Is uncertain logical-matching equivalent to conditional probability?

Logic-based Information Retrieval (IR) models represent the retrieval decision as a logical implication d->q between a document d and a query q, where d and q are logical sentences. However, d->q is a binary decision, we thus need a measure to estimate the degree to which d implies q, denoted P(d->q). In this study, we revisit the Van Rijsbergen's assumptions about: 1- the logical implication ->' is not the material one, and 2- P(d->q) could be estimated by the conditional probability P(q|d). More precisely, we claim that the material implication is an appropriate implication for IR, and also we mathematically prove that replacing P(d->q) by P(q|d) is a correct choice. In order to prove the Van Rijsbergen's assumption, we use the Propositional Logic and the Lattice theory. We also exploit the notion of degree of implication that is proposed by Knuth.