The fallacy of reaching an affirmative conclusion from a negative premise is a part of formal logic. That is, the problem with the fallacy is with how the argument is structured and not its content.
How this fallacy occurs is explained by its name. Nonetheless, in order to understand it better we have to call upon the father of the syllogism, Aristotle.
In Prior Analytics, Aristotle defines what a syllogism is what is its correct form. Basically, a syllogism is a speech in which something is stated, and something different necessarily follows from it. A correctly formed syllogism contains 2 premises, a conclusion and 3 terms (major, minor, middle).
A typical syllogism:
Premise 1: All triangles have three angles
Premise 2: This figure has three angles
Conclusion: This figure is a triangle
After laying the grounds on how a syllogism should look like, Aristotle proceeds to set some rules on how the conclusion follows from the premises. This is the part that’s important to us. Aristotle notes that the conclusion should be like the premise. That is, if there is a negative premise, the conclusion must also be negative, regardless of the other affirmative premise.
If Aristotle’s definition sounds confusing, look at it this way, in order to prove a negative conclusion, you must use a negative premise. Therefore, an affirmative conclusion from a negative premise is a fallacy; Aristotle calls this an illicit negative.
Formal fallacies are a bit difficult to understand as opposed to informal fallacies. Perhaps, some examples will make things clearer.
Example 1:
Premise 1: All politicians are liars
Premise 2: Citizens don’t like politicians
Conclusion: Citizens like liars
Example 2:
Premise 1: Hunters use guns
Premise 2: Hunters aren’t good people
Conclusion: Good people use guns
It’s obvious that both examples have hideous conclusions. We already said the problem with formal fallacies is with the structure of the argument. When the syllogism is incorrectly formed, even the content of it sounds ridiculous.
Both examples contain a negative premise, a premise that negates some connection between the subject and the predicate. Along with the affirmative, the negative premise leads to an affirmative conclusion. Hence, the fallacy occurs.
According to Aristotle, the conclusion should be the same kind as the premise. So, why do the affirmative premises don’t matter in Example 1 and 2? More precisely, why is one negative premise enough for the conclusion to be negative?
The rules of logic say that when one premise is negative, the other one must be affirmative. Otherwise, the conclusion becomes invalid. A valid syllogism can’t be formed with two negative premises. An example will make things clearer.
Example 3:
Premise 1: Witches don’t exist
Premise 2: Vampires don’t exist
Conclusion: ???
As you can see, nothing can necessarily follow from two negative premises. The only conclusion we can reach from Example 3 is a reiteration of the premises. That’s why a negative premise needs an affirmative premise with it. That way the terms (major, minor, middle) can be properly distributed.
Let us go back to Example 1 and 2. Now, instead of looking at the structure of the syllogism, let us look at its content. Is “Citizens like liars” a valid conclusion from “All politicians are liars” and “Citizens don’t like politicians”? Of course, not, it sounds absurd.
The same can be said for Example 2. It may be true that good people use guns but that doesn’t matter in this situation. A syllogism requires a conclusion that follows from necessity. However, when we see the correct conclusion, things are completely different. That is, when the conclusion is negative.
Example 1.1:
Premise 1: All politicians are liars
Premise 2: Citizens don’t like politicians
Conclusion: Citizens don’t like liars
Example 2.2:
Premise 1: Hunters use guns
Premise 2: Hunters aren’t good people
Conclusion: Good people don’t use guns
The problem with the illicit negative fallacy is evident. Poorly structured arguments lead to a poor conclusion. Our reasoning demands logically correct arguments. When we make mistakes like this fallacy, we render logic useless. More so, if the audience is not familiar with formal logic, this mistake will pass unnoticed and the argument will move to a useless position.