Remarks on remarks on many-sorted equational logic

data types may be challenged in at least three different ways: the need to accomodate empty carriers may be doubted; the correctness of our results on deduction may be questioned (see [Ehrig & Mahr 85], following [Loeckx & Mahr 85]); and, more fundamentally, the convenience of many-sorted algebra for applications to abstract data types may be doubted (e.g., [Goguen 78, Goguen & Meseguer 86]). We address each issue in a section below. 2 M u c h A d o a b o u t N o t h i n g We first address emptiness, following [Mesegner & Goguen 85b]. There are two basic approaches to avoiding the problems of empty carriers: (1) restricting the class of allowed signatures (for example, so that empty carriers are disallowed or restricted); or (2) directly restricting the class of models (e.g., requiring non-empty carriers, as in standard model theory}. However, neither approach is satisfactory. Restricting signatures so as to limit the possibility of algebras having empty carriers, unfortunately excludes much that is important for applications to parameterized abstract data types. Perhaps the least restrictive restriction is the "sensible" signatures of [Huet & Oppen 80]: first, define a sort s to be s t r ic t in a signature ~E iff either there is a constant symbol of sort s in ~ or else there is a function symbol in ~ of target sort s with all its source sorts strict; then E is sensible iff every non-constant function symbol with strict target sort has only strict source sorts. Now let us consider the theory PREORDER of preordered ordered sets, with the ordering relation represented as a Boolean valued-function on the sort E l t of elements; this theory might be used, for example, used to specify the interface of a parameterized sorting algorithm, stating that it is guaranteed to work correctly only if the elements to be sorted have an ordering relation which actually satisfies the transitive and reflexive laws. The signature of PREORDER is not sensible, since the sort Bool is strict while the sort E l i is not. There are also many other important interface theories which are not sensible; thus, the restriction to sensible signatures is not sensible in the study of parameterized abstract data types. If, following the second approach, one restricts models to those algebras having no empty