Critical Reasoning: Deductive Arguments Workshop

1. Philosophy 1100 Class #9 Title: Critical Reasoning Instructor: Paul Dickey E-mail Address: pdickey2@mccneb.edu Website: http://mockingbird.creighton.edu/NCW/dickey.htm Today: Submit Final Essay Turn in Exercise 9-14 Return Portfolios Discuss Chapter 10 Next Week: No class Final Exam (Take-home)

2. Final Exam will be posted on Quia by Noon, Tuesday, 11/17. Most questions are multiple choice but that there will be some questions that will require drawing & free form text. • Complete the exam in one of the following ways: • Download and print the document from the Quia site. Answer all questions on the hardcopy exam you print. After you are done, scan the exam as a jpg, pdf, doc, or docx file and email it to me at pdickey2@mccneb.edu; • or • Download and complete the exam by editing it within Microsoft Word and email it to me as a doc or docs file • to pdickey2@mccneb.edu. • E-mailed exams must be received no later than 6:00 P.M. on 11/23. Please Note: For every 8 hours (or partial) the exam is late, a full grade will be reduced. NO EXCEPTIONS.

3. As before, you may consult your notes, your laminated cards, or textbook while taking the exam, but not other students. • On your exam somewhere, please discuss which 3 (at least 3, more if you like) questions you found the most difficult and what other answers did you find “plausible.” This will help me to evaluate the exam for the future. Your exam will not be considered complete without this feedback. • Congratulations, you have completed the course. You should be proud. After you have finished the exam and e-mailed it to me, take the evening off and do something nice for yourself. I have enjoyed the class and hope you have also.

4. Chapter NineDeductive Arguments:Categorical Logic Class Workshop Exercise 9-14

5. Chapter TenDeductive Arguments:Truth-Functional Logic

6. 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.

7. 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

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

9. 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

10. 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

11. 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?

12. 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.

13. 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.

14. 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.

15. 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.

16. 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

17. Laziness is the Mother of Invention However there is often an easier way to demonstrate validity with truth tables. It is called the short truth-table method. The basic principle of this method simply is to look for a row that makes the argument invalid. As soon as you find one, you are done. If you exhaust all opportunities and can’t, then the argument is valid. Consider the argument: P -> Q ~Q -> R ~P -> R The argument could be invalid only if the conclusion is false while the premises are true. P Q R F T F Thus, the argument is invalid.

18. Now, consider the argument: (P v Q) -> R S -> Q S -> R The argument could be invalid only if the conclusion is false while the premises are true. To make the conclusion false -- P Q R S F T To make the second premise true -- P Q R S T F T But there is no way now to make the first premise true, so the argument is valid.

19. Exercises 10-6 1. K -> (L & G) M -> (J & K) B & M B & G To make the third premise true – B M T T But to make premise 2 true B M J K T T T T But to make premise 1 true, B M J K L G T T T T T T But there is no way now to make the conclusion false, so the argument is valid.

20. Class Workshop (Short Method) Translation Exercise: If Scarlet is guilty of the crime, then Ms. White must have left the back door unlocked and the colonel must have retired before ten o’clock. However, either Ms. White did not leave the back door unlocked, or the colonel did not retire before ten. Therefore, Scarlet is not guilty of the crime. Now, is is this valid? Look at pages 301 & 302 for the long proof. Wow! Now, let’s be “lazy”…

21. Deductive Arguments:Rules of Induction

22. 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

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

24. 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

25. 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

26. 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

27. Exercises 9-10 #1 • R -> P • Q -> R / .’. Q -> P • Q -> P 1,2, CA #2 • P -> S • P v Q • Q -> R / .’. S v R • S v R 1,2,3, CD #10 • (T v M) -> ~Q • (P -> Q) & (R-> S) • T / .’. ~P • T v M 3, ADD • ~ Q 1, 4, MP • P -> Q 2, SIMP • ~P 5, 6, MT