Quiz #6 -- ANSWER KEY with comments Name ________________________________________PHIL 110, Spring 2014 (10 pts)HAMILTON, 3/3/2014PART 1: Circle TRUE or FALSE for each of the following statements. (3 pts each)1. T A valid argument may have premises that contradict each other.Not only may a valid argument have contradictory premises, but any argument with contradictory premises will automatically be valid, because anything and everything follows from (i.e., is a logical consequence of) a contradiction. Refer tothe discussion of why the proof rule ⊥ Elim works.2. F The proof rule ⊥ Intro is also known as “proof by contradiction”.Actually, it’s the proof rule ¬ Intro that is known as proof by contradiction.3. F The proof rule ¬ Elim involves the use of a subproof, the premise of which is the opposite (or negation) of what you actually want to prove.No, this actually describes to the proof rule ¬ Intro.4. F The schema for the use of the proof rule ⊥ Elim involves first reaching a contradiction as the conclusion of a subproof, then closing the subproof and asserting anything at all one level back (toward the main proof level).It’s the last part of this description that makes it false. In actuality, with the proof rule ⊥ Elim you are allowed to assert ‘anything at all’ only at the same proof level as (i.e., directly underneath) the contradiction symbol, not ‘one level back’. (Don’t confuse this rule with ¬ Intro, which also involves the use of a contradiction symbol. With ¬ Intro, you do close the subproof immediately after the contradiction symbol and then assert the negation of the premise one level back.) 5. T Once a subproof has been discharged, only the subproof as a whole may be cited as input to a proof rule.That’s right, because the entire subproof hangs on its premise, meaning that anything stated within the subproof is ‘true’ only if the premise of the subproof istrue. In most cases you’ll either not know whether the premise of the subproof isactually true (as with some or all of the premises of subproofs used with a proof by cases—i.e., disjunction elimination—strategy) or you’ll actually know that the premise you’re ‘supposing’ is, in fact, not true at all (as in all cases of the subproof used with a proof by contradiction—i.e., negation introduction—strategy).Part 2: Compose a proof demonstrating the validity of the following argument. Be sure to format the proof correctly, including providing any necessary fitch bars, numbering your steps, and justifying eachstep (as needed) with a specific rule and any required step citations. (partial credit up to 15 pts)6. PREMISE: Cube(b) ᴧ Dodec(c)CONCLUSION: ¬¬Dodec(c)1 Cube(b) ᴧ Dodec(c)2 Dodec(c) ᴧ Elim: 13 ¬Dodec(c)4 ⊥ ⊥ Intro: 2,35 ¬¬Dodec(c) ¬Intro: 3-4Many of you made the above proof harder than it actually is. The first key was to realize that theonly ‘piece’ of the main premise that you need is the final conjunct Dodec(c), which you can access only by eliminating the conjunction in which it’s embedded (you could either assert this individual conjunct at the main proof level before starting the subproof, as I’ve shown above, or you could wait and assert it inside the subproof right before the contradiction symbol, as some of you did). The other key was to recognize that the only way to generate the double negation in the goal is to use a proof by contradiction strategy via the rule ¬ Intro. If you remembered how that rule works, you knew that you would need to premise the opposite of the goal and show that this premise leads to a contradiction, so that you could then close the subproof and assert the negation of the premise back at the main level. The subproof premise you needed forthis is ¬Dodec(c), which when negated at step 5 becomes the desired ¬¬Dodec(c). Some of you were on the right track but used Dodec(c) as the premise of your subproof (i.e., without the negation)—but that won’t work, because you needed the opposite of the goal you want to prove, whereas Dodec(c) is instead equivalent to the goal (not it’s opposite). Perhaps those of you who made that mistake were reluctant to premise something you knew to be false in light of the main premise—but keep in mind that this is precisely how proof by contradiction (negation intro) works: the premise of your subproof will be the opposite of the goal you’re actually working toward, which means you intentionally premise something that you know will lead to a contradiction … something that you know is actually false. Finally, a few students tried opening two subproofs, as though attempting a ‘proof by cases’ strategy. But notice that step 1 is a conjunction, not a disjunction, so you can’t base a proof by cases off of it.PART 3: Fill in the missing rules and step citations for the following proof. Only fill in those blanks that require a rule or step citation (not all of the blanks should be filled in). (2 pts each blank)7. 1 P ᴠ Q ᴠ R rule? ______ _ ___ step(s)? ___________2 ¬¬¬P ᴧ ¬¬¬R rule? ______ _ ___ step(s)? ___________3 P rule? ______ _ ___ step(s)? ___________4 ¬¬¬P rule? __ ᴧ Elim_ __ step(s)? __2________5 ¬P rule? __¬ Elim____ step(s)? __4________6 ⊥ rule? __ ⊥ Intro ___ step(s)? __3, 5______7 Q rule? __ ⊥ Elim_ ___ step(s)? __6________8 Q rule? ______ _ ___ step(s)? ___________9 Q rule? __Reit_ _ ___ step(s)? __8________10 R rule? ______ _ ___ step(s)? ___________11 ¬¬¬R rule? __ᴧ Elim _ ___ step(s)? __2________12 ¬R rule? __ ¬ Elim_ ___ step(s)? __11_______13 ⊥ rule? __ ⊥ Intro ___ step(s)? __10, 12____14 Q rule? __ ⊥ Elim_ ___ step(s)? __13_______15 Q rule? __ᴠ Elim _ ___ step(s)? 1, 3-7,8-9, 10-14The overall structure of the above proof is a proof by cases (hence, the premises of the three subproofs are taken from the three disjuncts of the main premise, and the rule ᴠ Elim justifies the last step of the proof). Some students (despite my many warnings) tried to justify the premises in the proof—especially the premises of the subproofs at steps 3, 8, and 10. But remember that premises of any sort are never justified (they are simply ‘supposed’ or ‘given’ to be true). If you had trouble with parts of this proof, study the rules and step citations above carefully to understand how each step is justified, and refer back to the
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