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Unit 2 Module 2 Analyzing Premises, Forming Conclusions



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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



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