A specific field of data extraction termed "logic mining" is important for retrieving insightful information from intricate datasets by generating logical representations. These logical frameworks are explainable and significant for knowledge-driven technologies in computational optimization. However, existing logic mining models suffer from key limitations, including inadequate attribute selection, rigid logical rule structures, inefficient training processes, and storage constraints that often lead to overfitting. To address these challenges, this study proposed an explainable logic mining framework that integrated four key components: At first, a log-linear based attribute selection method to identify significant features; second, a non-systematic higher-order logic structure using random k satisfiability (for k $\le$ 3) to enhance flexibility; after that, a multi-objective hybrid election algorithm for efficient and adaptive training; and, finally, an expanded retrieval phase employing a permutation operator to optimize the synaptic weight space in the discrete Hopfield neural network. The proposed framework was validated through comparative analyses against eight baseline models using real-world multidisciplinary datasets. Performance was rigorously evaluated across four evaluation metrics, where the experimental results demonstrated that the proposed model achieved a maximum accuracy of 97.73%, a precision of 100%, a specificity of 99.17%, and a matthews correlation coefficient (MCC) of 0.95 across 20 real-world datasets. Moreover, the proposed model's efficiency was also statistically validated through Nemenyi's post-hoc test and Cohen's d effect sizes, confirming its superior classification capability, stability, and reliability in logic-based knowledge.