In recent years, the pursuit of social sustainability has intensified, especially in urban development. This has resulted in numerous definitions of 'social sustainability' from stakeholders including researchers, non-governmental organizations (NGOs) and both private and public sector entities. These definitions vary widely, encompassing values such as well-being, good governance, equality, trust, diversity, and accessibility. Despite extensive contributions, a universally accepted definition remains elusive, indicating the need for further conceptual work. The primary issue is the lack of explicit criteria for adequacy, leading to a stalemate in the literature. To address this issue, de Fine Licht and Folland, in 2019, proposed a framework with nine conditions of adequacy for evaluating definitions. Building on this framework, our paper systematically analyses and critiques existing definitions and proposes new, formal definitions for both 'sustainability' and 'social sustainability'. Our approach identifies central challenges in existing definitions, demonstrates their inadequacy and presents an enhanced formal definition that avoids these pitfalls. Although our definitions do not specify particular values, they offer substantial results relevant to the sustainability discourse. The novelty of our work lies in our formal definition, engagement with the conditions of adequacy and systematic critique of existing definitions.