On the interrelation between systems of spheres and epistemic entrenchment relations

In this article, we present several results concerning the deep interconnection between systems of spheres and epistemic entrenchment relations, which, in a certain sense, complete the studies on this subject that have been presented in [Peppas and Williams (1995, Notre Dame J. Formal Logic, 36, 120–133); Hansson (1999); Rott and Pagnucco (1999, J. Philosophical Logic, 28, 501–547)]. The main contributions of this work are the following: First we prove that the condition used in Hansson (1999) to define an epistemic entrenchment relation by means of a system of spheres is a necessary and sufficient condition for two such structures to give rise to the same contraction function. Afterwards, we show in a direct way that such condition is equivalent to the one presented in Peppas and Williams (1995, Notre Dame J. Formal Logic, 36, 120–133) as a necessary and sufficient condition for an epistemic entrenchment relation and a system of spheres to yield the same revision function. Moreover, we show, by means of a constructive proof, that for any epistemic entrenchment relation there is a system of spheres such that the mentioned condition holds. We notice yet that, by combining some of those results, we obtain a direct and constructive proof for the well-known fact that the class of system of spheres-based contractions coincides with the class of epistemic entrenchment-based contractions, which differs from all the other proofs so far provided in the literature for