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.

Thirty-two possible correct answers: EEA, EEI, EOA, EOI, OEA, OEI, OOA, OOI (all four figures of each)

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, especially numbers 1 and 4.
  3. Let me know if you catch me in an error, or if there are possible correct answers that I have missed.

Two questions on Intro Logic Exercise 22

Mr. Nance,

I am loving Logic, and have understood the lessons up until now, but the syllogisms and validity has me a bit overwhelmed.

You lost me in the 256 challenge when you started using the same term (dogs) for the major, minor and middle terms. I thought we needed to use different terms when testing for validity. I went back and tried putting dogs into Exercise 22 to see how that worked, and now I’m even more confused. It looks to me that it doesn’t prove the syllogisms to be valid or invalid (when using only one term).

Could you also let me know if I am on the right track on somethings else? Can you test for validity by using the relationships between statements when going from the minor premise into the conclusion? For example, in exercise 22, #1 would be false by contradiction, #2 would be false by contrariety, #3 would be false by superimplication etc…#5 would be true by subimplication.

Thank you for considering my questions! Continue reading Two questions on Intro Logic Exercise 22

On Logical Independence

Mr. Nance,

On Test 2b there were two questions on the issue of statements being logically independent that I found myself tripping on a little. Can you help me understand them more clearly?

The first is Test 2b, 11a: “It is later than 1:00 pm. / It is later than 2:00 pm.”
The next is Test 2b, 11c: “Some siblings are twins. / Some siblings are not twins.”

Both are said to be not logically independent. I would appreciate if you could help me see that more clearly than I do.

Thank you! Continue reading On Logical Independence


Counterexample challenge

Mr. Nance,

My question has to do with Lesson 24. Exercise 22 has the students do a challenge, testing the validity of all 256 forms. I understand it’s to practice counterexample. Is there another reason not to put this challenge after lesson 26 after they’ve learned all the rules? (Which I think would make it easier.) Continue reading Counterexample challenge


Common errors to avoid: Don’t sweat Lesson 23!

Introductory Logic Lesson 23 introduces the concepts of validity and soundness. The lesson says that a syllogism is valid if and only if the premises imply the conclusion. If a syllogism can have true premises and a false conclusion, the argument is invalid. A sound argument is a valid argument with all true statements.

The only purpose of Lesson 23 is to introduce the concepts of validity and soundness. This lesson does not explain how to determine validity. So if after studying this lesson you have trouble knowing whether a given syllogism is valid or invalid, don’t worry about it. You will learn how to do that in the next three lessons. Lessons 24-26 are dedicated to teaching the methods for determining the validity of a syllogism.


Why Standard Form?

Teaching students how to translate syllogisms into standard categorical form occupies several lessons in Introductory Logic. Lessons 11 and 12 explain how to translate categorical statements into standard form, which is then emphasized while learning about the Square of Opposition in Lessons 13-18. Lesson 19 teaches students how to distinguish premises and conclusions, in part so that in Lesson 20 they will understand how to identify the major and minor premises, so that they may know how to arrange a syllogism in standard order. Finally, Lessons 21 and 22 teach them how to identify the syllogism form using mood and figure. All this occurs before the students begin to learn how to determine the validity of a syllogism. Continue reading Why Standard Form?

The Square of Opposition (for logic nerds only!)

The Square of Opposition is a useful tool for representing and understanding the relationships between categorical statements with the same subject and predicate:


The relationships are defined this way:

Contrariety: The statements cannot both be true, but can both be false.
Contradiction: The statements cannot both be true, and cannot both be false.
Subcontrariety: The statements can both be true, but cannot both be false.
Subimplication: If the universal is true, the particular must be true.
Superimplication: If the particular is false, the universal must be false.

Students might be interested to discover that the square of opposition can be created for non-categorical statements as well. Continue reading The Square of Opposition (for logic nerds only!)


Common errors to avoid: I’s don’t imply O’s

Students who are first learning about categorical statements may mistakenly think that any I statement, Some S is P, necessarily implies the O statement, Some S is not P. This is a reasonable error, since it seems to accord with our common use. For example, if I say “Some astronauts are men,” it is reasonable for you to think I also believe that some astronauts are not men.

But this is not always the case. Statements of the form Some S is P logically allow for the possibility that All S is P. When a theology student first learns that some books of the Old Testament speak about Jesus, he may not be surprised to later discover that all books of the Old Testament speak about Jesus (Luke 24:27). Or when a physics student first learns that some forms of usable energy end up as thermal energy, she is well on her way to acknowledging that eventually all usable energy ends up as thermal energy. Astronomers once knew only that some gas giants in the solar system are ringed planets (e.g. Saturn). They eventually discovered that all gas giants in the solar system are ringed planets.

These examples show that Some S is P does not necessarily imply that Some S is not P.