To deal with such a predicate hierarchy, which is more intricate than a sort-hierarchy, Nitta et al. implemented a typed (sorted) logic programming language extended to include a hierarchy of verbal concepts (corresponding to predicates).
In a many-sorted logic distinct variables are used for the different kinds of objects, perhaps m, n,… for numbers and p, q,… for people.
However, Carnap and others who advocated a many-sorted logic for reasons that were more ontological than pragmatic introduced different variables only for what they believed were metaphysically distinct kinds of entities, e.g., numbers, physical objects, sense data.
A related use of many-sorted logic is to provide a kind of translation from second-order logic into a first-order logic.
Our order-sorted logic contains types (rigid sorts), non-rigid sorts, and unary predicates that distinctly express essential sorts, non-essential sorts, and non-sortal properties.
To accommodate the new styles of variables in the model theory for many-sorted logic, we may add a domain for each style of variable.
The games described in this section adapt almost trivially to many-sorted logic: for example the quantifier ∀xσ, where xσ is a variable of sort σ, is an instruction for player ∀ to choose an element of sort σ.
However, the inference system lacks a theoretical foundation because its hierarchical expressions exceed the formalization of order-sorted logic.
Swinging types provide an integrated framework for specifying software on the basis of many-sorted logic in terms of "static" functions and relations as well as "dynamic" transition systems.
Order-sorted logic has been formalized as first-order logic with sorted terms where sorts are ordered to build a hierarchy (called a sort-hierarchy).
