What is Paradox?

Legal Definition
A paradox is a statement that, despite apparently sound reasoning from true premises, leads to a self-contradictory or a logically unacceptable conclusion. A paradox involves contradictory yet interrelated elements that exist simultaneously and persist over time.

Some logical paradoxes are known to be invalid arguments but are still valuable in promoting critical thinking.

Some paradoxes have revealed errors in definitions assumed to be rigorous, and have caused axioms of mathematics and logic to be re-examined. One example is Russell's paradox, which questions whether a "list of all lists that do not contain themselves" would include itself, and showed that attempts to found set theory on the identification of sets with properties or predicates were flawed. Others, such as Curry's paradox, are not yet resolved.

Examples outside logic include the Ship of Theseus from philosophy (questioning whether a ship repaired over time by replacing each of its wooden parts would remain the same ship). Paradoxes can also take the form of images or other media. For example, M.C. Escher featured perspective-based paradoxes in many of his drawings, with walls that are regarded as floors from other points of view, and staircases that appear to climb endlessly.

In common usage, the word "paradox" often refers to statements that may be both true and false i.e. ironic or unexpected, such as "the paradox that standing is more tiring than walking".
-- Wikipedia
Legal Definition
Any statement which is contradictory within itself, or has absurd assertions. It might be proven correct, but is usually confusing and incorrect.