: LAST DAY TO WITHDRAW FROM CLASS Re-submit Midterm Examination (Optional) Student Portfolio is Due Discuss Class Evaluation Conclude Discussion on Chapter Nine
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Philosophy 1100 Title: Critical Reasoning Instructor: Paul Dickey E-mail Address: pdickey2@mccneb.edu Today: LAST DAY TO WITHDRAW FROM CLASS Re-submit Midterm Examination (Optional) Student Portfolio is Due Discuss Class Evaluation Conclude Discussion on Chapter Nine Chapter Ten Next Week (8/3/15) Final Essay is Due Exercise as assigned later Exercises 9-19, p. 276-277, Problem #5 Extra Credit Editorial Essay – The New Standards: The Case for Intellectual Discipline in the Classroom 1
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.
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 super heroes (Conclusion) Some super heroes 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. Super heroes) 3) the middle term (M) – the term that occurs in both premises but not in the conclusion (e.g. Muppets).
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.
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.
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.
Class Workshop: Exercise 9-13, #6 More from 9-13?
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
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.
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.
Class Workshop: Exercise 9-15, 9-16, & 9-17, 9-18
Team 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.
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
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.
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.
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.
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 9-21, p. 278, Problem #5.
DUE NEXT WEEK: Perform all steps in the “Team Game” -- Exercises 9-19, p. 276-277, Problem #5
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.
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
Truth Functional Logic: The Basics • Perhaps the simplest truth table operation is negation: P ~P T F F T
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
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
Using Truth Tables To Test Validity • Now, consider the following argument: Premise: If Paula goes to work, then Quincy and Rogers will get a day off. Conclusion: If Paula goes to work and Quincy gets a day off, then Rogers will get a day off. • We symbolize the conclusion as (P & Q) -> R • Thus, the argument is: P -> (Q & R) (P & Q) -> R • Is this a valid argument?
