Research On The Anti-foundation Axiom AFA And Its Applications In Solving The Paradoxes

In ZFC system, due to the foundation axiom FA, we only discuss thewell-founded sets, not the non-well-founded sets. However, with the cognitive needsof the computer science, economics, cognitive science and philosophy, we need thedomain both including the well-founded sets and the non-well-founded sets, and thisis AFA. We replace the foundation axiom FA in ZFC with the anti-foundation axiomAFA and call the new system ZFA. Peter Aczel and Jon Barwise proved theconsistency of ZFA in different method. In the philosophy of logic, an importantapplication of the anti-foundation axiom AFA is to solving the paradoxes.The paradoxes are ancient conundrums of logic, deeply attracting the scholars toexplore them. In1980s, Jon Barwise and others used the anti-foundation axiom AFA,founded the situation semantics solutions to the paradoxes and the anti-foundedmodel theory method, and made some reasonable explanations for some paradoxes.This paper studies the anti-foundation axiom AFA in solving the paradoxes, and hastheoretical and practical significance.This paper introduces the basic theory of the anti-foundation axiom AFA fromthe solutions to the equations, and focuses on its applications in solving the paradoxes.First, on the basis of modeling the games, it constructs the supergame and thehypergame. Second, it explores the situation semantics solutions to the paradoxes andthe anti-founded model theory method, and gives reasonable explanations for theliar-like paradoxes and the paradoxes of denotation. This paper involves the followingfour aspects:Firstly, it sketches the basic theory of the the anti-foundation axiom AFA and theconsistency of ZFA system from the solutions to the equations.Secondly, on the basis of introducing the supergame, the hypergame, and thehypergame paradox, this paper gives the model of the games. It proves every opengame is determined, and constructs the supergame and the hypergame.Thirdly, this paper treats the solution lemma as a mathematical tool to introduce the situation semantics solutions to the paradoxes. It analyzes the like-like paradoxesfrom the Russellian account and the Austinian account respectively, and uses theReflection Theorem to characterize the relationship between the Russellian accountand the Austinian account. Finally, it points out that the confusion of negation anddenial is the real source of the paradoxes.Fourthly, this paper introduces the basic theory of the anti-founded model theory,and discusses how to solve the paradoxes in the case of adding predicates to thelanguage. In the case of adding a truth predicate to the language, it proposes the liartheorem, and solves the liar, the strengthened liar, the truth-teller, the card paradox,the contingent liars. In addation, it introduces the fuzzy logic and the intuitionisticfuzzy logic to partial model and Kleene evaluation, and explains the liar sentencedoes not generate the paradox in the fuzzy logic and the intuitionistic fuzzy logic. Inthe case of adding a new formation rule and a denote predicate to the language, itsolves the paradoxes of denotation.