Reply to David Ripley’s Comments on Comparing context updates in delineation and scale based models of vagueness

In his comments to my contribution ‘Comparing Context Updates in Delineation and Scale Based Models of Vagueness’ David Ripley points out some severe technical flaws. As he shows by a counter-example, the main result of the paper does not hold. The goal of this reply is to explain where exactly the source of the problem lies and to point out a possible way to solve it. In the paper I take two contextual approaches to vagueness both describing the notion of context update very precisely, namely one by Kyburg and Morreau and one by Barker. Besides analyzing strengths and weaknesses of these approaches I try to prove that they both make the same predictions, once the same situation is modeled twice using each approach. This is done by introducing an ‘intermediate representation’ of contexts and showing how contexts as defined by each approach can be translated into this representation and also the other way round. Finally, once this connection is established one can observe that a context update (with consistent information) has exactly the same effect on the intermediate representation in either approach. However, as Ripley points out, these translation functions are not perfectly reversible, they rather form antitone and monotone Galois connections between contexts and their intermediate representations. In order to illustrate this I will stick to the same notation as in the paper and in Ripley’s comment: the mappings from Kyburg and Morreau’s models into the intermediate representation and back will be denoted as Tkm and T−1 km , and Tb as well as T−1 b for Barker’s models. For directly translating between Kyburg and Morreau’s and Barker’s contexts I use the mappings K = T−1 km ◦ Tb and B = T −1 b ◦ Tkm. Assume two corresponding models, one according to Kyburg and Morreau and the other one according to Barker, let the initial contexts be denoted as p0 and C0, respectively, and let the resulting contexts after a (successful) update be called p and C. Theorem 8 then states that the intermediate representations of p and C coincide: Tkm(p) = Tb(C). As Ripley shows by giving a counter-example this claim does not hold due to the non-reversibility of these mappings. However, it is still true—at least for his example—that K(C) = p and that B(p) =C. One might be tempted to reformulate Theorem 8 in such a way that it does not directly refer to the intermediate representation but makes use of these connections instead and thus try to save Theorem 9. As it turns out, this does not suffice; one has to go deeper and rethink the notion of corresponding models. Intuitively, two models of the same situation, one as defined by Kyburg and Morreau and the other as