**Teaching: Logic**

This morning, whilst trying to come up with as dreadfully confusing a hypothetical syllogism as I could, I wrote this:

If it is opposite day, then the opposite is true.

It is opposite day.

Therefore it is not opposite day.

It is opposite day.

Therefore it is not opposite day.

Now is this a

*modus ponens*or a

*modus tollens*?

*Ponens*, if you remember, is constructed like this:

if P then Q

P

therefore, Q

Whereas

*tollens*is

if P then Q

~Q (not Q)

therefore ~P (not P)

One would quickly assume that the restatement of "It is opposite day," P, in the second statement would mean it is

*ponens*, even if the first appearance of Q is misleading. Theoretically, Q can be rewritten "it is not opposite day" and, you'd have:

If it is opposite day, then it is not opposite day.

It is opposite day.

Therefore it is not opposite day.

It is opposite day.

Therefore it is not opposite day.

Thus, you might also be able to prove that it's

*modus tollens*, because the second appearance of "it is opposite day" can be construed as a negation of the Q (the opposite of not-opposite day is opposite day) and the conclusion could be construed as a negation of the P (the opposite of not-opposite day is opposite day)!

The kids insisted this made no sense, and I'm actually inclined to agree, so far as anyone could usefully prove the apparent statement "If it is opposite day, then it is not opposite day" actually has no truth value. My colleague Mr. Clements thought it might be a valid

*tollens*provided it does not violate the Law of Non-Contradiction, which mandates that a statement cannot be true and false at the same time. The other suggestion is that the statement "It is opposite day" contains its own contradiction, and thus is inherently unstable and should not be inflicted on 8th Graders.

This next example does makes sense, however. Honest.

**If you ask Simon for the right answers, he'll give you the wrong ones.**

These are the right answers.

Therefore you didn't ask Simon for them.

*Ponens*or

*tollens*? It initially resembles

*ponens*, because "right answers" appears in the P and in the second statement. However, the first time "right answers" is

*part of the*P, while the second time it forms the Q. Said again, "the wrong ones" is negated to "the right answers" and the "right answers" is a red herring in the first statement-- you should realize that "you didn't ask Simon" is a negation of "if you ask Simon." Thus, the syllogism is

*modus tollens:*

if P then Q

~Q therefore ~P

That, I remember, was a rather confusing Logic lesson:)

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