While large language models have demonstrated impressive reasoning abilities, their extension to the audio modality, particularly within large audio-language models (LALMs), remains underexplored. Addressing this gap requires a systematic approach that involves a capable base model, high-quality reasoning-oriented audio data, and effective training algorithms. In this work, we present a comprehensive solution for audio logical reasoning (ALR) tasks: we introduce SoundMind, a dataset of 6,446 audio-text annotated samples specifically curated to support complex reasoning. Building on this resource, we propose SoundMind-RL, a rule-based reinforcement learning (RL) algorithm designed to equip audio-language models with robust audio-text reasoning capabilities. By fine-tuning Qwen2.5-Omni-7B on the proposed SoundMind dataset using SoundMind-RL, we achieve strong and consistent improvements over state-of-the-art baselines on the SoundMind benchmark. This work highlights the benefit of combining high-quality, reasoning-focused datasets with specialized RL techniques, and contributes to advancing auditory intelligence in language models. The code and dataset introduced in this work are publicly available at https://github.com/xid32/SoundMind.