 A grade function \(G_{i}\) maps the set \(X\) of alternatives into some given grade language \(L\). Intuitively, \(G_{i}(x)\) is \(i\)’s grade for alternative \(x\). A grade function \(G_{i}\) contains as much information as some weak ordering \(R_{i}\) of the alternatives. This can be seen by putting \(xR_{i}y\) if \(G_{i}(x) \ge G_{i}(y)\): \(i\) weakly prefers \(x\) to \(y\) if \(i\)’s grade for \(x\) is at least as high as \(i\)’s grade for \(y\). In general \(G_{i}\) contains more information than the corresponding ordering because we cannot always go in reverse: different grade functions correspond to the same ordering. For instance, one way to convey that \(A\) is preferable to \(B\) is to say that whereas \(A\) is good, \(B\) is merely acceptable; another is to say that whereas \(A\) is good, \(B\) is terrible, but this doesn’t come to the same thing because acceptable and terrible don’t. The additional information in grades is in general not cardinal information because for instance the expressions  good, acceptable and terrible, ordinary adjectives of English, do not carry cardinal information about how good \(A\) and \(B\) are.