On Evans's Vague Object from Set Theoretic Viewpoint

Abstract  Gareth Evans proved that if two objects are indeterminately equal then they are different in reality. He insisted that this contradicts the assumption that there can be vague objects. However we show the consistency between Evans's proof and the existence of vague objects within classical logic. We formalize Evans's proof in a set theory without the axiom of extensionality, and we define a set to be vague if it violates extensionality with respect to some other set. There exist models of set theory where the axiom of extensionality does not hold, so this shows that there can be vague objects.