Despite the remarkable progress in neural models, their ability to generalize, a cornerstone for applications such as logical reasoning, remains a critical challenge. We delineate two fundamental aspects of this ability: compositionality, the capacity to abstract atomic logical rules underlying complex inferences, and recursiveness, the aptitude to build intricate representations through iterative application of inference rules. In the literature, these two aspects are often conflated under the umbrella term of generalization. To sharpen this distinction, we investigate the logical generalization capabilities of LLMs using the syllogistic fragment as a benchmark for natural language reasoning. We extend classical syllogistic forms to construct more complex structures, yielding a foundational yet expressive subset of formal logic that supports controlled evaluation of essential reasoning abilities. Our findings on this non-trivial benchmark show that, while LLMs demonstrate reasonable proficiency in recursiveness, they struggle with compositionality. This disparity is not uniform, as a more detailed analysis reveals substantial variability in generalization performance across individual syllogistic types, ranging from near-perfect accuracy to significantly lower performance. To overcome these limitations and establish a reliable logical prover, we propose a hybrid architecture integrating symbolic reasoning with neural computation. This synergistic interaction enables robust and efficient inference, neural components accelerate processing, while symbolic reasoning guarantees completeness. Our experiments further show that high efficiency is preserved even when using relatively small neural components. Overall, our analysis provides both a rationale for hybrid neuro-symbolic approaches and evidence of their potential to address key generalization barriers in neural reasoning systems.