All and Only

This paper is a note on a very simple and elementary fact having to do with reasoning about relations. The kinds of reasoning that we have in mind come from two different areas: extended syllogistic logics and description logics. In extended syllogistic logic, one wants to reason about terms formed from binary relations and terms. We illustrate with some examples from the recent paper van Rooij [11]. He symbolizes “Some man loves every woman” (with the wide-scope reading: some man is such that he loves every woman) as Mi(WaL). In this example, M and W are the terms for “man” and “woman”, and L is the binary relation “loves.” The i and a are from traditional logic: i for “some” and a for “all.” So the term WaL is intended to denote the set of things who love all women. And then our sentence Mi(WaL) would be true if some man belongs to the denotation of WaL. For us, the crucial point is that WaL has the semantics that we said it has. This kind of semantics is the source of later work, including presentations of logical calculi for the relational syllogistic in works such as [10]. We contrast this with the situation in description logics (e.g., [1]). In that field, we have atomic concepts such as W for woman, and binary roles such as L for loves. And then ∀R.W would denote the set of objects with the property that everything which they love is a woman. To compare and contrast, and to anticipate notation which we shall use later in the paper, WaL is love all women, and ∀R.W is love only women.