A Paradox is a choke rifled gun made by Holland & Holland. Other makers made them under various names, but today, the tendency is to refer to them all collectively as Paradox guns much as we tend to call all facial tissues a Kleenex. They can be a single or double barrel, but nearly all the ones I've known about were doubles.

The concept is to make a gauge sized barrel that is rifled in the last few inches where a shot gun would have its choke. hence the term "choke rifled". If a gauge sized rifle cartridge (Not a shotgun slug load, the gauge sized rifle cartridge is a rifle cartridge in every sense of the term.) is fired, the choke rifling imparts spin to the bullet as in a conventional rifle. If loaded with a shot cartridge the main part of the barrel is smooth bored and does not disturb the shot charge. The last few inches of rifling impart a relatively small disturbance to the shot charge so the pattern is decent. The idea is to have one gun that can be used as either a rifle or shot gun.

Of course some compromises have to be made and such guns tend to be more rifle than shotgun. One compromise is the stock, as a rifle wants more drop than a shotgun. You have the same compromise with a gun with two sets of barrels, one for a rile, the other for shot. Others peculiar to the rifled choke gun is that you are stuck with a fairly open shot pattern and the barrels are a bit heavy for a shotgun.