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Unformatted text preview: Intro. to Logic Model Solutions for Midterm Exam 16 February 2012 (1) (a) Consider this sentence: ((P1 & (P1 P2)) (P1 (P1 & P2))) Is it a tautology or not? Justify your answer in this case by giving a complete truth table for it, together with a brief explanation. Truth Table (main connective shown in bold): P1 T T F F P2 ((P1 T T F T T F F F & (P1 T T T T F F F F P2)) T T T F T T F F (P1 T T T T T F T F (P1 T T T T F F F F & P2))) T T F F F T F F Answer and Explanation: This sentence is a tautology because it is true in all rows. (b) What can you conclude from (1a) about the following two sentences? (P1 & (P1 P2)) and (P1 (P1 & P2)) Do not points for it.) Instead, justify your answer by appealing to the result of (1a). Answer and Justification: These two sentences are logically equivalent because we can see them in the truth table for (1a) as the left and right sides of the biconditional, and they have the same truth values in all rows. In general, (A B) is a tautology if and only if A and B are logically equivalent. In this case, A stands for (P1 & (P1 P2)) and B stands for (P1 (P1 & P2)). Since we showed in (1a) that ((P1 & (P1 P2)) (P1 (P1 & P2))) is a tautology, we know that (P1 & (P1 P2)) and (P1 (P1 & P2)) must be logically equivalent. or (2) Consider the following argument: (Q1 Q2), (Q2 Q3), Q3 / Q1 Is this a valid argument or not? Justify your answer here either by giving a complete truth table or by using one of our shortcut methods, with brief explanation. Truth Table (main connectives shown in bold): Q1 T T T T F F F F Q2 T T F F T T F F Q3 (Q1 T T F T T T F T T F F F T F F F T T T T T T F F Q2) (Q2 T T T T F F F F T T T T F F F F Q3) T T T F T T F F T T T F T T F F F T F T F T F T Q3 T F T F T F T F Q1 T T T T F F F F Answer and Explanation: This argument is invalid because there is at least one row (row #6, highlighted in gray) where all the premises are true and the conclusion false. (3) Consider this list of sentences: P1, (P1 P2), (P2 P3), P3 (a) Is this list consistent or not? Justify your answer here either by giving a complete truth table or by using one of our shortcut methods, with brief explanation. Truth Table (main connectives shown in bold): P1 T T T T F F F F P2 T T F F T T F F P3 T F T F T F T F F F F F T T T T P1 (P1 T T T T T T T T F F F F F F F F P2) (P2 T T T T T T T F F T F F T T T T T T F F F F F F P3) T T F F T T T F T T F F T T T F F T F T F T F T P3 T F T F T F T F Answer and Explanation: This list is inconsistent because there is no row where they are all true. (b) What can you conclude from (3a) about the following argument: P1, (P1 P2), (P2 P3) / P3 Do not points for it.) Instead, justify your answer by appealing to the result of (3a). Answer and Justification: The premises of the argument are the first three sentences of the list in (3a); the conclusion of the argument is the negation of the fourth sentence of the list in (3a). As we saw in (3a), the list is inconsistent, i.e. there are no rows where P1, (P1 P2), (P2 P3), and P3 are all true. But this means that there are no rows where P1, (P1 P2), and (P2 P3) are all true and P3 is false. In other words, there are no rows where all premises of the argument are true and the conclusion is false. Thus, the argument is valid. Solution for (a): Translations of Basic Sentences: P1: Unemployment will go down. P2: Production will go up. P3: Investment will increase. (4) Consider the following two arguments: (a) Unemployment will go down if and only if either production will go up or investment will increase (or both). Investment will not increase. Hence unemployment will go down if and only if production will go up. (b) The murderer will be caught only if Sherlock Holmes will take over the investigation. Holmes will take over the investigation if Watson will help him. Another person will be killed if the murderer will not be caught. No other person will be killed. Thus Holmes will take over the investigation and Watson will help him. For each of them, first translate the argument into SENTENTIAL. Be explicit about which English sentences are translated into which basic sentences of Sentential. Then, again in each case, use a truth table a complete truth table or a shortcut method, with corresponding explanation to determine whether it is valid or not. Translation of Argument: (P1 (P2 P3)) P3 (P1 P2) Truth Table: P1 P2 T T T T T F T F F T F T F F F F P3 (P1 T T F T T T F T T F F F T F F F (P2 T T T T T F F F F T F T F F T F P3)) T T T F T T F F T T T F T T F F P3 (P1 F T T T F T F T T T F T F T F T F F F T F T F F P2) T T T T F F F F F T F T T F T F Answer and Explanation: This argument is valid because there are no rows where all the premises are true and the conclusion false. Solution for (b): Translations of Basic Sentences: Q1: The murderer will be caught. Q2: Sherlock Holmes will take over the investigation. Q3: Watson will help Sherlock Holmes. Q4: Another person will be killed. Translation of Argument: (Q1 Q2) (Q3 Q2) ( Q1 Q4) Q4 (Q2 & Q3) Truth Table: Q1 T T T T T T T T F F F F F F F F Q2 T T T T F F F F T T T T F F F F Q3 T T F F T T F F T T F F T T F F Q4 (Q1 T T F T T T F T T T F T T T F T T F F F T F F F T F F F T F F F Q2) (Q3 T T T T T T T T F T T F F F T F F T F F F F F F T T T T T T T T F T T F T F T T F T T F F T F F Q2) T T T T T T T T F F F F T F T F T T T T T T T T F F F F T F T F ( F F F F F F F F T T T T T T T T Q1 T T T T T T T T F F F F F F F F T T T T T T T T T F T F T F T F Q4) T F T F T F T F T F T F T F T F F T F T F T F T F T F T F T F T Q4 (Q2 T T F T T T F T T F F F T F F F T T F T T T F T T F F F T F F F & Q3) T T T T F F F F F T F T F F F F T T T T F F F F F T F T F F F F Answer and Explanation: This argument is invalid because there is at least one row (row #4, highlighted in gray) where all the premises are true and the conclusion false. (5) Suppose that the list of sentences A1 An is consistent. Suppose also that the sentence B is a contradiction. What can you conclude about the validity of the following argument: A1 An / B ? Justify your answer by talking about truth tables etc. Answer and Justification: If A1 An are consistent, then there is at least one row where they are all true. If B is a contradiction, then it is false in all rows. So it is false in the row(s) where A1 An are all true. But this means there is at least one row where A1 An are all true and B is false. Thus, the argument A1 An / B is invalid. EXTRA CREDIT PROBLEMS (i) Suppose you have an argument of this form: A, (B & C), D / C. Can you conclude anything about the validity of the argument? Explain briefly why or why not. Answer and Explanation: In any row where the conclusion is false, the second premise is also false, since (B & C) is false whenever C is false. This means there will be no row where the conclusion is false and all the premises are true. Thus, the argument is valid. or In any row where all the premises are true, the second premise, (B & C), will also be true. But if (B & C) is true, then C must be true, but this means that the conclusion is true. This means that in any row where all premises are true, the conclusion will also be true. Thus, the argument is valid. (ii) Consider a sentences of this form: A only if B. Is B a sufficient condition for A; is B a necessary condition for A; or both? No explanation is required here, just an answer. Answer: B is a necessary condition for A. Explanation (not required): A only if B A is true only if B is true. In other words, A can only be true if B is also true. This means that in order for A to be true, B must also be true. That is, in order for A to be true, it is necessary that B is also true. In other words, it is a necessary condition for A B is also true. Thus, B is a necessary condition for A. ...
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