Intro to Logic HW2

Intro to Logic HW2 - HW#2 Wednesday, July 18, 2007 10:25 AM...

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Unformatted text preview: HW#2 Wednesday, July 18, 2007 10:25 AM (PQ)((~P v Q)&(~Q v P)) P T T F F Q T F T F (PQ) T F F T T T T T ((~P v Q) & T F T T T F F T (~Q v P)) T T F T The conditional connective is a tautology so A logically implies B. ((~P v Q)&(~Q v P))(PQ) P T T F F Q T F T F ((~P v Q) & T F T T T F F T (~Q v P)) T T F T T T T T (PQ) T F F T The conditional connective is a tautology so B logically implies A. It is easy to see without making another truth table that the truth values for (PQ) are always the same as the conjunction connective so inserting a biconditional in place of the conditional will also result in a tautology. This shows that A and B are logically equivalent. Additionally it can be said that since A implies B and B implies A they are logically equivalent. ((P&Q)R)((P v Q)R) P T T T T F F F F Q T T F F T T F F R T F T F T F T F P&Q T T F F F F F F T F T T T T T T R T F T F T F T F T T T F T F T T P v Q T T T T T T F F T F T F T F T T R T F T F T F T F A does not logically imply B. The main connective is not a tautology. B T F T F T T T T A T F T T Intro to Logic Page 1 T F T T T T T T T T T T B logically implies A because the conditional is a tautology. Since A does not imply B, they can not be logically equivalent. P T T F F A T F T T Q T F T F T T T T (A) PQ (B)~Q~P T F T T B T F T T T F T T The main connective is a tautology so A logically implies B. B T F T T T T T T A T F T T Again the main connective is a tautology so B now logically implies A. Since B logically implies A and A logically implies B, they are logically equivalent. P T T T T F F F F A T F T T T T T T Q T T F F T T F F T T F F T F T F R T F T F T F T F B T F F F T F T F (P&Q) T T F F F F F F (A) T F T T T T T T R T F T F T F T F PQ T T F F T T T T &(B) T F F F T F T F R T F T F T F T F This is not a tautology so A does not logically imply B. B A Intro to Logic Page 2 T F F F T F T F T T T T T T T T T F T T T T T T This is a tautology so B logically implies A. Since A did not logically imply B, A and B can not be logically equivalent. P T T T T F F F F Q T T F F T T F F R T F T F T F T F PQ T T F F T T T T ; QR T F T T T F T T / ~P~R T T T T F T F T This is an invalid argument because there exist cases where the premises are true, but the conclusion is false. For an argument to be valid the conclusion must follow from its premises. If it is raining then the ground is wet. If the ground is wet, then the ground is slippery. /If it is not raining, then the ground is not slippery. P T T F F Q T F T F P~Q F T T T ; Q v P T T T F / ~PQ F T T F We can see that in any case where the premises are true, the conclusion holds true as well. This argument is valid. P T T T T F F F F Q T T F F T T F F R T F T F T F T F P v Q T T T T T T F F ; PR T F T F T T T T ; QR T F F T T F F T / ~~R T F T F T F T F We can see that all cases where the premises are true have true conclusions as well. This argument is valid. Intro to Logic Page 3 ((P~~Q)&(~Q v P))(PQ) P T T F F Q T F T F (P~~Q) & T F T T T F F T (~Q v P) T T F T T T T T (PQ) T F F T When we put all the premises together with conjunctions we are effectively checking for cases where all premises are true. These cases will result in a true value for the final conjunction. Once we have all cases where the premises are true, the conditional connective will check whether or not the conclusion is false. If the conditional results in a tautology (as in this problem), then the argument is valid. This process utilizes the connectives we are already familiar with to check for a false conclusion, given true premises. (((P&~Q)&(QP))&(P v P))~Q P T T F F Q T F T F ((P&~Q) & F T F F F T F F (QP)) & T T F T F T F F (P v P) T T F F T T T T ~Q F T F T The conditional forms a tautology so this argument is valid. See explanation above. (((P v R)&~(PR))&(~P&~Q))(R v Q) P T T T T F F F F Q T T F F T T F F R T F T F T F T F P v R T T T T T F T F & F T F T F F F F ~(PR) & F T F T F F F F F F F F F F F F ~P&~Q F F F F F F T T T T T T T T T T (R v Q) T T T F T T T F The conditional forms a tautology so this argument is valid. See explanation above. Intro to Logic Page 4 ~(J v R) or ~J & ~R (K&~O)M (L&P)(O v T) I wasn't sure whether you meant Tegan was in the Tardis or the Doctor is in the spaceship. (L&P)(O v D) PS ~J(T&D) ~(K v O)&R Dictionary: A = Sally is destined to be a fearless adventurer G = Sally is destined to be a great psychiatrist P = Robert is paranoid (A v G); ~P~G; ~A/P A T T G T T P T F A v G T T ; ~P~G ; T F ~A F F / P T F Intro to Logic Page 5 T T F F F F F F T T F F T F T F T F T T T T F F T T T F T T F F T T T T T F T F T F The only case in which all premises are true has a true conclusion. Thus, this argument is valid. Intro to Logic Page 6 ...
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This homework help was uploaded on 04/07/2008 for the course PHILOSOPHY 201 taught by Professor Morgan during the Spring '08 term at Rutgers.

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