Fuzzy Sets and Systems 160 (2009) 306 – 324 www.elsevier.com/locate/fss T-conditional possibilities: Coherence and inference Giulianella Coletti a , ∗ , Barbara Vantaggi b a Dip. Matematica e Informatica, Università di Perugia, viaVanvitelli, 06123 Perugia, Italy b Dip. Metodi e Modelli Matematici, Università “La Sapienza” Roma, via Scarpa 16, 00161 Roma, Italy Available online 3 May 2008 Abstract In this paper we refer to an axiomatic definition of T-conditional possibility, where T is any t-norm. We characterize a full T-conditional possibility in terms of a suitable set of unconditional possibilities. Starting from this characterization we are able to manage coherent conditional possibility assessments and their enlargements. To compare T-conditional possibility related to different t-norm T, we study binary relations locally representable by a T-conditional possibility. © 2008 Elsevier B.V. All rights reserved. Keywords: T-conditional possibility; Locally representable relations; Coherence; Inference 1. Introduction Any process of knowledge acquisition or inference starts from the choice of the (numerical or qualitative) model for representing the partial information and the relevant uncertainty. Nevertheless, independently of the chosen model, to make inference consists in the following two basic steps: • conditioning (consider, under different hypotheses,the events of interest); • enlarging the basis of knowledge to new objects of interest (conditional or unconditional events). In the classical probabilistic framework to make inference is strictly related to the Bayesian procedure: the aim is, as well-known, to compute P (H i |E) under the evidence P (E) > 0, starting from a probability assessment on a set of exhaustive and incompatible (prior) hypotheses H i and on conditional events E|H i . Due to the particular logical conditions (exhaustivity and incompatibility of H i ’s) any assessment {P (H i ), P (E|H i )} is coherent and the procedure gives a unique value. This could induce the idea that it is not necessary to test the con- sistence of the assessment {P (H i ), P (E|H i )} and that any inferential procedure leads to a unique result. Nevertheless, when the logical conditions are different (as often real problems require) we must rediscover the actual general proce- dure of inference. In particular, it is necessary to study for every (partial) assessment its coherence (consistency) with the model of reference and so to find the possible assessments for a new (conditional) event maintaining coherence, and having into account all the available information (see, e.g., [4]). In this paper we focus our attention on possibility measures and we study for them all the aspects regarding inference: conditioning, coherence and enlargement. ∗ Corresponding author. E-mail addresses: coletti@dipmat.unipg.it (G. Coletti), vantaggi@dmmm.uniroma1.it (B. Vantaggi). 0165-0114/$ - see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2008.04.006