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lec0113 - Methods to Prove Logical Implications The...

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Methods to Prove Logical Implications The standard form of an argument (or theorems) can be represented using the logical symbols we have learned so far: (p 1 p 2 p 3 ... p n ) q Essentially, to show that this statement is always true (a tautology), we must show that if all of p 1 through p n are true, then q must be true as well. Another way to look at this is that we must show that if q is ever false, then at least ONE of p 1 through p n must be false as well. (This is the contrapositive of the original assertion.) Let’s look at an example of how you might go about proving a statement in a general form, given some extra information. Let p, q, are r be the following statements: p: Sam Madison returns an interception a touchdown. q: The Dolphins rush for under 100 yards. r: The Dolphins will beat the Jaguars. Let the premises be the following: p 1 : If the Dolphins rush for over 100 yards, then Sam Madison will return an interception for a touchdown. p 2 : If Sam Madison returns an interception for a touchdown, the Dolphins will beat the Jaguars. p 3 : The Dolphins did rush for over 100 yards against the Jaguars.
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Now, I want to show that (p 1 p 2 p 3 ) r First, I must express p 1 , p 2 , and p 3 in terms of p, q, and r. p 1 : ¬ q p p 2 : p r p 3 : ¬ q Thus, we are trying to show the following: [( ¬ q p) (p r) ( ¬ q)] r
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