That's interesting. I'll have to read up on that. You're right, I am thinking about boolean algebra.

In the mean time though, I'll note that Boolean algebra on Wikipedia also refers to this operation, so I'm not alone:

Material conditional

The first operation, x → y, or Cxy, is called material implication. If x is true, then the result of expression x → y is taken to be that of y (e.g. if x is true and y is false, then x → y is also false). But if x is false, then the value of y can be ignored; however, the operation must return some Boolean value and there are only two choices. So by definition, x → y is true when x is false (relevance logic rejects this definition, by viewing an implication with a false premise as something other than either true or false).

It also uses the second interpretation that I mentioned in my earlier comment (4 above this one), with true being default, rather than the one we've been discussing.