[dvt]

> 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.

[fizixer]

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.

[dvt]

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?