  One such disjunctive definition, Longworth and Scarantino’s, adapts Gaut’s list of ten clustering properties, where that list (see 3.5 above) includes institutional properties (e.g., belonging to an established art form) and traditional ones (e.g., possessing positive aesthetic properties); see also Longworth and Scarantino 2010. The core idea is that art is defined by a disjunction of minimally sufficient and disjunctively necessary conditions; to say that a disjunct is a minimally sufficient constitutive condition for art-hood, is to say that every proper subset of it is insufficient for art-hood. An account of what it is for a concept to have disjunctive defining conditions is also supplied.  The definition of art itself is as follows: ∃Z∃Y (Art iff (Z ∨ Y)), where (a) Z and Y, formed from properties on Gaut’s cluster list, are either non-empty conjunctions or non-empty disjunctions of conjunctions or individual properties; (b) there is some indeterminacy over exactly which disjuncts are sufficient; (c) Z does not entail Y and Y does not entail Z; (d) Z does not entail Art and Y does not entail Art. Instantiation of either Z or Y suffices for art-hood; something can be art only if at least one of Z, Y is instantiated; and the third condition is included to prevent the definition from collapsing into a classical one. The account of what it is for concept C to have disjunctive defining conditions is as follows: C iff (Z ∨ Y), where (i) Z and Y are non-empty conjunctions or non-empty disjunctions of conjunctions or individual properties; (ii) Z does not entail Y and Y does not entail Z; (iii) Z does not entail C and Y does not entail C. A worry concerns condition (iii): as written, it seems to render the account of disjunctive defining conditions self-contradictory.  For if Z and Y are each minimally sufficient for C, it is impossible that Z does not entail C and that Y does not entail C. If so, then nothing can satisfy the conditions said to be necessary and sufficient for a concept to have disjunctive defining conditions.