Early detection of Alzheimer's disease using brain MRI image data can substantially improve clinical intervention and patient management. Our study evaluates the performance of an Alzheimer's classification system based on Fuzzy Inference Systems (FIS), specifically for the Mamdani and Sugeno models, in identifying four patient categories: (1) Non-Dementia, (2) Very Mild Dementia, (3) Mild Dementia, and (4) Moderate Dementia. In addition, this study compares the classification performance and computational efficiency of several deep learning architectures, including a traditional CNN (VGG16), a modern model (EfficientNet-B0), and a hybrid Fuzzy Convolutional Inference Engine (FCIE) that integrates CNN-based feature extraction with fuzzy logic reasoning. The dataset used consists of normalized and augmented Alzheimer's MRI images, and each model was trained and validated using a 70%:15%:15% split for training, validation, and testing. Experimental results show that the Mamdani and Sugeno FIS models achieve validation accuracies of about 32% and 35%, respectively, which highlights the limitations of pure texture-based features in capturing complex classification patterns. In contrast, VGG16 and EfficientNet-B0 produced validation accuracies of 82.81% and 85.22%, respectively, with AUC values of 0.95 and 0.96, respectively. However, when both schemes were combined into a hybrid model FCIE achieved the highest validation accuracy of 98.03% and AUC of 0.99. Comparative analysis of metrics, including precision, recall, F1-score, AUC, and training duration, shows a clear trade-off between accuracy and computational efficiency. This study recommends the FCIE model for clinical applications requiring high diagnostic accuracy, while EfficientNet-B0 is suggested for medical environments with moderate GPU resource constraints.