On the interrelation between systems of spheres and epistemic entrenchment relations
Maurício D. L. Reis · Logic Journal of IGPL · 2013
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 that fact.