The article discusses the concept of non-well-founded sets and their implications for understanding the liar paradox. The author, W.D. Hart, begins by comparing the lack of denotation for singular terms like "Santa Claus" to the paradoxes in set theory, suggesting that not all predicates have sets as extensions. He then explores the idea that no proposition answers to a liar sentence, similar to how "Santa Claus" does not denote a person. Hart introduces the concept of propositions as ordered pairs, where the first member is the denotation of the subject and the second member is the extension of the predicate. This framework is used to analyze self-referential sentences like "This sentence is false," which leads to the idea of a self-referential proposition being an ordered pair that is its own first member. To address the liar paradox, Hart turns to the iterative conception of sets, which is the dominant view in contemporary set theory. However, this conception rules out infinitely descending chains of membership, which is necessary to resolve the liar paradox. He then introduces alternatives to the axiom of foundation, such as Peter Aczel's theory of non-well-founded sets, which allows for sets to be members of themselves. Hart explains that non-well-founded sets can be represented using graphs and arrows, where a graph can be a picture of a set. He discusses different criteria for identifying non-well-founded sets, including extensionality, uniqueness of decoration, and tree structures. These criteria are shown to be inconsistent with each other, leading to the conclusion that there is no clear basis for choosing a single criterion. Finally, Hart reflects on the implications of these findings for understanding propositions and non-well-founded sets, suggesting that both concepts are subject to similar levels of obscurity and uncertainty. He concludes by noting that while there may be no definitive answers, the exploration of these concepts can provide valuable insights into the nature of truth and self-referential statements.The article discusses the concept of non-well-founded sets and their implications for understanding the liar paradox. The author, W.D. Hart, begins by comparing the lack of denotation for singular terms like "Santa Claus" to the paradoxes in set theory, suggesting that not all predicates have sets as extensions. He then explores the idea that no proposition answers to a liar sentence, similar to how "Santa Claus" does not denote a person. Hart introduces the concept of propositions as ordered pairs, where the first member is the denotation of the subject and the second member is the extension of the predicate. This framework is used to analyze self-referential sentences like "This sentence is false," which leads to the idea of a self-referential proposition being an ordered pair that is its own first member. To address the liar paradox, Hart turns to the iterative conception of sets, which is the dominant view in contemporary set theory. However, this conception rules out infinitely descending chains of membership, which is necessary to resolve the liar paradox. He then introduces alternatives to the axiom of foundation, such as Peter Aczel's theory of non-well-founded sets, which allows for sets to be members of themselves. Hart explains that non-well-founded sets can be represented using graphs and arrows, where a graph can be a picture of a set. He discusses different criteria for identifying non-well-founded sets, including extensionality, uniqueness of decoration, and tree structures. These criteria are shown to be inconsistent with each other, leading to the conclusion that there is no clear basis for choosing a single criterion. Finally, Hart reflects on the implications of these findings for understanding propositions and non-well-founded sets, suggesting that both concepts are subject to similar levels of obscurity and uncertainty. He concludes by noting that while there may be no definitive answers, the exploration of these concepts can provide valuable insights into the nature of truth and self-referential statements.