The paradox of analysis is roughly this: If a conceptual analysis of a term like justice was successful, then the two sides of the analysis should mean the same thing, so it should also be trivial.
The notions of cognitive triviality (analyticity?) and sameness of meaning are infamously hard to spell out, but I think we can get much of the intuitive puzzlement of the paradox of analysis by rephrasing it as follows:
If you know already what 'justice' means, how can it be useful to you to have a conceptual analysis that says an act is just if and only if it is ____?
If you accept this restatement of the problem, I propose the answer is this:
Your "knowledge of what `justice' means" consists in something like a disposition to accept some collection of methods of inference, which - under favorable conditions- tend lead to your beliefs about what's just correctly tracking the facts about what's just. Call the particular algorithm for making and revising judgements about what's just α. So your understanding of the word justice consists in the fact that your brain implements α.
The potential usefulness of conceptual analysis comes from the fact that your brain can implement α without:
a) your knowing what algorithm α is (e.g. some processes in your brain recognize grammatical english sentences, but you don't know what these processes are).
b) your knowing that the descriptions of actions which algorithm α ultimately gives a positive verdict on are exactly those which have property B. (this is useful when your usual methods of checking for B-hood are faster/easier to deploy than your usual methods of checking for justice)
c) your knowing that property C applies to most of the things which A would ultimately give a positive verdict on, but C is easier to apply, and all the purposes normally served by considering which actions are just would be served even better by thinking about which actions have property C. (the classic definitions of computability and limit are examples of this kind)