Enabling natural communication through brain-computer interfaces (BCIs) remains one of the most profound challenges in neuroscience and neurotechnology. While existing frameworks offer partial solutions, they are constrained by oversimplified semantic representations and a lack of interpretability. To overcome these limitations, we introduce Semantic Intent Decoding (SID), a novel framework that translates neural activity into natural language by modeling meaning as a flexible set of compositional semantic units. SID is built on three core principles: semantic compositionality, continuity and expandability of semantic space, and fidelity in reconstruction. We present BrainMosaic, a deep learning architecture implementing SID. BrainMosaic decodes multiple semantic units from EEG/SEEG signals using set matching and then reconstructs coherent sentences through semantic-guided reconstruction. This approach moves beyond traditional pipelines that rely on fixed-class classification or unconstrained generation, enabling a more interpretable and expressive communication paradigm. Extensive experiments on multilingual EEG and clinical SEEG datasets demonstrate that SID and BrainMosaic offer substantial advantages over existing frameworks, paving the way for natural and effective BCI-mediated communication.