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# Class Note for PHIL 110 at UMass(20)

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Date Created: 02/06/15

INTRO LOGIC DAY 04 Schedule for Unit 1 Day l Intro Day2 Chapter l Day3 ChapterZ Day4 Chapter3 Day5 Chapter4 Day6 Chapter4 Issac1 Day7 Chapter4 Day8 EXAM 1 warmup 40 of Exam 1 CHAPTER 3 VALIDITY IN SENTENTIAL LOGIC Validity in General an argument is valid if and only if it is Mpossible for the conclusion to be false while the premises are true an argument is invalid if and only if it is possible for the conclusion to be false while the premises are true What are Possibilities in Sentential Logic at least a possibility is a case which is a possible combination of truthvalues assigned to the atomic sentences Example 1 If an argument form has 2 atomic sentences then there are 4 cases R S case 1 T T case 2 T F case 3 F T case 4 F F Example 2 If an argument form has 3 atomic sentences then there are 8 cases Q R 8 case 1 T T T case 2 T T F case 3 T F T case 4 T F F case 5 F T T case 6 F T F case 7 F F T case 8 F F F In General If an argument form has n atomic sentences then there are 2quot cases Example 1 Modus Tollens premise1 premise2 I conclusion if R then S not S I not R R gt S S I R 9 TruthTable case R S R a S S I R 1 T T T F F 2 T F F T F 3 F T T F T 4 F F T T T Is there a case in which the premises are all true NO but the conclusion is false Is the argument form valid or invalid VALID 10 Example 2 Evil Twin of Modus Tollens premise1 premise2 I conclusion if R then S not R I not S R gt S R I S 11 Counterexample if R then S not R I not S if I am in Boston I am not in I I am not in then I am in Mass Boston Mass T T F 12 TruthTable case R S R a S NR 1 Ms 1 T T T F F 2 T F F F T 3 F T T T F 4 F F T T T Is there a case in which the premises are all true YES but the conclusion Is false Is the argument form valid or invalid INVALID Example 3 Modus Ponens premise1 premise2 conclusion ithhenS R I S RAS R IS TruthTable case R S R a S R I S 1 T T T T T 2 T F F T F 3 F T T F T 4 F F T F F Is there a case in which the premises are all true NO but the conclusion Is false Is the argument form valid or invalid VALID 15 Example 4 Evil Twin of Modus Ponens premise1 premise2 I conclusion if R then S S I R R a S S I R 16 Counterexample if R then S S I R if I am in Boston I am in I I am in then I am in Mass Mass Boston T T F 17 TruthTable case R S R a S S I R l T T T T T 2 T F F F T 3 F T T T F 4 F F T F F Is there a case in which the premises are all true YES but the conclusion is false Is the argument form valid or invalid INVALID 18 Example 5 Modus Tollendo Ponens disjunctive syllogism premise1 premise2 Iconclusion RorS notR I S R v s R I s 19 TruthTable case R S R v S R I S 1 T T T F T 2 T F T F F 3 F T T T T 4 F F F T F Is there a case in which the premises are all true NO but the conclusion is false Is the argument form valid or invalid VALID 20 1O Example 6 Evil Twin of MTP premise premise I conclusion R or S R I not S R v S R I S 21 TruthTable case R S R v S R I S 1 T T T T F 2 T F T T T 3 F T T F F 4 F F F F T Is there a case in which the premises are all true YES but the conclusion Is false Is the argument form valid or invalid INVALID 22 11 Example 7 notR not R and S RampS 23 TruthTable R R amp s F T F T T T F T T T F F T F T F F T T F T F F F Is there a case in which the premises are all true N0 but the conclusion Is false Is the argument form valid or invalid VALID 24 12 Example 8 notRand S not R RampS R 25 TruthTable R amp S R F T T T F T T T F F F T T F F T T F T F F F T F Is there a case in which the premises are all true YES but the conclusion Is false Is the argument form valid or invalid INVALID 26 13 Logical Equivalence Two formulas are logically equivalent if and only if they have the same truthvalue no matter what in every case 27 Examples 7 and 8 ZOMBIE REASONING not R and S notR and notS 3 not R or S notR or notS 3 IT E JUST LIKE MATH X 302 X2 y2 3 l x y x y 5 28 14 Examples 7 and 8 cont not R and S i not R and not 8 not R or 8 not R or notS H 29 TruthTable for 7 RampSll R amp 3 F T T T F T F F T T T F F F T F T F T F F T T F F F T T F F F T F T T F Do the formulas match in truth value N0 Are the two formulas logically equivalent N0 30 15 TruthTable for 8 R v S N R v N S F T T T F T F F T F T T F F T T T F F F T T T F T F T T F F F T F T T F Do the formulas match in truth value N0 Are the two formulas logically equivalent N0 31 Valid Equivalence 1 RampSll R v 8 F T T T F T F F T T T F F F T T T F T F F T T F T F T T F F F T F T T F Do the formulas match in truth value YES Are the two formulas logically equivalent YES 32 16 Valid Equivalence 2 RvS Ramp 3 F T T T F T F F T F T T F F T F T F F F T T T F F F T T F F F T F T T F Do the formulas match in truth value YES Are the two formulas logically equivalent YES 33 34 17

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