Earlier I explained the fallacies of Undistributed Middle and Illicit Major/Minor. But what about the fallacies regarding the quality of the statements? One such rule of validity states,
A valid syllogism cannot have two affirmative premise and a negative conclusion.
Why is this the case? What prevents two affirmative statements from implying a negative one? The easiest way to show this is to consider counterexamples for syllogisms with two affirmative premises and a negative conclusion, in which the premises are necessarily true, and the conclusion necessarily false. We will do this with a trick. To understand the trick, we must also recall that if a particular form is shown to be invalid, then every syllogism of that form (regardless of the specific terms used) is also invalid.
Here is the trick: for the desired counterexample, we simply make every term identical. Consider the truth values of the following statements:
All dogs are dogs. – True
Some dogs are dogs. – True
No dogs are dogs. – False
Some dogs are not dogs. – False
Notice that, when the terms are identical, the affirmative statements are true, and the negative statements are false. Therefore, if a syllogism has two affirmative premises and a negative conclusion, using the same terms produces a counterexample with two true premises and a false conclusion. And if a single counterexample can be written for a given form of syllogism, then every syllogism of that form is invalid.
Thus we can write a counterexample for any syllogism that makes this fallacy, for example AIO-2, using this same trick:
All dogs are dogs.
Some dogs are dogs.
Therefore, some dog are not dogs.
So, there you go.