This paper focuses on the improvement of the conceptual structure of FrameNet (FN) for the sake of applying this resource to knowledge-intensive NLP tasks requiring reasoning, such as question answering, information extraction etc. In this paper we show that in addition to coverage incompleteness, the current version of FN suffers from conceptual inconsistency and lacks axiomatization which can prevent appropriate inferences. For the sake of discovering and classifying conceptual problems in FN we investigate the FrameNet-Annotated corpus for Textual Entailment. Then we propose a methodology for improving the conceptual organization of FN. The main issue we focus on in our study is enriching, axiomatizing and cleaning up frame relations. Our methodology includes a data-driven analysis of frames resulting in discovering new frame relations and an ontological analysis of frames and frame relations resulting in axiomatizing relations and formulating constraints on them. In this paper, frames and frame relations are analyzed in terms of the DOLCE formal ontology. Additionally, we have described a case study aiming at demonstrating how the proposed methodology works in practice as well as investigating the impact of the restructured and axiomatized frame relations on recognizing textual entailment.