Good question! In this context, we're relating a system of proofs to the properties of a desired model. If you start from a sufficiently expressive model and try to design a proof system where valid proofs correspond to true properties, then you will either miss some properties (an unproveable truth) or you will have too many proofs (proveable falsehoods).

Without biasing for a particular model, we could simply say that every sufficiently expressive proof system admits either zero models or more than one model.