HACKING MATHEMATICS UNIT 2 MODULE 2 ANALYZING PREMISES FORMING CONCLUSIONS First we define a Trivial Valid Conclusion No matter how poorly formulated an argument may be it is always possible to form a valid conclusion by merely restating one of the premises and calling it the conclusion Such a conclusion is called a trivial valid conclusion EXAMPLES OF ARGUMENTS HAVING TRIVIAL VALID CONCLUSIONS 1 It is raining Therefore it is raining 2 My feet hurt and I m having a bad hair day Therefore my feet hurt 3 If I work hard then I will succeed I succeeded Therefore I succeeded In the work that follows we will analyze collections of premises in order to recognize whether valid conclusions are possible In all cases we will exclude trivial conclusions 18 HACKING MATHEMATICS We begin this module with a summary of important results observed in Unit 2 Module 1 SUMMARY SOME COMMON PATTERNS OF VALID REASONING DIRECT REASONING p q CONTRAPOSITIVE REASONING p q p q q p DISJUNCTIVE SYLLOGISMS TRANSITIVE REASONING p q p q p q q p q r p q p r SUMMARY SOME COMMON PATTERNS OF INVALID REASONING FALLACY OF THE CONVERSE FALLACY OF THE INVERSE p q q p q p q p DISJUNCTIVE FALLACIES p q p q p q q p FALSE CHAINS p q p q p r r q q r p r 19 UNIT 2 MODULE 2 20 HACKING MATHEMATICS These patterns of reasoning are especially useful for exercises like the following example EXAMPLE 2 2 1 Select the statement that is a valid conclusion from the following premises if a valid conclusion is warranted I use my computer or I don t get anything done I get something done A I use my computer B I don t use my computer C I use an abacus D None of these is warranted EXAMPLE 2 2 1 Solution This argument problem differs from the earlier examples in that we aren t given a conclusion for the argument This means that if we try to use a truth table to analyze the argument we may not be sure what statement to use in the conclusion column However it is easily solved by reference to the patterns of reasoning summarized above We symbolize the premises Let p be the statement I use my computer Let q be the statement I don t get anything done Then the symbolic representation for the two premises has this form p q q Now we observe that this is the premise arrangement for one form of Disjunctive Syllogism which is a form of valid reasoning The pattern tells us that we can form a non trivial valid conclusion p q q p That is when Therefore p is attached as the conclusion we will have a valid argument This means that a valid conclusion is warranted namely I use my computer which is choice A 21 UNIT 2 MODULE 2 EXAMPLE 2 2 2 Select the statement that is a valid conclusion from the following premises if a valid conclusion is warranted If I win the Lotto then I ll reform my life I reformed my life A I didn t win the Lotto B I ll run for President as the Reform Party nominee C I won the Lotto D None of these is warranted EXAMPLE 2 2 2 solution Let p be the statement I win the Lotto Let q be the statement I reform my life The premise arrangement has this symbolic form p q q We recognize that this is the premise arrangement for an invalid argument Fallacy of the Converse This tells us that the best sounding choice C I won the Lotto is not correct because that choice would result in an invalid argument More importantly because we have the premise arrangement for an invalid argument it is not possible to produce a non trivial valid conclusion This tells us that the correct choice must be D EXAMPLE 2 2 3 Select the statement that is a valid conclusion from the following premises if a valid conclusion is warranted If we strive then we excel We didn t strive A We excelled B We didn t excel C We didn t inhale D None of these is warranted 22 HACKING MATHEMATICS 23 UNIT 2 MODULE 2 EXAMPLE 2 2 4 Select the statement that is a valid conclusion from the following premises if a valid conclusion is warranted If we win then we celebrate We aren t celebrating A We won B We didn t win C We stink D None of these is warranted EXAMPLE 2 2 5 Given i If my car doesn t start then I ll be late for work and ii I m not late for work select the statement that is a valid conclusion if a valid conclusion is warranted A My car started B I rode the bus C I m late for work D None of these is warranted EXAMPLE 2 2 6 Given i All nurses are kind and ii Florence isn t a nurse select the statement that is a valid conclusion if a valid conclusion is warranted A Florence is a city in Italy B Florence isn t kind C Florence is kind D None of these is warranted EXAMPLE 2 2 7 Given i No kittens are fierce and ii Fluffy isn t fierce 24 HACKING MATHEMATICS select the statement that is a valid conclusion if a valid conclusion is warranted A Fluffy is a kitten B Fluffy has fleas C Fluffy isn t a kitten D None of these is warranted WORLD WIDE WEB NOTE For practice on problems like these visit the companion website and try THE DEDUCER SPECIAL CASES INVOLVING TRANSITIVE REASONING EXAMPLE 2 2 8 Select the statement that is a valid conclusion from the following premises if a valid conclusion is warranted If you want a better grade then you bring an apple for the teacher If you bring an apple for the teacher then you expose the teacher to dangerous agricultural chemicals A If you expose the teacher to dangerous agricultural chemicals then you want a better grade B If you don t expose the teacher to dangerous agricultural chemicals then you don t want a better grade C You want a better grade D None of these is warranted EXAMPLE 2 2 8 solution Let p be the statement You want a better grade Let q be the statement You bring an apple for the teacher Let r be the statement You expose the teacher to dangerous agricultural chemicals The premise arrangement has this form p q q r We see that this is the arrangement of premises for Transitive Reasoning which is a form of valid reasoning This means that we will be able to form a valid conclusion namely p q q r p r 25 UNIT 2 MODULE 2 In words the valid conclusion is If you want a better grade then you expose the teacher to dangerous agricultural chemicals Unfortunately this isn t one of the listed choices We may now refer to the following fundamental fact …