RE: Ford Explorer 4wd/tire size question
All four tires should be the same size. This tests my memory, but I think that should select rear wheel, four wheel auto, four wheel high, and four wheel low? If the auto-four wheel drive can be disabled (two wheel drive only position), then there is no damage being done to the four wheel drive system in any way while you're in two wheel drive. I find it hard to fathom that anyone would willingly install two tires only on a vehicle with an "auto four wheel drive" that didn't have a full two wheel mode.
Listen to them when they tell you that you need four matched tires on the truck in order for it to live up to it's potential, and so that you (she) can have a small piece of mind in knowing that you don't have to cringe about using the four wheel drive when it is needed.
What specific problem(s) is the vehicle having? If it's in the "rear end" being the differential, or even if "rear end" means everything behind the transfer case, be skeptical about blaming the tires too quickly. Mismatched tires and uninformed drivers can tear a driveline into little tiny pieces in short order, however it does what it does and doesn't do what it doesn't do...
Now, if I'm remembering wrong about the two wheel only mode availability and you can't lock out the auto engagement of the front drive, or if she's prone to throwing it into four wheel drive before the traction gets really, really bad, then maybe you should consider the matching of the tires to be a much more critical item.