The Shandy family only has a few wheels, but all those wheels are set in motion by different principles and impulses.
One of the strange things about the family is that whatever is being discussed in the parlour is also being discussed by the servants in the kitchen. Mr. Shandy actually thinks this is okay, always leaving the door slightly open so that he doesn't have to go to the bother of actually running the household.
So, while Mr. Shandy is drawing on ancient authorities to philosophize about his son's death, Corporal Trim is drawing on his own experience to cope with it. Kind of like parallel universes.