In logic, a conditional is an if-then statement. “If it rains, then I will go to the movies,” that’s a conditional. The question is, how should we assign a truth value to such a statement? This is a question of some importance to mathematicians, since every theorem is ultimately an if-then statement.

In some situations it seems easy enough. If it actually rains and I do go to the movies, then we would say the statement is true. If it rains and I don’t go to the movies then the statement was false. Simple! But what if it doesn’t rain? You might be inclined to say that we just should not assign a truth value in this case, since the statement was never properly tested. Sadly, classical logic does not permit this option. Every proposition must be either true or false. No exceptions!

Actually, it gets worse. Classical logic, by which I mean the sort of logic we teach to beginning math majors in discrete mathematics classes, treats conditionals as “truth-functional.” That’s a fancy way of saying that the truth value of the full if-then statement should depend solely on the truth values of the individual parts. This means that some convention has to be adopted for assigning a truth value to the if-then statement in each of the four possible cases. This includes those cases where the first part is false.

The standard convention is that a conditional statement in which the first part is false is automatically true, regardless of whether the second part is true or false. Put differently, the only way for a conditional to be false is if the first part is true and the second part is false. We figure that if the first part of an if-then is false then I didn’t actually lie to you.

Now, this has some weird consequences. It implies that the statement, “If Santa Claus exists then the Moon is made of green cheese,” is true. That’s weird, but whatever. This is the kind of thing you just accept when you’re doing logic.

But there are other problems. Quite a few of them, actually. Here in the real world, as opposed to on planet abstraction, both parts of an if-then could be true while the whole statement is false. Keeping in mind that I am writing this in Virginia, how about the statement, “If I am not in France then I am not in Spain.” Both parts of the conditional are true, but it would be absurd to say the whole statement is true.

Here’s another amusing consequence of the standard convention: All historical counterfactuals have to be considered true. For example, “If Neil Armstrong had not walked on the moon, then someone else would have.” Making such a statement presupposes that Neil Armstrong really did walk on the moon. If we adopt the standard convention, then we would have to declare the if-then to be true since its antecedent is false. Of course, that would apply with equal force to the statement, “If Neil Armstrong had not walked on the moon, then no one else would have.” This is all very strange.

Here’s another problem. In classical logic we have the “principle of strengthening the antecedent.” In other words, if you grant the statement “If P then R,” then you must also grant the statement “If (P and Q) then R.” Since (P and Q) can only be true if P is true all by itself, and since P implies R, we must grant that this enhanced conditional is true. But that’s not how it works in the real world, is it? “If it rains, then I will go to the movies,” therefore, “If it rains and I break my leg falling down the stairs, then I will go to the movies.” Or how about, “If I put sugar in my tea it will taste good, so if I put sugar and diesel oil in my tea it will taste good.” That doesn’t seem quite right.

Treating conditionals as truth-functional also pays no attention to whether the antecedent has any relevance to the conclusion. “If 2+2=4 then Paris is in France.” Are you really comfortable saying that’s true?

This is all good fun. The question is what to do about it. Mathematicians take the easy way out. We don’t worry about it. The classical account works perfectly well when applied to the sorts of conditionals we care about. Good enough! Philosophers, though, are hunting bigger game, and I’m sure you will be shocked to learn that they have produced a vast and dense literature trying to figure out what’s going on. Maybe we need to make a distinction between indicative conditionals and subjunctive conditionals. Maybe there’s a distinction between truth and assertibility, so that a statement could be true in some technical logical sense but still be unassertible given various conversational conventions. Maybe conditional probability can come to our aid, so that the truth of “If P then Q,” has to do with the conditional probability of Q given P. Maybe modal logic is the way to go, so that we need to refer to possible worlds to get a grasp on what’s going on.

Good stuff! But that’s all definitely to be saved for a different post…