Wrt defining art, I offer my definition:
"An artifact whose purpose is to be perceived and thereby produce in its perceiver a positive experience of no direct practical value to the perceiver."
"Artifact" here is meant in the sense of being appropriate for Daniel Dennett's design stance. It is not neccessarily tangible or durable.
This is what's called a Genus-differentia definition, or type-and-distinction definition. "Artifact" is the type, the rest is the distinction.
This lets me build on existing understandings about artifacts. They have a purpose, but they remain artifacts even when they are not accomplishing that purpose. They are constructed by human beings, but this is a pragmatic fact about human ability and not a part of their definition.
I avoided terms that make no definitional progress such as "beauty" and "aesthetic". Using them would just be passing the buck.
This definition seems to include birdsong. Make of that what you will. One could reasonably say that birdsong is a fitness signal of direct practical value to the intended perceiver, though.
Under this definition, throw-the-paint art is not so much excluded as it is a marginal, failed, or not serious example, much the way that a hammer (which is another type of artifact) constructed of two twigs scotch-taped together at right angles is a failure as a hammer
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It's usually given as "Have you stopped beating your wife yet?" (Emph mine). The problem is the presupposition that you have been beating your wife. Either answer accepts (or appears to accept) that presupposition.
It's a different sort of bad question than the underconstrained questions. The Liar Paradox OTOH is a case of underconstrained question because it contains non-well-founded recursion.