Large Vision-Language Models (LVLMs) have achieved impressive results across various multi-modal tasks. However, hallucinations, i.e., the models generating counterfactual responses, remain a challenge. Though recent studies have attempted to alleviate object perception hallucinations, they focus on the models' response generation, overlooking the task question itself. This paper discusses the vulnerability of LVLMs in solving counterfactual presupposition questions (CPQs), where the models are prone to accept the presuppositions of counterfactual objects and produce severe hallucinatory responses. To this end, we introduce “Antidote,” a unified, synthetic data-driven post-training framework for mitigating both types of hallucination above. It leverages synthetic data to incorporate factual priors into questions to achieve self-correction and decouple the mitigation process into a preference optimization problem. Furthermore, we construct “CP-Bench,” a novel benchmark to evaluate LVLMs' ability to correctly handle CPQs and produce factual responses. Applied to the LLaVA series, Antidote can simultaneously enhance performance on CP-Bench by over 50%, POPE by 1.8-3.3%, and CHAIR & SHR by 30-50%, all without relying on external supervision from stronger LVLMs or human feedback and introducing noticeable catastrophic forgetting issues.