The recent development of fact verification systems with natural logic has enhanced their explainability by aligning claims with evidence through set-theoretic operators, providing faithful justifications. Despite these advancements, such systems often rely on a large amount of training data annotated with natural logic. To address this issue, we propose a zero-shot method that utilizes the generalization capabilities of instruction-tuned large language models. To comprehensively assess the zero-shot capabilities of our method and other fact verification systems, we evaluate all models on both artificial and real-world claims, including multilingual datasets. We also compare our method against other fact verification systems in two setups. First, in the zero-shot generalization setup, we demonstrate that our approach outperforms other systems that were not specifically trained on natural logic data, achieving an average accuracy improvement of 8.96 points over the best-performing baseline. Second, in the zero-shot transfer setup, we show that current systems trained on natural logic data do not generalize well to other domains, and our method outperforms these systems across all datasets with real-world claims.