 Perhaps because of the importance of the rule of cut in proof theory, nontransitive approaches are often studied via proof systems rather than via models. The essential use of transitivity properties in paradoxical derivations was noted in Tennant 1982; an approach to paradoxes that rejects both cut and simple transitivity in a general setting can be found in Hallnäs 1991; Hallnäs and Schroeder-Heister 1991; and Schroeder-Heister 2004. There are helpful philosophical remarks on cut in Schroeder-Heister 1992, which also notes some relations between noncontractive and nontransitive approaches.