Zero-shot image classification, which aims to predict unseen classes whose samples have never appeared during the training phase, is crucial in the Web domain because many new web images appear on various websites. Attributes, as annotations for class-level characteristics, are widely used semantic information for this task. However, most current methods often fail to capture discriminative image features between similar images from different classes, leading to unsatisfactory zero-shot image classification results. This is because they solely focus on limited visual-attribute feature alignment. Therefore, we propose a Zero-Shot image Classification with Logic adapter and Rule prompt method called ZSCLR, which utilizes logic adapter and rule prompts to encourage the model to capture discriminative image features and achieve reasoning. Specifically, ZSCLR consists of a visual perception module and a logic adapter. The visual perception module extracts image features from training data. At the same time, the logic adapter utilizes the Markov logic network to encode the extracted image features and rule prompts for refining the discriminative image features. Due to predicates of rule prompts representing symbolic discriminative features, the proposed model can focus more on these discriminative features and achieve more precise image classification. Additionally, the logic adapter enables the model to adapt from recognizing images in seen classes to those in unseen classes through the reasoning of the Markov logic networks. We implement experiments on three standard zero-shot image classification benchmarks, and ZSCLR achieves competitive performance. Furthermore, ZSCLR can provide explanations for its predictions through rule prompts.