I don’t believe that your definition of “reference” is correct: perhaps you could persuade me. “Reference” in the relevant sense is “the act of referring”. We should dig deeper into what things “refer”, but as a start, expressions refer. Not all expressions are concepts. “The new occupants of the White House” refers to real people, and those people are the referents of the expression, but “The new occupants of the White House” is not a concept. If you want a set-theoretic definition of “reference”, it should be the set of all expressions of any type, paired with their referents (plural or singular). You might coin a word “word-reference” which specifically refers to just concepts and the things they refer to. In that case, r is a set, not an individual (it’s not a singular referent, it’s all of the referents).
We can mostly set aside the concept of “reference” (though not the matter of what refers), because it is irrelevant to cooking up and evaluating the invalid concept “anti-reference” (it’s relevant to the proof of contradiction). “Anti-reference” could almost qualify as a label, although again it should be “word-anti-reference” if the goal is to only look at a kind of referring relation of concepts, and not those of everything that refers (briefly: denial of a proposition is not invalid). Because we need to evaluate the potential legitimacy of the putative concept qua concept, the label needs to be replaced so that there is no surreptitious smuggling in of ideas from other, valid concepts. For the sake of clarity, we should call this concept “glank”.
A glank is the complement of the referents of a concept – everything that a concept does not refer to. An example of a glank would be a relationship between “dog” and the universe (not just things, but also abstractions, and any other fact such as the fact that adding baking soda to vinegar causes the mix to foam up) – it refers to everything except for dogs.
It is cognitively valid to assert the proposition “this is a dog”, and it is equally valid to deny that proposition. The denial of a proposition is not automatically a concept. We do have valid method-concepts that pertain to denial – “denial, exclusion, contradiction, complement”. We can easily construct an expression which identifies the glank of a concept, using ordinary language expressions such as “everything that is not a dog”. The question is whether the word “glank” does something that makes it superior to the compositional expression “everything that is not”.
In order for this monster glank to be elevated to the status of a concept, it needs cognitive validity, some purpose. There may be a narrow professional context (anti-cognitivist logicians) where it is useful to be able to quickly say “the complement of the concept C with respect to all existence”, so that instead of constantly saying “the cardinality of the complement of the concept ‘dog’ with respect to all existence is identical to the cardinality of the complement of the concept ‘run’ with respect to all existence”. Instead, philosophers could more efficiently say “the cardinality of the glank of dog is identical to the cardinality of the glank of run”.
This would not suffice. “Glank” was cobbled together to relate concepts and things that they don’t refer to, but the complement relationship is broader, so we need to create “florn”, which is the complement of the facts that any expression identifies. Thus the florn of “everything that is not a dog with blue eyes and grey fur” is, simply, the universe, minus those dogs that have both blue eyes and grey fur. A glank is a florn where the expression is a word.
The florn of “a dog with blue eyes and grey fur” includes all expressions (sentences, clauses and words are not actual dogs of that type), all actions (running is not a dog), all cats, rocks etc., and all dogs which don’t have blue eyes or don’t have grey fur. The florn of a dog (an actual dog) is undefined, because an actual dog is not a linguistic expression, and “florn” takes an expression as its argument. Similarly, “reciprocal of blue” is undefined. Since the florn of “dog” is not an expression, the florn of the florn of “dog” is likewise undefined. In your proof of contradiction, you don’t distinguish between A and “A”, which is a problem.
Since we can identify what a concept refers to, we can evaluate the proposition “the concept A does not refer to X”. But we are not directly aware of all existents that a concept refers to, nor are we directly aware of all existents that are not instances of that concept. Regarding your final conclusion, is your point that we are not aware of all referents (are not omniscient)? If not, I don’t see wherein lies the problem with evaluating the denial of a proposition.