First off, by a "proposition" I simply mean an expression that has a clear Boolean value. Propositions can be combined into larger propositions by using the "not, "and" and "or" operations that we are all familiar with. "Socrates is a cat or it is raining" is true if either of its sub-propositions are true, and false otherwise.
Those are operations are all uncontroversial. But we run into big problems when we try to come up with a definition for a proposition of the form "if P then Q", or, as it is traditionally notated, P -> Q. P here is the "antecedent" and Q is the "consequent".
The first thing we have to emphasized is that the implication operation does not suggest that there is a causal relationship. We usually speak of implications that have a causal relationship: "if it is raining then the streets are wet". But the implication proposition "if it is raining then Minerva is a cat" needs to have a truth value irrespective of the fact that there is no obvious causal connection between the weather in Seattle and the name of my elderly cat.
Explanation
So what then should be the rules for the truth value of P -> Q, given truth values for P and Q?
Plainly if P is true and Q is false then the implication was false. If we have a situation where the antecedent is true but the consequent is false then the implication was a false statement. That is, if some day it is raining but the streets are, bizarrely, not wet, the "if it is raining the streets are wet" must have been false.
Similarly if P is true and Q is true then the implication was true. If it is the case that it is raining and Minerva is a cat then we can conclude that the implication proposition is true, end of story. Again, remember, the truth of the implication is simply that if the antecedent was true then the consequent turned out to be true, not that there was a causal connection.
That's pretty uncontroversial. The problems arise when the antecedent is false. What is the truth value of P -> Q when P is false? In traditional propositional logic an implication proposition with a false antecedent is considered a true statement, regardless of the truth of the consequent. That is, "if Socrates is immortal then the streets are wet" is a true proposition, regardless of whether the streets are wet or not.
Many people find this unsettling. I did myself for some time. But now it seems very natural to me. Rather than ask about the truth of a proposition, let me tweak it slightly. Suppose I tell you "if it is raining then I wear rubber boots". Under what circumstances could you reasonably call me a liar? If it is raining and I am not wearing rubber boots then clearly I am a liar. If it is raining and I am wearing rubber boots then clearly I am not a liar. If it is not raining then you have nothing upon which to deduce that I am a liar; my statement is true.
So what does this have to do with programming languages?
I now think of material implication like this:
if (p)
{
debug.assert(q);
...
}
We will say that the program is correct (true) if the assertion never fires and incorrect (false) if the assertion fires. Under what circumstances would we say that the program is incorrect? Only if p is true and q is false. If p is false then the assertion never fires, and if both are true then the assertion never fires. So again, it makes sense to say that P -> Q is true when P is false.