Audio-language models have shown promising results in various sound understanding tasks, yet they remain limited in their ability to reason over the fine-grained semantics of sound. In this paper, we present AudSemThinker, a model whose reasoning is structured around a framework of auditory semantics inspired by human cognition. To support this, we introduce AudSem, a novel dataset specifically curated for semantic descriptor reasoning in audio-language models. AudSem addresses the persistent challenge of data contamination in zero-shot evaluations by providing a carefully filtered collection of audio samples paired with captions generated through a robust multi-stage pipeline. Our experiments demonstrate that AudSemThinker outperforms state-of-the-art models across multiple training settings, highlighting its strength in semantic audio reasoning. Both AudSemThinker and the AudSem dataset are released publicly.