More Answers for Exercise 25

One of the difficulties in writing a textbook like Introductory Logic is that, for most of the questions, there are often several possible correct answers. Rather than writing “Answers may vary” every time, I elected in the answer key to give a typical correct answer to each question that could have more than one possible answer.

But all the possible correct answers for Exercise 25 are worth a little more thought. In this exercise, I ask students to write schemas of syllogisms that have a given set of fallacies. If for each problem I only allow those fallacies and no others, there are a reasonably small number of identifiable answers for each problem. Here they are (for the sake of space, I gave the answers as mood & figure, rather than schema):

1.  Illicit major, illicit minor.

Six possible correct answers:  AOE-1; IOE-1, 2; OIE-2, 4; OAE-4

2. Two negative premises, undistributed middle.

Two possible correct answers: OOO-3; OOE-3

3. Two negative premises, negative premise and affirmative conclusion.

Twenty-six possible correct answers: EEA, EEI, EOA, EOI, OEA, OEI, OOA, OOI (all four figures of each, except EOA-1, 2 and OOA-1, 2, which also have an Illicit Minor, and OOA-3, OOI-3, which also have an Undistributed Middle).

4. Two affirmative premises and a negative conclusion, illicit major.

Seven possible correct answers: AAE-1; IAO-3, 4; AIO-1, 3; AAO-1, 3

5. Illicit major, illicit minor, undistributed middle, and two affirmative premises and a negative conclusion.

Four possible correct answers: IIE-1, 2, 3, 4

A few notes on the above:

  1. If you allow the syllogisms to make more fallacies than those given, then there are even more possible correct answers.
  2. It was a fun challenge to determine all these possibilities.
  3. Let me know if you catch me in an error, or if there are possible correct answers that I have missed.

2 thoughts on “More Answers for Exercise 25

  1. Hello Mr. Nance,
    I am in Introductory Logic and have had trouble with lesson 26. To me the correct and incorrect sound the same, so i have a difficult time telling the difference between the right and wrong answers. Even with the five rules i am having trouble understanding the difference. for example,

    Some Christians are not Bible-readers
    No Bible-reader is an ignorant person
    Therefore, no ignorant person is a Bible-reader
    All water is clear liquid
    no clear liquid is a solid object
    Therefore, some water is not a solid object

    Both of these examples sound the same to me? could you possibly clarify the rules or explain what i am missing?

  2. Hi Katrina,

    When you say “the correct and incorrect” or “right and wrong answers” I take it that you mean valid and invalid, and that your struggle is that you cannot immediately (or intuitively) tell the difference between valid and invalid syllogisms; they “sound the same” to you. Well, the fact that you cannot intuitively distinguish an invalid syllogism from a valid syllogism helps to demonstrate why we must study formal methods — that is, counterexamples and rules of validity — that allow us to distinguish them every time.

    The first syllogism is invalid because it has two negative premises, and no statement can be validly inferred from two negative premises. (Please note that you wrote the conclusion incorrectly; the text has “Therefore, no ignorant person is a Christian.”) But to show that it is invalid, you can write a counterexample (substituting terms to make the premises true and the conclusion false), like this:

    Some candy is not bubble gum.
    No bubble gum is jelly beans.
    Therefore, no jelly beans are candy.

    This should “sound” invalid, because the premises are clearly true, and the conclusion is clearly false. But it has exactly the same form as the original argument about Christians, Bible readers, and ignorant persons, namely OEE-4. Thus they are both invalid, even though one perhaps “sounds” more invalid than the other.

    The second syllogism is valid. It passes all five rules. No counterexample can be written for it. If you keep the same form (AEO-4), you cannot substitute terms to make the premises true and the conclusion false.

    Mr. Nance

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