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3.4: Chapter Exercises

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    Chapter Exercises

    1. Put each of the following conditional arguments into standard form. Then say which form the argument has, and whether it is valid or invalid. Remember that we have four types of conditional arguments:
      1. Affirming the antecedent (always valid)
      2. Affirming the consequent (always invalid)
      3. Denying the antecedent (always invalid)
      4. Denying the consequent (always valid)
    1. If Arturo had gotten the job, he would have called me to brag about it by now. But he hasn’t called. So, he didn’t get it.
    2. If Nekesa passed the bar exam, she would have called me to brag about it by now. And she did pass it.
    3. If Nekesa passed the bar exam, she would have called me to brag about it by now. But she didn’t pass it.
    4. Your mom offered to buy you a new car if you got an A in your critical thinking class. And I hate to tell you, but you did not get an A.
    5. You will do well in Critical Reasoning if you keep up with the assignments, and you do keep up.
    6. Tiffany’s car only dies when the temperature is below freezing. But it’s below freezing today, so it will die.
    7. If Smith embezzled the money, then Jones was involved in the crime. But Smith didn’t embezzle it. So, Jones wasn’t involved.
    8. If Wilbur is an uncle, he couldn’t have been an only child. And he is an uncle.
    9. If wishes were horses, then beggars would ride. But beggars don’t ride. So, it looks like wishes aren’t horses.
    10. If everyone at the school were doing well with the current course requirements, there would be no need to change them. But some people are not doing well.
    11. If OU has a winning record in the Big 12, then if all their players are healthy, they will do well in the tournament. And they have a winning record.
    12. If Tom’s prints are on the gun, then he is guilty. So, he must be innocent, because those weren’t his prints on the weapon.
    13. If morals could be taught simply on the basis that they are necessary to society, there would be no social need for religion. But morality cannot be taught in that way (Patrick Lord Devlin, The Enforcement of Morals).
    14. Wilbur is guilty of first-degree murder only if he intended to kill the victim. But he was in such a rage he couldn’t really have intended anything. So, he isn’t guilty in the first degree.
    15. The jury must vote not guilty if they have a reasonable doubt about the guilt of the defendant. And they can’t help but have a reasonable doubt in this case.
    16. Suppose that you have a pack of special cards, each of which has a letter [either a consonant or a vowel] on one side and a number [either even or odd] on the other. If you have some of the cards lying flat on a table, which ones should you turn over to determine whether cards with vowels on one side always have odd numbers on the other side (this exercise is harder)?
      1. cards with consonants and cards with even numbers on them
      2. cards with vowels and cards with even numbers on them
      3. cards with consonants and cards with odd numbers on them
      4. cards with vowels and cards with odd numbers on them
      5. you need to turn over all the cards, in order to determine whether or not this is so
      6. None of the above
    1. What is the relationship between sufficient conditions and the rule of affirming the antecedent? What is the relationship between necessary conditions and the rule of denying the consequent?
    1. Here are a few more exercises on necessary and sufficient conditions:
    1. “You will not pass this course unless you pass the final.”

    Here the claim is that passing the final is a __________ condition for passing the course.

    1. “Rafael will marry Akiko only if she agrees to have three children.”

    Here the claim is that agreeing to have three children is a __________ condition for Rafael agreeing to marry Akiko.

    1. Vixens are female foxes.

    Being a vixen is a __________ condition for being a female fox?

    1. “You won’t be happy, if you buy it at Sturdley’s.”

    Here the claim is that buying it at Sturdley’s is a __________condition for being unhappy.

    1. “A person is a brother just in case he is a sibling that identifies as male.”

    Here the claim is that being a sibling that identifies as male is a __________condition for being a brother.

    1. If A is a sufficient condition for B, then the negation of B is also a sufficient condition for the negation of A. For example, being a dog is a sufficient condition for being an animal. And not being an animal is a sufficient condition for not being a dog. Explain why this holds in general and draw a diagram to illustrate your points.
    1. If A is a necessary condition for B, then the negation of B is also a necessary condition for the negation of A. For example, being an animal is a necessary condition for being a dog. And not being a dog is a necessary condition for not being an animal. Explain why this holds in general and draw a diagram to illustrate your points.
    Answer

    Section 1:

    1. If Arturo had gotten the job, he would have called to brag about it by now. But he hasn’t called. So, he didn’t get it.
      1. If Arturo had gotten the job, he would have called to brag about it by now.
      2. He hasn’t called.
      3. Therefore, he didn’t get it.

    The conclusion is that Stan didn’t win the race. The argument denies the consequent, so it is valid.

    1. If Nekesa passed the bar exam, she will call me to brag about it. And she did pass.
      1. If Nekesa passed the bar exam, she will call me to brag.
      2. She did pass it.
      3. Therefore, she will call me to brag about it.

    The conclusion of this argument isn’t included in it. You must supply it. The conclusion is that she called me to brag. The argument affirms the antecedent, so it is valid.

    1. If Nekesa passed the bar exam, she would have called me to brag about it by now. But she didn’t pass it.

    Invalid. Why?

    1. You will do well in Critical Reasoning if you keep up with the assignments. You keep up with the assignments, so you will do well. The first premise says that if you keep up, you’ll do well. The second premise says that you do keep up. You must supply the conclusion, which is the claim that you will do well.

    The argument affirms the antecedent, so it is valid.

    1. Tiffany’s car only dies when the temperature is below freezing. But it’s below freezing today, so it will die. Hint: what is the necessary condition here?
    2. If Smith embezzled the money, then Jones was involved in the crime. But Smith didn’t embezzle it. So, Jones wasn’t involved.

    Denying the antecedent. Invalid.

    1. If Wilbur is an uncle, he couldn’t have been an only child. And he is an uncle.

    You must supply a missing conclusion here; then it should be easy.

    1. If wishes were horses, then beggars would ride. But beggars don’t ride. So, it looks like wishes aren’t horses.

    Denying the consequent. Valid

    1. If everyone at the school was doing well with the current course requirements, there would be no need to change the requirements. But some people are not doing well. Unstated conclusion: we should change the present course requirements.

    Denying the antecedent. Invalid.

    1. If OU has a winning record in the Big 12, then if all their players are healthy, they will do well in the tournament. And they have a winning record.

    Affirming the antecedent. Valid.

    1. If Tom’s prints are on the gun, then he is guilty. So, he must be innocent, because those weren’t his prints on the weapon.

    Denying the antecedent. Invalid (being innocent is the opposite of being guilty).

    Section 2:

    1. “You will not pass this course unless you pass the final.”

    Here the claim is that passing the final is a necessary condition for passing the course.

    1. “Rafael will marry Akiko only if she agrees to have three children.”

    Here the claim is that agreeing to have three children is a necessary condition for his agreeing to marry her.

    1. Vixens are female foxes. Being a vixen is both a necessary and a sufficient condition for being a female fox.

    It is a definition of ‘vixen’, and definitions typically involve both sorts of conditions.

    1. “You won’t be happy, if you buy it at Sturdley’s.”

    Here the claim is that buying it at Sturdley’s is a sufficient condition for being unhappy.

    1. “A person is a brother just in case he is a sibling that identifies as male.”

    See the answer to the question on vixens above.

    This page titled 3.4: Chapter Exercises is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Jason Southworth & Chris Swoyer via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.