Help with Exercises

Help with Exercises

Help with Exercises
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Below you will find help with selected exercises from the book.

9-2, 5. If Parsons signs the papers then if Quincy goes to jail Rachel will file an appeal.
P (Q R). Note the "then if" phrase. It tells you (1) that P will be set apart from everything that follows, and (2) that a conditional claim will come inside the parentheses.

9-4, 7.
(P & R) Q
~Q
~P
Invalid. Let's do this with the short truth-table method. Very quickly, a false conclusion and a true second premise give us P's truth and Q's falsity:
P     Q     R
T     F
Looking up to the first premise, we see that its antecedent must be true to make the whole premise true (since Q is false). P is already true; R must be true as well. We get:
P     Q     R
T     F     T
No contradictions here; so the argument is invalid.

9-6, 5.
(Q T) S
~S v ~P
R P
~(Q T) v ~R
Disjunctive dilemma. Note the negations of both consequents. Ignore the complexity of (Q T). (In such cases, it helps to think of parentheses as a nutshell: You treat the nut as a single object, no matter how many bits may be rattling around inside.)

9-7, 10.
         1. (T v M) ~Q
         2. (P Q) & (R S)
         3. T                           /_{<a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=jpg::::/sites/dl/free/0073386677/610541/therefore.jpg','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif"> (0.0K)</a>} ~P
Start by thinking strategically. The letter P will be in your conclusion, and premise 2 is the only line with a P in it, so you'll have to get the P from there. Premise 2 also contains an R and an S that you don't need; so use simplification to drop that whole part of the line. Now you have the line P Q. How can you get ~P from that? Modus tollens will do it for you, if you can find a ~Q. But where can you get that? Your only choice is premise 1, which will give you a ~Q if you can only find a T v M. Premise 3 supplies the T. Now you can see what to do:
         4. T v M          3, ADD
         5. ~Q             1, 4, MP
         6. P Q        2, SIM
         7. ~P              5, 6, MT

9-10, 7.
         1. (M v R) & P
         2. ~S ~P
         3. S ~M          /_{<a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=jpg::::/sites/dl/free/0073386677/610541/therefore.jpg','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif"> (0.0K)</a>} R
Two tricks here. First, looking at the conclusion, you realize that premise 1 will be vital to your deductive efforts. Second, you should be struck by the oddity of premise 2: both S and P negated, though they're not negated elsewhere. Use this as a clue that premise 2 will be more helpful with its negations gone—which means, if you take the contrapositive of it to produce P S. Then, if you can get S out of that conditional, premise 3 gives you ~M.
        One more thing. You now see that both conjuncts in premise 1 will be necessary, the first because it contains the vital R that's in the conclusion, and the second because you've produced a conditional containing P. There's nothing wrong with going back to premise 1 twice and simplifying it both times, once to use P on your new conditional and the other time to use M v R to get you the desired conclusion. So:
         4. P S      2, CONTR
         5. P              1, SIM
         6. S              4, 5, MP
         7. ~M           3, 6, MP
(Now we simplify premise 1 again:)
         8. M v R        1, SIM
         9. R              7, 8, DA

9-14, 4.
         1. P (Q v R)
         2. T (S & ~R) /_{<a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=jpg::::/sites/dl/free/0073386677/610541/therefore.jpg','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif"> (0.0K)</a>} (P & T) Q
It's obvious what your CP premise will be, namely P & T. But how will it lead to Q? The main strategy is to simplify that P & T once you've assumed it, and use the P and the T separately to get Q. Premise 1 will then do the trick; but premise 1 leaves you with a disjunction, which can't be simplified. That R needs to be eliminated with a ~R. Now you notice that premise 2's consequent contains a ~R that can be simplified out of the compound consequent. Now we begin:
         3. P & T                CP Premise
         4. P                      3, SIM
         5. T                      3, SIM
         6. Q v R               1, 4, MP
(The remaining work consists in also getting a ~R from the premises, which you'll then use in a disjunctive argument on step 6:)
         7. S & ~R              2, 5, MP
         8. ~R                    7, SIM
         9. Q                      6, 8, DA
       10. (P & T) Q     3–9, CP

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Home > Chapter 9 > Help with Exercises

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	Course-wide Content
		Appendix

	Quizzes
		Multiple Choice Quiz True and False Quiz Fill in the Blank Quiz
	More Resources
		Chapter Overview Chapter Objectives Frequently Asked Quest... Tips on Applications Help with Exercises Key Terms Flashcards