W.D. Hart discusses the nature of non-well-founded sets and their implications for the liar paradox. He argues that the liar paradox suggests that no proposition answers to a liar sentence, and that the structure of propositions must be such that they do not include liar sentences. He proposes that propositions can be understood as ordered pairs, where the first member is the denotation of the subject and the second member is the extension of the predicate. This leads to the idea that a self-referential proposition is an ordered pair where the first member is the proposition itself. Hart then explores different ways of understanding the identity of non-well-founded sets, noting that the standard iterative conception of sets, which is widely accepted, rules out infinitely descending membership chains. However, alternative conceptions, such as those proposed by Aczel, allow for non-well-founded sets. Hart examines various principles for determining the identity of non-well-founded sets, including those proposed by Forti and Honsell, Boffa, and Finsler. He argues that these principles differ in their treatment of identity, leading to inconsistencies. Hart concludes that the identity of non-well-founded sets remains a matter of debate, and that the nature of propositions and non-well-founded sets may be similarly obscure.W.D. Hart discusses the nature of non-well-founded sets and their implications for the liar paradox. He argues that the liar paradox suggests that no proposition answers to a liar sentence, and that the structure of propositions must be such that they do not include liar sentences. He proposes that propositions can be understood as ordered pairs, where the first member is the denotation of the subject and the second member is the extension of the predicate. This leads to the idea that a self-referential proposition is an ordered pair where the first member is the proposition itself. Hart then explores different ways of understanding the identity of non-well-founded sets, noting that the standard iterative conception of sets, which is widely accepted, rules out infinitely descending membership chains. However, alternative conceptions, such as those proposed by Aczel, allow for non-well-founded sets. Hart examines various principles for determining the identity of non-well-founded sets, including those proposed by Forti and Honsell, Boffa, and Finsler. He argues that these principles differ in their treatment of identity, leading to inconsistencies. Hart concludes that the identity of non-well-founded sets remains a matter of debate, and that the nature of propositions and non-well-founded sets may be similarly obscure.