The Belief-Desire-Intention (BDI) model is a cornerstone for representing rational agency in artificial intelligence and cognitive sciences. Yet, its integration into structured, semantically interoperable knowledge representations remains limited. This paper presents a formal BDI Ontology, conceived as a modular Ontology Design Pattern (ODP) that captures the cognitive architecture of agents through beliefs, desires, intentions, and their dynamic interrelations. The ontology ensures semantic precision and reusability by aligning with foundational ontologies and best practices in modular design. Two complementary lines of experimentation demonstrate its applicability: (i) coupling the ontology with Large Language Models (LLMs) via Logic Augmented Generation (LAG) to assess the contribution of ontological grounding to inferential coherence and consistency; and (ii) integrating the ontology within the Semas reasoning platform, which implements the Triples-to-Beliefs-to-Triples (T2B2T) paradigm, enabling a bidirectional flow between RDF triples and agent mental states. Together, these experiments illustrate how the BDI Ontology acts as both a conceptual and operational bridge between declarative and procedural intelligence, paving the way for cognitively grounded, explainable, and semantically interoperable multi-agent and neuro-symbolic systems operating within the Web of Data.