Truth Tables for Arguments

Transcription

1 ruth ables for Arguments 1. Comparing Statements: We ve looked at SINGLE propositions and assessed the truth values listed under their main operators to determine whether they were tautologous, self-contradictory, or contingent. In this section, we re going to compare the truth tables of WO propositions with EACH OHER. o do this, we take the columns of truth vales under the main operators for EACH of the two propositions, and COMPARE them. Sometimes, when we compare the truth tables for two different propositions, the truth values under the main operator will be exactly the same (when this happens, the two propositions are said to be logically equivalent ). Sometimes they will be exactly opposite (when this happens, the two propositions are said to be contradictory ). Sometimes, there will be one or more lines where both propositions come out true (in this case, the propositions are consistent ). Sometimes, there will NO be any lines where both propositions are true (in this case, the propositions are inconsistent ). Columns Under the Main Operators SAME ruth Values on EVERY Line DIEREN ruth Values on EVERY Line At Least One Line Where Both Are rue No Line Where Both Are rue Relation Between Logically Equivalent Contradictory Consistent Inconsistent Let s look at an example. We have learned that only if statements are translated in the following way: You will win the lottery only if you have a ticket. W H However, in a previous lecture, I noted that the following translation is ALSO acceptable: H W It turns out that the following statements are logically equivalent; that is, they have the same meaning: You will win the lottery only if you have a ticket. W H If you win the lottery, then you have a ticket. W H If you don t have a ticket, then you don t win the lottery. H W Let s prove their logical equivalence using truth tables. We ll determine their truth functions side by side, like this: 1

2 W H W H H W he left statement (W H) is just a conditional. We know its truth function immediately. Let s fill that in, as well as the four possible combinations for truth values for H and W for the right statement: W H W H H W We re done with the left statement. Let s focus on the right statement, beginning with the truth values under the negations: W H W H H W inally, let s find the final truth values under the main operator ( ): W H W H H W Compare the red truth values under each statement. hese two statements have exactly the same truth table! Since their truth values are the same on all four lines, we say that they are logically equivalent. ry another one on your own: On homework #2, part II, question 3 ( Neither Ardbeg nor Bobo fails to howl ) the class was largely split between the following two answers. I gave full credit for both. Why? It turns out that they re logically equivalent! ( A B) A B (Note: he symbol indicates logical equivalence) 2

3 Let s do another example. Imagine that an economist claims: he balance of payments will decrease if and only if interest rates remain steady; however, it is not the case that either interest rates will not remain steady or the balance of payments will decrease. here are two claims here. (he second claim begins with the word however ) Let D = he balance of payments will decrease, and let S = interest rates remain steady. hen, we get the following two statements: o compare the two propositions, we will need to do WO truth tables one for each statement. Since there are two letters, our truth table will have four lines. Like this: Next, we can completely finish the first statement on the left (it s just the truth table for ). or the one on the right, let s start by solving for the in front of the S : he negations are in green. Next, let s solve the disjunction by comparing the green letters with the black letters underneath the D. Remember that a disjunction is only false when BOH disjuncts are false. So, we get the following: inally, we should solve for the final negation, OUSIDE of the parenthesis. We do this by taking the blue letters and writing down the OPPOSIE of what is written in blue: 3

4 Now that we have the truth tables completed for both propositions, we can compare them. o do this, we simply compare the columns in red with one another. Like this: Are the two statements logically equivalent? No. o be logically equivalent, the truth values in red would need to be IDENICAL. Are they contradictory? No. o be contradictory, the truth values in red would need to be EXACLY OPPOSIE. hough lines 1, 3, and 4, are exactly opposite, on line 2 they are BOH ALSE (). Are they consistent? No. o be consistent, there would need to be at least one line where BOH statements are true. But there are NO lines with two s. Answer: he economist s two statements are inconsistent. hat is, there is NO possible circumstance in which both propositions could be true; i.e., there is NO line with two s. 2. ruth ables for Arguments: We have learned how to make truth tables for propositions. We have even learned how to construct truth tables to compare two propositions. In this section, we will continue to construct truth tables for multiple propositions but now we ll be constructing them in order to determine whether an argument is valid or invalid. Let s start with an easy one, with only one premise. Example #1: Imagine that I say: If you don t study, then you will not get good grades. herefore, if you do study, then you will get good grades. Let S = You study and G = You will get good grades. My argument may then be written as follows: Premise: 1. S G Conclusion: 2. S G Is this argument valid or invalid? In order to answer that question, we will need to draw up truth tables for BOH the premise AND the conclusion. Before we start, we first write up a four line truth table, with truth values for S and G, like this: 4

