I think it's safe to say that many don't get the distinction between a logical problem and a factual one. A logical problem involves the strength, plausibility, or validity of an inference from one fact to another fact; a factual problem concerns whether a given fact is in fact a fact. Here's an example (from Marc Ambinder's blog) apropos of yesterday's post:
The Logic Of George Will
John McCain probably was eager to return to the Senate as an avatar of bipartisanship, a role he has enjoyed. It is, therefore, a measure of the recklessness of House Democrats that they caused the stimulus debate to revolve around a bill that McCain dismisses as "generational theft."
P1: John McCain enjoyed being bipartisan in the past.
P2: [All people who enjoy things in the past will want to continue doing them in the future.]
C1: Therefore John McCain wanted to continue being bipartisan.
P3: John McCain did not continue being bipartisan.
P4: [Only recklessness by House Democrats could cause John McCain not to be bipartisan.]
C2: Therefore House Democrats are reckless.
There is nothing wrong with Will's logic here (there is almost everywhere else in yesterday's piece–such as his comparing the quantity of money spent on the stimulus with the size of the federal budget twenty five years ago). The problem with Will's argument is that P1, P2 and C1 are just false.
The argument however is something of a topical inference. A topical inference, on Boethius's definition (cf. De topicis differentiis), rests on an implied maximal proposition. I'm at a loss for the moment to find in Boethius's text the exact one (there are many of these maximal propositions) which would apply here. But it seems to me in the first place that this is not, as Ambinder suggests, an enthymeme with P2 as a supressed premise (besides, if it were it would still be valid). The inference here rests on the notion that McCain is maximally conciliatory such that to scare him away really means something. Here, perhaps, is an appropriate analogy. Imagine you have a brother who does not enjoy any kind of breakfast comestible, if he eats and enjoys the new one you offer him, it will really say something about that particular food. That's basically what Will is arguing, but it turns out that your brother likes everything, so your inference, while a good one, fails.
**edited for clarity.