Lessons in Logic: Hypothetical Syllogism

Pretty big words for a concept that you already understand, and it’s definitely something you already get. A hypothetical syllogism is a rule of inference which allows us to compound related material conditional statements. Maybe that wasn’t helpful either. I’ll show you. 

Here’s an example:

1. If snow is white, then frogs jump. (S → F)
2. If frogs jump, then Elvis walks among us. (F → E)
3. Therefore, if snow is white, then Elvis walks among us (S → E)

You use this all the time to make inferences about things. Note that what connects the two premises is that the consequent of the first (F) is the antecedent of the second. Also, as for any other rule of inference, the conclusion is only true if both of the premises are true. If one of them is false, then the conclusion has to be false. The actual content of the conditionals in the premises is irrelevant, except so far as it dictates the truth of the conditional.

We encounter bad hypothetical syllogisms all the time, and they aren’t always easy to spot. All it takes is one conditional that’s false, and the whole thing falls apart. Today’s example of bad logic comes via beantown mom at dailykos.com, talking about her daughter’s experiences after her school found out she uses birth control. The actual origin of this argument is renowned radio logician Rush Limbaugh, who drew a similar conclusion about law student Sandra Fluke. The argument goes something like this:

1. If a girl takes birth control, then she is sexually promiscuous. (B → P)
2. If a girl is sexually promiscuous, then she is a slut. (P → S)
3. Therefore, if a girl takes birth control, then she is a slut. (B → S)

Now, I’m simplifying it a bit for the sake of the example, but I think we can capture Rush’s and the students’ meaning if we define a slut as “someone worthy of blame” in this case. Still, it’s pretty clear that premise 1 is false. There are lots of reasons to take birth control, such as the treatment of ovarian cysts. And even if it’s to prevent pregnancy, it still doesn’t follow that one is sexually promiscuous. Premise 2 is also false, because it’s not at all clear that one should be blameworthy if one is sexually promiscuous. Since both premises are false, the conclusion is also false.

And that’s hypothetical syllogism. Use it to make brilliant connections, and amaze your friends! We’ve just got three more rules of inference to get through, and then we’re on to counter examples, or how to prove that your friends (or enemies) are wrong (so long as they are).