5 S G S G S G???????? he shading represents the divide between the premise and the conclusion. Recall our definition of the term validity : We said that a valid argument is one for which it is impossible for the premises to be true and the conclusion false. We then used the counter-example method to determine whether or not an argument is invalid. If we could come up with an argument with the same ORM, where the premises were obviously true and the conclusion was obviously false, then we concluded that the argument was INVALID. hat is basically what we ll be doing here, but with truth tables. We will write up truth tables for the premises and the conclusion, and if there is any line where the premises are ALL true, and the conclusion is false, then the argument is invalid. Otherwise (if there is no such line where this occurs), the argument is valid. Let s do the truth table for the argument above now. We can finish the right statement immediately. or the left, we simply fill in the truth values under S and G, like this: S G S G S G Next, let s get rid of the negations in the premise, like this: S G S G S G inally, we can solve for the conditional ( ). Remember that a conditional is ALWAYS true unless it has a true antecedent and a false consequent. 5

6 S G S G S G Now, in order to determine whether this argument is valid or invalid, we simply look for a line where the premise(s) are true and the conclusion is false. Look at each line. Is there any line where the red letter on the left is a and the letter on the right is an? here sure is! he second line does this: S G S G S G Because there IS a line where the premise is true and the conclusion is false (circled in blue above), this argument is INVALID. In other words, even if it is true that those who don t study will get bad grades, this does NO entail that those who DO study WILL get GOOD grades. he inference is invalid. Example #2: Let s try a more complicated argument. his next one has multiple premises and HREE different statement letters: An employer says, If racial quotas are adopted for promoting employees, then qualified employees will be passed over; but if racial quotas are not adopted, then prior discrimination will go unaddressed. Either racial quotas will or will not be adopted for promoting employees. herefore, either qualified employees will be passed over or prior discrimination will go unaddressed. Let R = Racial quotas are adopted, Q = Qualified employees are passed over, and P = Prior discrimination goes unaddressed. So, the argument can be written as: 1. R Q 2. R P 3. R R 4. Q P Note: We could have combined the first two statements separated by the semicolon and the but. In that case, there would only be 2 premises instead of 3, and premise 1 would be: (R Q) ( R P) his would make no difference. he truth table to be filled in will look like the following: 6

7 R P Q R Q R P R R Q P???????????????????????????????? Let s start by filling in all of the truth values for R, P, and Q, like this: R P Q R Q R P R R Q P Next, let s get rid of the negations in the second and third premise, like this: R P Q R Q R P R R Q P Next, we can solve for all of the premises and the conclusion. he first two premises are conditionals (which are ONLY false when they have a true antecedent and a false consequent), while the third premise and the conclusion are disjunctions (which are ONLY false when BOH disjuncts are false). Here is the result: 7

8 R P Q R Q R P R R Q P Now we look for lines where the conclusion is false and the premises are true. here are only 2 lines where the conclusion is false, so we should only be concerned with those: R P Q R Q R P R R Q P he two yellow highlighted lines are the ONLY lines where the conclusion is false. We should ignore all of the others. Now ask, are either of those lines ones where ALL of the premises are true? Nope On the first yellow line, premise 1 (R Q) is false. On the second yellow line, the second premise ( R P) is false. So, since there ARE NO any lines where all of the premises are true and the conclusion is false, the argument is valid. (Note: Alternatively, if there WERE a line where each of the premises was true and the conclusion was false, then the argument would be invalid.) 8