Beside formal approaches to semantic inference that rely on logical representation of meaning, the notion of Textual Entailment (TE) has been proposed as an applied framework to capture major semantic inference needs across applications in Computational Linguistics. Although several approaches have been tried and evaluation campaigns have shown improvements in TE, a renewed interest is rising in the research community towards a deeper and better understanding of the core phenomena involved in textual inference. Pursuing this direction, we are convinced that crucial progress will derive from a focus on decomposing the complexity of the TE task into basic phenomena and on their combination. In this paper, we carry out a deep analysis on TE data sets, investigating the relations among two relevant aspects of semantic inferences: the logical dimension, i.e. the capacity of the inference to prove the conclusion from its premises, and the linguistic dimension, i.e. the linguistic devices used to accomplish the goal of the inference. We propose a decomposition approach over TE pairs, where single linguistic phenomena are isolated in what we have called atomic inference pairs, and we show that at this granularity level the actual correlation between the linguistic and the logical dimensions of semantic inferences emerges and can be empirically observed.