Different rule semantics have been successively defined in many contexts such as functional dependencies in databases or association rules in data mining to mention a few. Given the well-known Armstrong's axioms, we are interested in defining on tabular datasets a class of rule semantics for which Armstrong's axioms are sound and complete, so-called well-formed semantics. The main contribution of this paper is to show that an equivalence does exist between some syntactic restrictions on the natural definition of a given semantics and the fact that this semantics is well-formed. From a practical point of view, this equivalence allows to prove easily whether or not a new semantics is well-formed. We also point out the relationship between our generic definition of rule satisfaction and the underlying data mining problem, i.e. given a well-formed semantics and a tabular dataset, discover a cover of rules satisfied in this dataset. This work takes its roots from a bioinformatics application, the discovery of gene regulatory networks from gene expression data.