Implicit Discourse Relation Recognition (IDRR), which infers discourse relations without the help of explicit connectives, is still a crucial and challenging task for discourse parsing. Recent works tend to exploit the hierarchical structure information from the annotated senses, which demonstrate enhanced discourse relation representations can be obtained by integrating sense hierarchy. Nevertheless, the performance and robustness for IDRR are significantly constrained by the availability of annotated data. Fortunately, there is a wealth of unannotated utterances with explicit connectives, that can be utilized to acquire enriched discourse relation features. In light of such motivation, we propose a Prompt-based Logical Semantics Enhancement (PLSE) method for IDRR. Essentially, our method seamlessly injects knowledge relevant to discourse relation into pre-trained language models through prompt-based connective prediction. Furthermore, considering the prompt-based connective prediction exhibits local dependencies due to the deficiency of masked language model (MLM) in capturing global semantics, we design a novel self-supervised learning objective based on mutual information maximization to derive enhanced representations of logical semantics for IDRR. Experimental results on PDTB 2.0 and CoNLL16 datasets demonstrate that our method achieves outstanding and consistent performance against the current state-of-the-art models.