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Critical Reasoning with Professor Paul Dickey

Submit your final essay and exam by email before 8/12, 5:30 PM. Understand and distinguish issues in Trayvon Martin and George Zimmerman events for logical reasoning. Class covers deductive and truth-functional logic exercises.

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Critical Reasoning with Professor Paul Dickey

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  1. Philosophy 1100 Title: Critical Reasoning Instructor: Paul Dickey E-mail Address: pdickey2@mccneb.edu Website:http://mockingbird.creighton.edu/NCW/dickey.htm Today: Exercise 8-11, all problems Portfolios due (at last!) Tomorrow (8/6): Final Exam will be posted on Quia. There will be some exercises that may require drawing. Next week: No Physical Class. Submit Final Essay & FINAL EXAM by email BEFORE 8/12, 5:30 P.M. For every 4 hours the essay and/or exam is late, a full grade will be reduced. NO EXCEPTIONS. 1

  2. COURSE EVALUATION • Electronic/Online Course/Instructor Feedback • 13/SS Availability until • August 9, 2013. • Instruction Sheet is on Quia site.

  3. Final Class Essay: Due – 8/12 Identify two related but logically distinguishable issues in the events regarding Trayvon Martin and George Zimmerman that many (e.g. your friends, the media, etc) might confuse when intuitively taking a “side,” but are necessary to an adequate and reasonable understanding of what may have happened or what should have happened. Make a good argument to address each issue. Clearly identify how your premises support your claims specifically.

  4. Chapter EightDeductive Arguments:Categorical Logic

  5. Three Terms of a Categorical Syllogism • For example, the following is a categorical syllogism: • (Premise 1) No Muppets are Patriots. • (Premise 2) Some Muppets do not support themselves financially as puppets. • (Conclusion) Some puppets that do not support themselves 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. Puppets that are non self-supporters) • 3) the middle term (M) – the term that occurs in both premises but not in the conclusion (e.g. Muppets).

  6. Class Workshop: • Exercise 8-11, ?

  7. 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.

  8. 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.

  9. 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.

  10. 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.

  11. 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.

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

  13. Chapter NineDeductive Arguments:Truth-Functional Logic

  14. 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.

  15. 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

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

  17. 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

  18. 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

  19. 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?

  20. Using Truth Tables To Test Validity • Is this a valid argument? We can determine whether or not it is by constructing a truth table that presents the premise(s) and conclusion. • In this case, to do so we add to the previous truth table the necessary columns to represent the conclusion. • P Q R Q&R P -> (Q & R) P & Q (P & Q) -> R • T T T T T T T • T T F F F T F • T F T F F F T • T F F F F F T • F T T T T F T • F T F F T F T • F F T F T F T • F F F F T F T • Now, remembering the definition of a deductive argument, we look for a row in the table in which the premise(s) is true but the conclusion is not true. If we find one, the argument is invalid. If there is none, then the argument is valid.

  21. Using Truth Tables To Test Validity • We can determine whether or not a deductive argument is valid or invalid by constructing a truth table that presents the premise(s) and conclusion. • A deductive argument is valid when if the premises are true, the conclusion has to be true. Or in other words, an argument is valid if there are no possible states or conditions in which the premises are true and the conclusion is false. • And, of course, a truth table represents all the possible states or conditions of the claims. • Thus, an argument is valid when there are NO rows of the truth table in which the premise(s) are true and the conclusion is not true. If there is even one, the argument is invalid.

  22. Consider the following argument: P -> Q ~P _________ ~Q • P Q • T T • T F • F T • F F • Construct the appropriate truth table to include all possible t-f scenarios for all variables in the argument. • If there are x (e.g. 2) variables, note that there with always be x(so in this case, 2) columns in the truth table at this point and there will be 2**x (or 2 to the x power) number of rows (in this case, 4). • P1 • P Q P->Q • T T T • T F F • F T T • F F T 2. Add a column to the truth table to express the first premise based on the truth tables for the basic operations. You may have to do this in multiple steps.

  23. P -> Q ~P _________ ~Q Consider the following argument: • P1 P2 • P Q P->Q ~P • T T T F • T F F F • F T T T • F F T T 3. For each remaining premise (there more may be more than one) add a column to the truth table to express the premise based on the truth tables for the basic operations. • P1 P2 C • P Q P->Q ~P ~Q • T T T F F • T F F F T • F T T T F • F F T T T 4. Add a column to the truth table to express the conclusion based on the truth tables for the basic operations. You may have to do steps #3 and #4 also in multiple steps.

  24. Consider the following argument: P -> Q ~P ______ ~Q • P1 P2 C • P Q P->Q ~P ~Q • T T T F F • T F F F T • F T T T F • F F T T T • Ask yourself “Are there any rows in the truth table that I have just created in which all premises are true and the conclusion is false?” 6. If the answer is yes, then write “invalid.” If the answer is no, write “valid.” Invalid

  25. Class Workshop: • Exercise 9-7, #3

  26. Deductive Arguments:Rules of Induction

  27. Deduction: Group 1 Rules • The basic valid argument patterns of deductive logic • (If doubted, all the rules we discuss below can be confirmed by the • truth-table method) is another method to prove a deductive • argument (that is, to show that it is valid). • Modus Ponens (MP) • P -> Q • P____ • Q • -- Affirming the antecedent • P Q P->Q • T T T • T F F • F T T • F F T

  28. Deduction: Group 1 Rules • Modus Tollens (MT) • P -> Q • ~Q____ • ~P • -- Denying the consequent

  29. Deduction: Group 1 Rules • Okay, now that we have two rules to play • with, let’s stop for a minute and see how we • prove an argument valid using the rules. • (P & Q) -> R • S • S -> ~R / .’. ~ ( P&Q) • ~R 2,3, MP • ~ (P & Q) 1,4, MT

  30. Chain Argument (CA) • P -> Q • Q -> R____ • P -> R • Disjunctive Argument (DA) • P v Q P v Q • ~P ~Q__ • Q P • Simplification (SIM) • P & QP & Q • P Q • Conjunction (CONJ) • P • Q__ • P & Q

  31. Addition (ADD) • P Q • P v Q P v Q • Constructive Dilemma (CD) • P –> Q • R -> S • P v R • Q v S • Destructive Dilemma (DD) • P –> Q • R -> S • ~Q v ~S • ~P v ~R

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