I had a fun exchange by text today. My father texted both my brother and myself the following message "Did I borrow [a particular book] from either of you?". This left me in a bit of a predicament; I don't have the information to be able to answer his question.

For those of you who are somewhat confused1 by this, let me explicate. The problem hinges on the word "either". Let's look at a table.

  Borrowed from me Not borrowed from me
Borrowed from brother Yes Yes
Not borrowed from brother Yes No

Suppose we're in the first column and thus he did get the book from me. Then I could answer "yes", and everything is fine2.

Now let's consider the case where I don't own the book. Now we're in the other column. Certainly I know that he didn't borrow the book from me. But I don't know what books my brother owns. Perhaps my brother did have the book and my father borrowed it; in this case I should say "yes". But perhaps my brother didn't have the book; in this case I should say "no". I have two plausible scenarios with contradicting responses; I can't respond.

In this situation I can only hope that my brother in fact owned the book and replied "yes". Otherwise, we'd both be stuck unable to respond. The question would go unanswered and it would stay in the front of my mind. Because I can only keep track of 7 ± 2 things in working memory, I would be working with at best 88% of my working memory for the rest of my life. I'm a forgetful person at the best of times, so with my diminished memory I'll probably forget to turn off my stove and burn down my apartment complex. All because of a single careless question.

Given that I'm not typing this from the smoking ruins of an apartment building, apparently things were resolved. Fortunately I recalled an old joke

Three logicians walk into a bar. The barkeep says, "Do all of you want a beer?"

The first logician says, "I don't know."

The second logician says, "I don't know."

The third logician says, "Yes!"

So I replied back with an allusion to this joke and, to my eternal pride, my brother (who studies math) figured it out and replied "no". So there was a happy ending after all.

How did my joke reference help? Well, it gave my brother some information. Namely, it told him that I didn't have enough information to answer the question myself. The only case in which that's true is when the book wasn't borrowed from me. This gave my brother the information needed to confidently answer "no".

I'll leave it to the reader to figure out the connection with the joke.



Yes, I basically wrote this up to explain a joke to my father


I didn't think of this until writing this up, but there's an unwarranted assumption lurking here. As a family, we've empirically shown that the fact that a book is sitting on my shelf does not prevent other family members from buying the same book. Thus, even if I had a copy, I still couldn't be certain that my father had borrowed it; he might have just bought it himself. There are, of course, no end to these sorts of quibbles (see philosophical skepticism).