Tree-based and rule-based machine learning models play pivotal roles in explainable artificial intelligence (XAI) due to their unique ability to provide explanations in the form of tree or rule sets that are easily understandable and interpretable, making them essential for applications in which trust in model decisions is necessary. These transparent models are typically used in surrogate modeling, a post-hoc XAI approach for explaining the logic of black-box models, enabling users to comprehend and trust complex predictive systems while maintaining competitive performance. This study proposes the Cost-Sensitive Rule and Tree Extraction (CORTEX) method, a novel rule-based XAI algorithm grounded in the multi-class cost-sensitive decision tree (CSDT) method. The original version of the CSDT is extended to classification problems with more than two classes by inducing the concept of an n-dimensional class-dependent cost matrix. The performance of CORTEX as a rule-extractor XAI method is compared to other post-hoc tree and rule extraction methods across several datasets with different numbers of classes. Several quantitative evaluation metrics are employed to assess the explainability of generated rule sets. Our findings demonstrate that CORTEX is competitive with other tree-based methods and can be superior to other rule-based methods across different datasets. The extracted rule sets suggest the advantages of using the CORTEX method over other methods by producing smaller rule sets with shorter rules on average across datasets with a diverse number of classes. Overall, the results underscore the potential of CORTEX as a powerful XAI tool for scenarios that require the generation of clear, human-understandable rules while maintaining good predictive performance.