I’ve decided that occasionally I am going to put up a post explaining a fallacy or bias, as a way to hopefully help a few people learn about some critical thinking errors. With that introduction, onto the topic of this post: Affirming the consequent.
I am currently finishing up a four-month term of working as a computer science tutor (basically a teacher’s assistant) for a second year computer science course at the University of Waterloo. The other day we marked final exams, and I was amazed at how many computer science students (who are supposed to be good at logic, considering the tight link between computer science, mathematics and logic) fell prey to a logical fallacy on one of the questions.
The question was about regular languages and context-free languages (both are types of formal languages). For those without computer science background, fear not. All you need to understand to appreciate the fallacy is that regular languages are a subset of context-free languages. That is, every regular language is a context-free language. Here’s a visual:
Here, A is a subset of B. That is, anything in A is also in B. So any regular language (A) would automatically be a context-free language (B). This means that something being a regular language implies it is also a context-free language. It is impossible to have a regular language that is not a context-free language. That does not mean it is impossible to have a context-free language that is not a regular language, and this is where the fallacy comes in.
Affirming the consequent is an invalid argument of this form:
P implies Q
Q
Therefore, P
It might be extremely obvious, when put this way, that it is not a valid argument. However, this comes up all the time in real life. Back to the formal language example I started with: Students were asked if a language L was regular after answering whether the language was context-sensitive (L was regular, and thus both). They were also required to give an explanation of their answer. A very common explanation (paraphrased) was: “All regular languages are context-free. L is context-free, and therefore L is regular”. One student even drew the Venn Diagram that showed he understood the relation between context-free and regular languages, yet still they affirmed the consequent, contradicting their diagram.
Perhaps this all seems too abstract to be relevant to real life. Here are some more concrete examples:
You are walking down a street towards your friend’s house. You don’t know the area well and it is dark, and only remember that there is a white fence around their house. You come to a house with a white fence and decide “This must be the house.” This is fallacious. It might be the house, but it could equally just be another house with a white fence. It is not necessarily the house (although with other information, i.e. it seems familiar and houses with white fences are rare in this part of town, it might be very likely that it is your friends house).
An example given by my friend after I taught her about affirming the consequent:
I learned something new and that was fun!
Learning is fun.
But not all forms of fun involve learning.
-Amanda Van Mierlo
Last but not least, a fallacy that I think permeates the minds of quite a few creationists:
If Goddidit, evolution is wrong (a false dichotomy, but that’s a fallacy for another day)
Evolution is wrong (cite some faulty pseudo-science for this)
Therefore Goddidit
Just as a small side note, affirming the consequent is closely related to an argument of the form:
P implies Q
Not Q
Therefore, not P
This argument is valid. This is because "Not(Q) implies Not(P)" is the contrapositive of "P implies Q" (see the link for more information and examples of contrapositives), and the contrapositive of a statement is logically equivalent to that statement.
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