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# Class Note for PHIL 110 at UMass(15)

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COURSE
PROF.
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TYPE
Class Notes
PAGES
8
WORDS
KARMA
25 ?

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This 8 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Massachusetts taught by a professor in Fall. Since its upload, it has received 14 views.

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Date Created: 02/06/15
INTRO LOGIC CHAPTER 3 DAY 04 VALIDITY IN SENTENTIAL LOGIC 1 3 Schedule for Unit 1 Validity in General Day1 Intro an argument is valid an argument is invaicl J D 2 Ch t 1 if and only if if and only if ay ap er it is Mpossible for it is possible for Day 3 Chapter 2 the conclusion to be the conclusion to be 40 of false false Day 4 Chapter 3 Exam 1 while while Day 5 Chapter 4 the premises are true the premises are true 60 of Day 6 Chapter 4 Exam 1 Day 7 Chapter 4 Day 8 EXAM 1 Validity in Sentential Logic Exampe1 an argument is valid an argument is analid If an argument form has 2 atomic sentences if and only if if and only if then there are 4 cases 4 22 there is there is m case at least one case R S in which in which the premises are true the premises are true case 1 T T and and case 2 T F the conclusion is the conclusion is false false case 3 F T case 4 F F 5 What is a Case Example 2 A case is a If an argument form has 3 atomic sentences 3 possible combination of truthvalues the there are 8 cases 8 39 2 139 assigned to the atomic formulas Q R 3 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 Example 1 Modus Tollens premise premise I conclusion if R then S not 8 I not R R a S MS I NR Example 2 Evil Twin of Modus Tollens TruthTable case R S R a S MS I NR 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 butthe conclusion is false Is the argument form valid or invalid VALID 10 premise premise I conclusion if R then S not R I not 8 R a S NR I NS 11 Counterexample ithhen S notR InotS ifl live in Boston I don39t live I don39t live then I live in Mass in Boston in Mass T T F 12 TruthTable TruthTable case R S RaS NR INS case R S RaS R IS 1 T T T F F 1 T T T T T 2 T F F F T 2 T F F T F 3 F T T T F 3 F T T F T 4 F F T T T 4 F F T F F Is there a case in which the premises are all true YES Is there a case in which the premises are all true No butthe conclusion is false butthe conclusion is false Is the argument form valid or invalid INVALID Is the argument form valid or invalid l VALID l 13 15 Example 3 Example 4 Modus Ponens Evil Twin of Modus Ponens premise premise I conclusion premise premise I conclusion if R then S R I S if R then S S I R R a S R I S R a S S I R 14 16 Counterexample Example 5 Modus Tollendo Ponens disjunctive syllogism premise premise I conclusion R or S not R I S R v S NR I S 19 if R then S S I R ifl live in Boston I live in I live in then I live in Mass Mass Boston T T F 17 TruthTable case R S R a S S I R 1 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 butthe conclusion is false Is the argument form valid or invalid INVALID 18 TruthTable case R S R v S NR 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 Example 6 Evil Twin of MTP premise premise I conclusion R or S R I not 8 R v S R I NS 21 Example 7 notR I notRand S RampS 23 TruthTable case R S R v S R I NS 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 butthe conclu5ion IS false Is the argument form valid or invalid INVALID 22 TruthTable R I 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 No but the conclusion is false Is the argument form valid or invalid VALID 24 Example 8 not R and 8 not R RampS 25 Logical Equivalence Two formulas are logically equivalent if and only if they have the same truthvalue no matter what in every case 27 TruthTable ampS I aaam mm i li lm 111a marme aamm mmaam butthe conclusion is false Is there a case in which the premises are all true YES Is the argument form valid or invalid INVALID 26 Examples 7 and 8 ZOMBIE REASONING not R and S not R and not 8 3 not RorS not Ror notS 3 IT IS JUST LIKE MATH x y2 x2 y2 a x y xx y 3 28 TruthTable for 7 Valid Equivalence 1 RampSII 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 T Do the formulas match in truth value T NO T T Are the two formulas logically equivalent T NO T 30 TruthTable for 8 RVSII R v 3 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 T Do the formulas match in truth value T NO T T Are the two formulas logically equivalent T NO T 31 RampSII R v 3 F T T T T 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 T Do the formulas match in truth value T YES T T Are the two formulas logically equivalent T YES T 32 Valid Equivalence 2 RVSII R amp 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 T Do the formulas match in truth value T YES T TArethe two formulas logically equivalent T YES T 33

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