In real life, data are often of poor quality as a result, for instance, of uncertainty, mismeasurements, missing values or bad inputs. This issue hampers an implicit yet crucial operation of every database management system: equality testing. Indeed, equality is, in the end, a context-dependent operation with a plethora of interpretations. In practice, the treatment of different types of equality is left to programmers, who have to struggle with those interpretations in their code. We propose a new lattice-based declarative framework to address this problem. It allows specification of numerous semantics for equality at a high level of abstraction. To go beyond tuple equality, we study functional dependencies (FDs) in the light of our framework. First, we define abstract FDs, generalizing classical FDs. These lead to the consideration of particular interpretations of equality: realities. Building upon realities and possible/certain answers, we introduce possible/certain FDs and give some related complexity results.