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Philosophy 1100

Join Paul Dickey's Philosophy class for a discussion on critical reasoning. Learn about deductive arguments, truth-functional logic, and how to test validity using truth tables. Contact Paul Dickey at pdickey2@mccneb.edu for more information.

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Philosophy 1100

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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: Student Portfolios Final Essay Questions? Exercise 8-11 Finish Chapter 8 & begin Chapter 9 Next Week: Class Essay is due. Read Chapter 9, pages 317-330 Exercise 9-7 1

  2. 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.

  3. 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.

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

  5. Chapter NineDeductive 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. Class Workshop: • Exercise 9-7, #3

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