Why You Failed Your Logic Test

An interesting parallel’s struck me between nonexistent things and the dead: you can say anything you want about them. At least in United States law it’s not possible to libel the dead, since they can’t be hurt by any loss of reputation. That parallel doesn’t lead me anywhere obviously interesting, but I’ll take it anyway. At least it lets me start this discussion without too closely recapitulating the previous essay. The important thing is that at least in a logic class, if I say, “all the coins in this purse are my property”, as Lewis Carroll suggested, I’m asserting something I say is true without claiming that there are any coins in there. Further, I could also just as easily said “all the coins in this purse are not my property” and made as true a statement, as long as there aren’t any coins there.

This is an echo — really, it’s a restatement — of something about null sets that most students look at, remember just about long enough for the exam on that chapter, and then don’t think of again. It’s in set theory. A set, well, call it a collection of stuff gathered by some rule, and trust me that any clever idea you’ve just had to show that definition doesn’t make sense was thought of long ago and the precise definition avoids that problem. Think with good will of a collection of things and that’s a set. The null set, the collection of things that hasn’t got anything in it (think the contents of an empty pocket, if that helps, or the collection of all people who live on your ceiling), is a subset of every set. So we can say the set that consists of all the unicorns is a subset of the set that consists of all the one-horned animals; or that it’s a subset of the set consisting of all blue things with names beginning “K”; or that it’s a subset of the set consisting of all animals that haven’t got one horn.

There’s also a way of restating this as the logical implication statement, which is the framing, “if (something) then (something else)”, which is the implication that in my experience students have the most trouble with. Probably it’s because we really, really want to look at the truth of the (something) or the (something else), and it feels abnormal to look instead at the truth of the whole statement, “if (something) then (something else)”. The statement looks like it’s calling for either (something else) to be true or false, and it takes time to get used to the idea that the statement, “if (something) then (something else)” can be true, or false, without particularly committing to whether (something) or (something else) is true.

In this case, the rephrasing would be “if (this is a unicorn) then (this is a one-horned animal)”. The implication, as opposed to the component parts of (this is a unicorn) and (this is a one-horned animal), is true if either the second part, (this is a one-horned animal), is true; or if the first part, (this is a unicorn), is false, or if both the first part is false and the second part is true simultaneously. So supposing there not to be any unicorns, the implication “if (this is a unicorn) then (this is a one-horned animal)” is true; but equally, the implication “if (this is a unicorn) then (this is not a one-horned animal)” is true; and building up the implication for the blue things starting with the letter “K” is left for the reader. I’m getting married this weekend; I haven’t time to do everything.

The only way the implication “if (something) then (something else)” can be false is if the (something) is true while the (something else) is simultaneously false. That is, the implication “if (this is a unicorn) then (this is a one-horned animal)” can only fail to be true if you find something that is a unicorn and that it’s not a one-horned animal. If there’s no unicorns to be found, the implication’s true. The set of all unicorns is contained in the set of all one-horned animals. If there were any unicorns around, a lot of propositions about them could be cleared up.

(I originally wrote “the implication’s just fine”, as though being true were the same thing as being good. There’s some interesting implication there, but it’s beyond my skill level to say just what it is.)

That implication part is, in my experience, one of the points on which Intro to Logic students crash. I believe what they see is the “if (something) then (something else)” part, and try to look for anything else in the problem that speaks to the (something), and then draw conclusions about what they suppose the (something else) then has to be. But the whole statement is itself a proposition, true or false as a whole, and it can be true or false even if the (something) is true or is false, or even if the (something else) is true or is false. That alone is good for maybe twenty points on the relevant exam.