# 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.)

Placing the challenge after Lesson 26 would certainly make it easier, since it would tell the student if a syllogism is valid or invalid, so they would know whether or not they could write a counterexample. But I have two reasons for putting the challenge after Lesson 24.

First, as you said, to give them lots of practice writing counterexamples, which is one of the more practical tools of learning, and one which teaches them through repetition the concept of validity. But secondly, I use this to allow the students to discover the rules on their own, or at least some of them, before I teach them the rules explicitly.

For example, consider this invalid AAO-4 schema:

All P is M
All M is S
∴ Some S is not P

Try writing a counterexample for this schema, and you will eventually discover that the only possible counterexample is one in which every term is identical or synonymous. Like this:

All dogs are dogs
All dogs are dogs
∴ Some dogs are not dogs.

This certainly fits the form, and the premises are clearly true, the conclusion clearly false. So this is an invalid form. But through careful discussion, the students can be led to see that any syllogism that has two affirmative premises and a negative conclusion can be shown to be invalid in the same way, by using identical terms. Thus we can give a counterexample for IAE-2 (actually any figure of IAE):

Some dogs are dogs
All dogs are dogs
∴ No dogs are dogs

Thus on their own they can discover this rule of validity: A valid syllogism cannot have two affirmative premises and a negative conclusion. With some careful guided discussion, students can also discover some of the other rules of validity as well, as they work on writing counterexamples of the mood and figure.

Note that I tell my students that there are 24 valid forms. Once they discover the valid ones (the forms for which no counterexample can be written), they do not have to keep writing counterexamples for the remaining forms.

I also make this a game, telling students things like this: “The EAE mood has two valid figures, and two invalid. Find the valid ones by writing counterexamples for the invalid ones.”

I also point out things like this: “Remember that the E statement and the I statement have equivalent converses. That means you can switch the terms around and get equivalent statements. This implies that if EIO-1 is valid, so is every other figure of EIO. And if EEE-4 is invalid, so is every EEE-1, 2 and 3.”