The general context of paradox may at first be assumed to consist of just the sorts of relationships which occur within the schema of categorical deduction: the bounded version of the Cartesian Coordinate System, in which opposite qualities oppose along the diagonal.
However, this is not precisely the case. For a standard categorical deduction does not represent a paradox at all, nor a contradiction, but rather, a non-trivial case.
The case in which a paradoxical solution is needed are only cases which are problematic. And that is not obvious, because ordinarily all terms are necessary to create a categorical deduction.
So, it must be clarified that there is an exceptional case which is paradoxical even for the categorical deduction. But how could this be the case?
The answer is that true opposites are always positive terms, because they have content. What is paradoxical is simply the existence of a concept solely as the negation of its opposite, a kind of contention with zero.
We can see, for example, that in categorical deduction... A. Arbitrary solutions have non-arbitrary problems, AND B. Arbitrary problems have non-arbitrary solutions.
...that both 'non-arbitrary' and potentially 'problems' pose Type 3 weaknesses, that is, weaknesses involving the assertability of single subset categories*
*(Each of the opposites is a subset category. For four categories there are two deductions, rendering it exponential or efficient).
Since there are two weaknesses, it is possible to see a solution to the problem through paroxysm: while one missing category is incoherent, two missing categories means there is a problem.
If there is a problem, however, there is a solution!
The solution in this case is simply to ignore the problematic categories, and accept that the remaining categories are coherent by themselves.
For example, the paroxysm of non-arbitrary problems involves arbitrary solutions. The paroxysm of problematic non-arbitration involves a solution to arbitrariness.
But although these paroxysms are helpful with incoherent cases in which positive data has gone missing, they are not so helpful with actual opposites, at least coherently!
For example, we would find it less helpful to realize that the paroxysm of the arbitrary solution to non-arbitrary problems is the non-arbitrary problem of arbitrary solutions! The explanation for the exception in which paroxysm is used instead of categorical deduction has two parts: 1. The data expressed is already coherent, and 2. The limit of the data has been reached. But in the vocabulary of coherent philosophy, these two explanations mean the same thing! In general, what I have described is a group of methods related to the topic of exceptional knowledge.