Radford's Rule

In honor of Ben Radford, I'm suggesting we define "Radford's Rule": When caught in a logical fallacy, just insist your logic is fine.

I've never heard of Ben Radford before, but apparently he's known as a debunker and skeptic. He blogs at Center for Inquiry. In a recent post on SkeptiXX, he picked apart a video in which 4-year-old Riley rants about the way stores market princesses to girls and superheroes to boys. Although many people pointed out the problems with Radford's piece, he first insisted he wasn't wrong, then belatedly issued a non-apology apology, in which he continued to insist on the following "syllogism" (yes, he actually called it a syllogism):

1) Most things girls play with are dolls; 
2) Most dolls are pink things; 
3) Therefore, most things girls play with are pink.

For everyone who has already spotted the problem with this argument, stop reading now and go outside and play. For Ben and anyone else who has difficulty seeing it, read on.

Try this argument, which has exactly the same form as Ben's:

1) Most adults in the US are women.
2) Most women in the US shave their legs.
Therefore, 3) Most adults in the US shave their legs.

Now, (2) may or may not be true, but let's assume it is for the sake of the argument. I hope you can see that (3) doesn't follow logically from (1) and (2). First of all, even though (1) is true, there aren't a whole lot more women than men. Secondly,  all those non-shaving men count under "most adults", and there may be enough unshaven women to make up an unshaven majority.

I think what Ben probably has in mind is the classic syllogism:

All men are mortal.

All Greeks are men.

Therefore, all Greeks are mortal.

But it just doesn't work to replace "all" with "most." It's not longer a syllogism - it's now a probabilistic argument. And it's an invalid one.

To make it perfectly clear how the probabilities work, let's look at Ben's argument with some probabilities that I just made up. Let's say that 60% of girls play with dolls, and 60% of dolls are pink things. Also, let's assume that these are independent probabilities. Then the probability that a girl is playing with a pink thing is at least the product of the two probabilities: 0.6*0.6 = 0.36 or 36%. Clearly, we cannot conclude that "most things girls play with are pink."

Some folks on Ben's thread objected that you can't just make up numbers like that. But, IF the conclusion follows logically from the premises, THEN it must follow for any set of numbers you make up. So when I show that there is a set of numbers for which the conclusion doesn't follow, I am also showing that the logical structure of the argument isn't valid.

This is just basic logic, and I'm amazed that a well-known "skeptic" like Ben Radford isn't aware of these things. I pointed this out in a comment on his thread, and several other commenters did, too. But in his most recent post, he still says he "stands by" his logic.

Brilliant.