Where the material conditional gets its truth conditions

Oh, the material conditional. Some love it, some hate it. But can we all agree that explaining it to the uninitiated is a perennial headache? If you’ve taught baby-logic, you know how this goes. There you are, giving a just lucid shpeel on deductive systems, until you get to this part:

A B A→B T T T T F F F T T F F T

and the tires screech to a halt. Why are those bottom two values True? — they demand. The first two rows don’t bother them. But if A is false, why should it be that A → B is true, regardless of the truth of B?

You could just say it’s a convention, get over it. But why is this the convention adopted in classical logic? My colleague Jonathan Livengood and I discussed this, and came up with a better answer:

Suppose we agree on the first two rows of the above truth table. If implication (→) is both non-trivial and asymmetric, then this its only possible truth table.

Here’s why. Start by writing down all the possibilities for these bottom two rows. There are only four, and A → B has to be one of them.

A B (1) (2) (3) (4) T T T T T T T F F F F F F T T F F T F F F T F T

Column 1 is trivial, because it has the same values as B. If this were the correct column, then saying A → B would mean the same thing as just saying B. So, assuming → is not trivial, we can throw this column out.

Column 2 has a symmetry property that implication doesn’t. Namely, it stays the same if we reverse the A and B cells. If this were material implication, then A → B would be true if and only if B → A is true. So, assuming → is asymmetric, we can throw this column out too. (This column is actually the usual truth table for A ↔ B.)

Column 3 has exactly the same problem: it stays the same when we reverse the cells containing A and B. So we can throw it out for the same reason. (This column is actually the usual truth table for A&B. So, plausibly, we can also observe that “implies” should mean something different than “and.”)

And that’s it! If → is non-trivial and asymmetric, then Column 4 is the only option left: the standard, not-just-conventional truth table for material implication.