Is a catch-22 a specific subtype of circular logic? How are they different? Also, is the catch-22 paradox known by any other names?
I think that Catch-22s (Catches-22?) are different from paradoxes, contradictions, or circular reasoning, though they do resemble these things, and are interesting for this reason.
It seems to me that a Catch-22 consists essentially of:
The similarity to circular reasoning is, I think, in the cycle of conditions.†
The classic example is drawn from the novel Catch-22, with respect to Orr being allowed not to fly anymore in the Allied campaign in Italy during WWII. Without commenting on whether or not these premises are valid propositions in themselves, the structure of that Catch-22 is as follows: Orr will only be relieved of pilot duty if he is crazy; but if he were crazy, he wouldn't request to be relieved of pilot duty. We may analyse this by:
As D ⇒ N1 ⇒ N2 ⇒ N3 ≡ ¬D, we have a logical Catch-22.
The other, "chronological" Catch-22 involving a cycle of preconditions is actually a special case of a logical Catch-22; albeit one which is more commonly found in practise, for instance in bureaucratic situations involving approval or identification. The "logical" contradiction in this case arises from the need for the temporal ordering: if a condition Nj is initially not satisfied, and if for some time t the condition Nj can only be satisfied if (through a cycle of preconditions) it is also satisfied at some earlier time t − t', and if we suppose that there is some lower limit to the amount of time required to satisfy each subsequent condition given that its preconditions have been satisfied, then we can conclude that Nj and any condition depending on it will never be satisfied, by induction. This chronological Catch-22 differs from circular reasoning in that we do not suppose that we can actually achieve any of the preconditions; in fact we conclude that we can achieve none of them.
The logical status of a Catch-22 is therefore a reductio ad absurdum which demonstrates that the desideratum D cannot be achieved, because a logical contradiction would be required to achieve it. The only reason to confuse this with an actual contradiction or circular reasoning is if you assume that it is possible to achieve; but that is merely a bad assumption. The emotional power of a Catch-22 is when the desideratum D is more obviously compelling than one or more of the preconditions Nj, so that although D is impossible (and so 'technically' absurd), it is the logical structure which renders D impossible which seems absurd instead. This usually arises because the logical structure of the Catch-22 is established by some entity — again, usually a bureaucracy — whose priorities are more strongly aligned with the logical structure that it has created than with the desideratum D. The absurdity then comes from a clash of priorities between the agent trying to achieve the desideratum D, and the agent or more general setting which they find themselves in which is not very sympathetic to their priorities.
† This can also be used to capture a more complicated and Byzantine collection of preconditions which together form a set of necessary conditions, by taking the disjunction of preconditions and tracing them backwards until the collection of preconditions converge on a fixed set.
Catch-22 refers to a logical dilemma in which both outcomes are either equal or undesirable, rendering it unsolvable. It's not circular logic, but a logical complication of the inevitable outcome(s) of a set of premises.