Proofs of existence.
Proofs of existence are proofs of the existence of something. It can be shown that all proofs are proofs of existence. And all proofs are proofs by implication. However, what sets what is meant by proofs of existence here from all other proofs by implication is that these proofs do not prove a whole exists, but only its part. Such proofs are not uncommon in mathematics. A mathematician might be able to prove that a solution to an equation exists, and be unable to tell you what it is.
Proofs of uniqueness.
Proofs of uniqueness are proofs that a set of wholes for a particular part is made of one thing only. For our topic, they are of interest, because proofs of existence prove the existence of a part and not the whole, and therefore they are incomplete when the proof of the existence of a particular is required. In mathematics proofs of uniqueness are often easier than proofs of existence. After proving the solution exists, the mathematician might proceed to prove that it is unique, that there is only one of it. Like proofs of existence, these proofs still do not establish what the whole is.
Realizability is the question of whether the whole can be known, and for our topic, whether the whole, or wholes, can be known after a proof of existence. Proofs of existence in mathematics are all realizable. The mathematician might not know the solution to the equation now even though he knows that it exists, but he, or some one else, can one day capture the solution in full. It might be a number, a data set, an object, an equation even if an infinite series, a formula, or an algorithm. The mathematician can then take the realized and use it else where. However, not all proofs of existence are realizable. An example is the existence of the primary cause. Its non existence is a contradiction to the present, and therefore it exists, but it is not realizable. Proofs of existence that are not realizable operate in the context of fact. (See Lesson 01).
The part of the world that is realizable is of interest, and is called, the realizable.