Introductory Logic Lesson 24 challenges you to find the 24 valid forms of mood and figure (out of 256) by writing counterexamples. In these two videos, I show you how it can be done.

5 thoughts on “Challenge Accepted! Finding the valid forms by counterexample”

Hello James, I am a 1st year Challenge B director in Bismarck, ND. I am loving Logic, and have understood the lessons up until now. 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! I so appreciate your extra time in helping us newbies!

Hi Julie,

A counterexample needs to have true premises and a false conclusion, and the form must remain the same. This last requirement is not violated when the terms are identical throughout the syllogism. Imagine you are writing a counterexample to this invalid syllogism:

All pigs are mammals.
All mammals are animals.
Therefore, some animals are not pigs.

The premises do not imply the conclusion, but the only way a counterexample can show this is to substitute identical terms or exact synonyms. For example, you could substitute people for “pigs,” homo-sapiens for “mammals” and humans for “animals” and get this counterexample:

All people are homo-sapiens.
All homo-sapiens are humans.
Therefore, some humans are not people.

This should be a clear counterexample. But there is nothing stopping you from substituting dogs for “pigs,” dogs for “animals” and dogs for “mammals” and getting this counterexample:

All dogs are dogs.
All dogs are dogs.
Therefore, some dogs are not dogs.

Like the previous example, the premises are true and the conclusion false, and you have made a consistent substitution. Thus the counterexample works.

However, you cannot substitute an identical word or synonym for all the terms with any of the questions on Exercise 22, because it will not result in true premises. In each case, every negative premise would be false (i.e. “Some dogs are not dogs” or “No dogs are dogs”). But to have a good counterexample, every premise must be true. Thus you are quite correct (in the case of every syllogism in Exercise 22) when you say, “It looks to me that it doesn’t prove the syllogisms to be valid or invalid (when using only one term).”

The answer to your second question, “Can you test for validity by using the relationships between statements when going from the minor premise into the conclusion?” is No. The Square of Opposition relationships only apply to statements that have the same subject and predicate. Also, it is possible to have a valid syllogism with a minor premise that is, say, an A statement, and the conclusion an O statement (which might appear to contradict). Here is an example:

Some men are not fathers.
All men are people.
Therefore, some people are not fathers.

This is valid, despite the fact that the minor premise is an A statement, and the conclusion an O statement.

Blessings!

Mr. Nance,

Thank you for your extra help. I have read this over and over, and I think it’s beginning to click.

Hmm it looks like your blog ate my first comment (it was super long) so I guess I’ll
just sum it up what I submitted and say, I’m thoroughly enjoying your blog.
I as well am an aspiring blog writer but I’m still new to everything.
Do you have any recommendations for first-time blog writers?
I’d really appreciate it.

Thank you for the note. Just a bit of advice: write regularly, keep it focused on a single theme (so that if you take a tangent it is obvious), and use social media to share it with potential readers who are likely to appreciate what you have to say. People will read a blog post if they think they will benefit (laugh, learn) from it, and if it’s not too long. My most popular posts have been those that have challenged my readers without offending them.
Blessings!

Hello James, I am a 1st year Challenge B director in Bismarck, ND. I am loving Logic, and have understood the lessons up until now. 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! I so appreciate your extra time in helping us newbies!

Hi Julie,

A counterexample needs to have true premises and a false conclusion, and the form must remain the same. This last requirement is not violated when the terms are identical throughout the syllogism. Imagine you are writing a counterexample to this invalid syllogism:

All pigs are mammals.

All mammals are animals.

Therefore, some animals are not pigs.

The premises do not imply the conclusion, but the only way a counterexample can show this is to substitute identical terms or exact synonyms. For example, you could substitute

peoplefor “pigs,”homo-sapiensfor “mammals” andhumansfor “animals” and get this counterexample:All people are homo-sapiens.

All homo-sapiens are humans.

Therefore, some humans are not people.

This should be a clear counterexample. But there is nothing stopping you from substituting

dogsfor “pigs,”dogsfor “animals” anddogsfor “mammals” and getting this counterexample:All dogs are dogs.

All dogs are dogs.

Therefore, some dogs are not dogs.

Like the previous example, the premises are true and the conclusion false, and you have made a consistent substitution. Thus the counterexample works.

However, you cannot substitute an identical word or synonym for all the terms with any of the questions on Exercise 22, because it will not result in true premises. In each case, every negative premise would be false (i.e. “Some dogs are not dogs” or “No dogs are dogs”). But to have a good counterexample, every premise must be true. Thus you are quite correct (in the case of every syllogism in Exercise 22) when you say, “It looks to me that it doesn’t prove the syllogisms to be valid or invalid (when using only one term).”

The answer to your second question, “Can you test for validity by using the relationships between statements when going from the minor premise into the conclusion?” is

No. The Square of Opposition relationships only apply to statements that have the same subject and predicate. Also, it is possible to have a valid syllogism with a minor premise that is, say, an A statement, and the conclusion an O statement (which might appear to contradict). Here is an example:Some men are not fathers.

All men are people.

Therefore, some people are not fathers.

This is valid, despite the fact that the minor premise is an A statement, and the conclusion an O statement.

Blessings!

Mr. Nance,

Thank you for your extra help. I have read this over and over, and I think it’s beginning to click.

Hmm it looks like your blog ate my first comment (it was super long) so I guess I’ll

just sum it up what I submitted and say, I’m thoroughly enjoying your blog.

I as well am an aspiring blog writer but I’m still new to everything.

Do you have any recommendations for first-time blog writers?

I’d really appreciate it.

Thank you for the note. Just a bit of advice: write regularly, keep it focused on a single theme (so that if you take a tangent it is obvious), and use social media to share it with potential readers who are likely to appreciate what you have to say. People will read a blog post if they think they will benefit (laugh, learn) from it, and if it’s not too long. My most popular posts have been those that have challenged my readers without offending them.

Blessings!