Given an arbitrary piece of work, defined over the n-dimensional linguistic space, this qualifies as art if and only if it explores the human condition by being incomplete and ill-defined.
The encrypted message
The theorem being enunciated, let us now elaborate more in detail on how incompleteness and ill-definition underlie each piece of art. To that end, I will resort to an artistic device, namely, a metaphor.
A piece of art is an encrypted message that comes without a key (incompleteness). In other words, it is an ensemble of signs and symbols, which appeals to one or more senses, that constitutes a motif whose rationale is at most hinted at and at any rate never revealed by the artist. The moment a piece of work has an expressed purpose, it is immediately demoted and ceases to be an object of art. Hence the notion that art is an end in itself. Yet, put it this way, one could mistake the artist for an obscure plotter hiding behind his tools or, even worse, as a charlatan who hides his lack of virtue behind unintelligible constructs. Instead, the true artist is not the one who hides his rationale, but rather the one who gives form to shapeless matter moved by the ephemeral ardor of creation, which transcends any rationale. In fact, the artist, upon returning on worldly grounds after a brief ascension to the creative dimension, simply does not possess the key to his/her own creation. Finally, the artist qualifies as such whenever he/she manages to sketch a motif composed by signs that taken singularly are intelligible to the spectator and yet as whole constitute an incomplete picture that forces the spectator to "fill in the blanks". Therefore, the keystones that support the edifice of art are provided by its interpreters rather than by the artist. It follows that each piece of art is inextricably entangled with its recipient/interpreter: the former does not exist without the latter and both the artist and the interpreter are creators in the sense that they confer form to shapeless matter at different times. The fundamental role of the spectator/recipient/interpreter brings us to the ill-defined nature of each piece of art.
Each piece of art is an ill-defined encrypted message in the sense that it can be decrypted with multiple keys that in turn lead to multiple messages. Therefore, by improperly paraphrasing Pirandello, each piece of art is one, none, and one hundred thousand, since there exists one for each interpreter (and none when there is no interpreter). Indeed, the motif of a piece of art is better described as evoking rather than saying; suggesting rather than dictating; alluding to something rather than pointing at something; provoking rather than inquiring. A real piece of art is never definitive or conclusive in nature, but rather unresolved, faceted, and even shape-shifting. Each piece of art, by tapping into the imagery of the interpreter, in the desperate attempt to reach completion, takes on the most disparate forms. This is because one's imagery is the complex product of experience and innate nature, which not only vary on an individual basis but also in time, as one slowly yet restlessly mutates over time. As the interpreter, in experiencing the piece of art, adds his/her own material to it, the piece of art can even act as the mirror of one's soul at a given time. To some extent, a piece of art resembles the vague and ambiguous prophesies given by the oracle of Delphi, which were often self-fulling because the recipient himself would be interpret them according to the whims of his own mind. It comes as no surprise that the famous maxim inscribed in the pronaos of the Temple of Apollo at Delphi ran: know thyself. This brings us to the incommensurable importance of art in each human era.
In virtue of its incomplete and ill-defined nature, each piece of art let one partake in the pleasure of creation ('filling the blanks') and in doing so one has also the chance to catch a glimpse in one's mirrored image, or in other words, to explore one's human condition. Moreover, the incompleteness makes each piece of art eternal, in the same way a love affair that has not reached closure stirs our soul endlessly. Finally, like the Aristotelian unmoved mover, a piece of art can move the hearths of different interpreters, in different ways and at different times, while being immutable over the centuries.
A short note on the language of the arts
In composing a piece of art the artist makes use of some sort language, which can appeal to one of the senses or even be synesthetic. It follows that for the interpreter to enjoy the piece of art, he/her has to be accustomed to the language of the artist. Otherwise, the interpreter will fail to enjoy it in the same way one would fail to fill in the blanks in a text written in an unknown language. This is the reason why certain art trends may appeal to people with a certain background and not others, or even why many people do not appreciate art in general. In fact, learning to appreciate art is a process that is not dissimilar from learning how to read and write.