Sloppy Sam is trying to prove a certain proposition P. He defines two related propositions and...
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Sloppy Sam is trying to prove a certain proposition P. He defines two related propositions and R, and then proceeds to prove four implications: R⇒Q, Q⇒R, R⇒P, P⇒ Q. He then reasons as follows If Q is true, then since I proved Q⇒ R, I can conclude that R is true. Now, since I proved R⇒ P, I can conclude that P is true. Similarly, if R is true, then P is true and so Q is true. Likewise, if P is true, then so are Q and R. So any way you look at it, all three of P, Q, and R are true. (a) Exhibit truth tables for and for (PQ) ^ (Q⇒ R) ^ (R⇒P), (1) PAQAR. (2) Use these tables to find a truth assignment for P, Q, R so that (1) is T and (2) is F. (b) You show these truth tables to Sloppy Sam and he says "OK, I'm wrong that P, Q and R all have to be true, but I still don't see the mistake in my reasoning. Can you help me understand my mistake?" How would you explain to Sammy where the flaw lies in his reasoning? Sloppy Sam is trying to prove a certain proposition P. He defines two related propositions and R, and then proceeds to prove four implications: R⇒Q, Q⇒R, R⇒P, P⇒ Q. He then reasons as follows If Q is true, then since I proved Q⇒ R, I can conclude that R is true. Now, since I proved R⇒ P, I can conclude that P is true. Similarly, if R is true, then P is true and so Q is true. Likewise, if P is true, then so are Q and R. So any way you look at it, all three of P, Q, and R are true. (a) Exhibit truth tables for and for (PQ) ^ (Q⇒ R) ^ (R⇒P), (1) PAQAR. (2) Use these tables to find a truth assignment for P, Q, R so that (1) is T and (2) is F. (b) You show these truth tables to Sloppy Sam and he says "OK, I'm wrong that P, Q and R all have to be true, but I still don't see the mistake in my reasoning. Can you help me understand my mistake?" How would you explain to Sammy where the flaw lies in his reasoning?
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Related Book For
Management and Cost Accounting
ISBN: 978-1405888202
4th edition
Authors: Alnoor Bhimani, Charles T. Horngren, Srikant M. Datar, George Foster
Posted Date:
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