The FAIR Principles aim to make data and knowledge Findable, Accessible, Interoperable, and Reusable, yet current digital infrastructures often lack a unifying semantic framework that bridges human cognition and machine-actionability. In this paper, we introduce the Grammar of FAIR: a granular and modular architecture for FAIR semantics built on the concept of semantic units. Semantic units, comprising atomic statement units and composite compound units, implement the principle of semantic modularisation, decomposing data and knowledge into independently identifiable, semantically meaningful, and machine-actionable units. A central metaphor guiding our approach is the analogy between the hierarchy of level of organisation in biological systems and the hierarchy of levels of organisation in information systems: both are structured by granular building blocks that mediate across multiple perspectives while preserving functional unity. Drawing further inspiration from concept formation and natural language grammar, we show how these building blocks map to FAIR Digitial Objects (FDOs), enabling format-agnostic semantic transitivity from natural language token models to schema-based representations. This dual biological-linguistic analogy provides a semantics-first foundation for evolving cross-ecosystem infrastructures, paving the way for the Internet of FAIR Data and Services (IFDS) and a future of modular, AI-ready, and citation-granular scholarly communication.