Logic-based information retrieval (IR) models represent the retrieval decision as an implication d→q between a document d and a query q, where d and q are logical sentences. However, d→q is generally a binary decision, thus we need a measurement to estimate the degree to which d implies q, denoted U(d→q)⁠. Most of the existing logic-based IR models either do not precisely define the implication d→q or use non-classical definitions. Some models also define the uncertainty U in informal ways. More importantly, they use two non-related frameworks to define d→q and its uncertainty U, even though the two notions are intrinsically related. The goal of this study is to propose a new logic-based IR model, which overcome these shortcomings. To this end, we first propose to replace the implication d→q by the validity of material implication⊨d⊃q⁠. Second, we redefine and adapt the mathematical relationship between logics, lattices and probability. Our new IR model presents a possible formalism for van Rijsbergen's intuition about replacing U(d→q) by P(q∣d)⁠.

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.