There is a problem that we cannot prove that someone else's judgment of taste is wrong, but we can still argue over whether or not something is a great work of art. What is the point of arguing if there is no proof? This is what Kant calls the antinomy of taste. To be more precise, the antinomy consists of a thesis and an antithesis, both of which seem true, and yet they contradict each other.
"Thesis: The judgment of taste is not based upon concepts, for otherwise it would admit of controversy (would be determinable by proofs.)
Antithesis: The judgment of taste is based on concepts, for otherwise, despite its diversity, we could not quarrel about it (we could not claim for our judgment assent of others.)" (Bernard translation)"
The solution to the antinomy is simply that both are true if we consider "concept" to mean something different in each case. In the first, we are talking about definite concepts and in the second about indefinite concepts. But how can the antithesis be true (the thesis is not really problematic)? The relevant indefinite concept is that of what Kant rather obscurely calls "the supersensible substratum of humanity."
There are three hints as to what is meant by this term and how it might apply to resolving the problem. (1) Kant says that "the transcendental rational concept of the supersensible, which lies at the basis of all sensible intuition, is [undertermined and undeterminable], and therefore cannot be theoretically determined further." So we are talking about a "transcendental rational concept" but not one that can be defined. (2) We also know that we cannot know anything through this concept and it cannot provide a proof of a judgment of taste. Rather, it is"the mere pure rational concept of the supersensible which underlies the object (and also the subject judging it), regarded as an object of sense and thus as phenomenal." I take it that this does not mean that it is a rational concept of the noumenal realm or of the thing-in-itself. It is transcendental, not transcendent. So it is like the idea that "everything has a cause" and not exactly like the idea of God or the Soul, i.e. not an idea of reason. Ironically, however, it is the idea of something that goes beyond sense experience. But we must keep in mind that this idea happens a priori as a transcendental, not transcendent, aspect of the world of experience. It lies therefore at the basis of all experience. (It may be the same as the transcendental unity of apperception as found in the Critique of Pure Reason, and yet it refers in this case to the activity of the genius artist, something that does not happen in the earlier book.) That it underlies "the subject judging it" means that it is the basis for the possibility of judgment, for example that this work of art is great. The universality of the judgment of taste is, then, not just based on the shared structure of our human experience, i.e. that we all have "common sense," which is to say the shared faculties of the imagination and understanding that can go into free play. The universality and necessity of the judgment of taste, and the demand that others agree with us, also requires that there must also be this supersensible substratrum. (It seems to be neither clearly just a subjective substratum nor clearly an objective one...something in between) (3) Also, the concept of the supersensible substratum is "the concept of the general ground of the subjective purposiveness of nature for the judgment" which is to say that this is the basis as what Kant previously referred to as purposiveness without purpose, which we attend to when disinterested. Even the botanist, when looking to experience of beauty, must regard the flower without thinking about the actual purpose of its parts but only attending to the look of purpose, the look of design. So, if we have a concept that there is a ground for this subjective purposiveness, this is the concept of the supersensible substratum. Similarly, the judgment of taste has validity for everyone because the ground lies in the concept of the supersensible substrate of humanity. (By all rights this should be of any experiencing being whatsoever, and not just of humanity.) (4) Another hint is that aesthetical ideas, which Kant had previously explained to us were not determinate, might well be of the same sort, or related to this notion of the supersensible substratum. Perhaps it is an aesthetic idea or the ground of aesthetic ideas, and perhaps aesthetic ideas refer to it. (5) The final hint is that the antinomy forces "us to look beyond the sensible and to seek in the supersensible the point of union for all our a priori faculties, because no other expedient is left to make our reason harmonious with itself..." Again, the point of union for all our a priori faculties would be the transcendental or perhaps the transcendental unity of apperception, and not the transcendent reality of God or even the rational idea of God.
Two thoughts about this (1) It is not clear how the existence of the idea of the supersensible can provide the basis for anything like argument. At best it can provide a basis for why I might expect, or as is sometimes crudely put, "demand" that others agree with me in my judgment of taste when I am being disinterested when when my judgment is direct to an object insofar as it exhibits a look of purpose. There is still not answer to the question "what is the point of arguing?" (2) The argument seems to hint at an argument from design although with no reference to a person-like God: so it may be that Deism is the result of this (or the assumption?). If we assume Deism (or Pantheism), rather than Theism, then there is some supersensible basis for the universal validity of the experience of the beautiful both in nature and in fine art. It is found in Nature, not in a transcendent realm. We cannot prove that God exists, but at least the requirement of assuming a supersensible ground that unifies the a priori points towards such an existence, but, again, not one that is radically different from the grounds of Nature herself, since the ground of this is transcendental, not transcendent.