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> But, since John has never left office before 5:00 pm, and let's assume he never will, according to the rules of logic, Jennifer's claim is True: It is True that if "John left office by 4:55 pm" then "it'll take 1 minute for him to get home".

A-ha! What logic are you thinking of? Jennifer is making an inductive -- a probabilistic -- claim. So right off the bat, we throw out our deductive rules. There might be an "if" and there might be a "then", but there is no material implication (=>) here.

There's a weaker form of something we can call "probabilistic implication" which will naturally lend itself to plenty of holes: there's a nonzero chance that John might be abducted by aliens, for example, and he never makes it home. These probabilities are built into Jennifer's original claim. There's nothing to refute.

A: John's office leaving time is 4:55 pm or less.

B: John's travel time is 1 minute.

C: John's travel time is 30 minutes or more.

A is False, B is False, C is True (as given). Hence A => B is True (a ridiculous conclusion).

I don't see any probability or induction in this problem.

Let me give you a simpler example we can work with.

   1: If the sun sets, it will rise.
   2: The sun set.
   Conclusion: The sun will rise.

Even though the above might seem like an application of modus ponens, it's actually not. This argument (like yours with John and Jennifer) is inductive -- probabilistic -- in nature. There's some debate when you truly get down to it (is causality empirical?), but almost all claims about the natural world: be they about John's driving or the Sun rising, are inductive; and as such, deductive rules do not apply. In the case above, the sun can, technically, disappear out of existence. Obviously, this is very unlikely but, in fact, guaranteed it will happen at some point by Poincaré and Boltzmann.

In Jennifer's case, her argument is inductive because there are a few (actually an infinite number of) assumptions she's making: that he has a full tank of gas, that lightning won't strike him, that he doesn't get abducted by aliens, etc.

This differs from a deductive claim, e.g.:

   1: Two sets are equal if and only if they contain the same elements.
   2: A is a set that contains {a, b, c}
   3: B is a set that contains {a, b, c}
   Conclusion: A = B

See the difference?

> The question is, how do we incorporate this extra information, C, in our problem, consisting of A and B, if we wish to refute Jennifer's claim of 'A => B' being True?

If you want an intuitive result (Jennifer's claim is false) why do you start with unintuitive preconditions (John will never leave by 4:55 pm.)? If you allow for the possibility that John might leave earlier, then Jennifer can easily be proven wrong by A being true and B still being false.

If you really only want to consider the case A = false, B = false, then you can somewhat counter "A => B" by noting that "A => not B" is also true. Both statements only give you additional information when A is true, in which case "A => B" is false and "A => not B" is true. If A is never true, neither tells you anything interesting.

I think I might be confusing 'material implication' with 'logical implication' (also known as entailment, or logical consequence).

Because from your last sentence, I thought, if A => B is True and A => not B is True, then A doesn't imply B, or B doesn't (necessarily) follow from A. I think this notion of "B follows from A" is something represented by entailment, not by material implication. (?)

Would it be correct to say that material implication is just a formula (in which case it shouldn't even be called an implication or a conditional, but something like simply a 'material formula'), while entailment is the one that has a real world interpretation? (Also entailment cannot be encoded as a formula but has to be proven on a case-by-case basis?)

edit: But this is again a problem. Because in order to prove entailment we'd invoke a logical proof, which would be a sequence (or a tree or a graph) of logical statements with the chain of reasoning connected by, surprise surprise, material implication, which we have already discarded as just a formula with no convincing logical interpretation! (hence our proof is not convincingly logical!)

edit 2: And that is my main issue with how logicians try to justify material implication. On one hand, they try to convince you that MI is nothing but a formula. On the other hand, they use MI as a connecting glue in mathematical proofs which to me sounds like they're using it as 'entailment'. This feels like a double standard at best.

> I think I might be confusing 'material implication' with 'logical implication' (also known as entailment, or logical consequence).

Both are intended to work essentially identical, but when talking about logic, it might be helpful to make the distinction. Implication as a formula is usually written with →, semantic entailment between formulas with ⊨. That they are essentially identical is because, if A → B is entailed unconditionally ( ⊨ A → B ), then A entails B ( A ⊨ B ).

When you apply this distinction to

> if A => B is True and A => not B is True

you have ( ⊨ A → B ) and ( ⊨ A → ¬B ). That is not very different from ( A ⊨ B ) and ( A ⊨ ¬B ), or ( A ⊨ B ∧ ¬B ), where the consequence is contradictory. Unless logic is broken, you can only arrive at contradictions when you already started with them, so A itself must be contradictory. When you said that A is false, that's exactly what you required: A is contradictory.

Maybe the above doesn't help you much, but it should make clear that your problem isn't just with the difference between implication and entailment.

EDIT: I didn't see your edits. I think I already somewhat responded to edit 1 (implication does have an interpretation, as an encoding of entailment). Regarding edit 2: Implication is justified by mostly working like our intuition would expect. Maybe you should try coming up with alternatives and test how well they work.

MORE EDIT: I think what you're actually confused about is the difference between a formula being true only under certain circumstances vs. always. Implication being true when the premise is false is a bit like a broken clock being sometimes right: if you only focus on that one case, it's not what you expect, but when you take all cases into account, it's the only way it could be.

I think of it in programming language terms.

The syntax 'A => B' is just a mechanical rule, which is macro-expanded into:

if A then B else true

where A and B are both bools, and the 'if..else' is an expression, not a statement. It evaluates to B when A is true, otherwise it returns true.

> as a behind-the-scenes knowledge

You can't reason about information you haven't included in the logic. To say it another way, if a predicate doesn't have a letter representing it, you won't get any conclusion about the predicate.