Editing Ontological commitment (section)

== Background ==
The sentence “Napoleon is one of my ancestors” apparently commits us only to the existence of two individuals (i.e., [[Napoleon]] and the speaker) and a line of ancestry between them. The fact that no other people or objects are mentioned seems to limit the “commitment” of the sentence. However, it is well known that sentences of this kind cannot be interpreted in [[first-order logic]], where individual variables stand for individual things. Instead, they must be represented in some [[second-order logic|second-order]] form. In ordinary language, such second-order forms use either grammatical plurals or terms such as “set of” or “group of”.

For example, the sentence involving Napoleon can be rewritten as “any group of people that includes me and the parents of each person in the group must also include Napoleon,” which is easily interpreted as a statement in second-order logic (one would naturally start by assigning a name, such as ''G'', to the group of people under consideration). Formally, collective noun forms such as “a group of people” are represented by [[second-order variable]]s, or by [[first-order variable]]s standing for [[set (mathematics)|sets]] (which are [[well-defined]] objects in [[mathematics]] and logic). Since these variables do not stand for individual objects, it seems we are “ontologically committed” to entities other than individuals — sets, classes, and so on. As [[Willard Van Orman Quine|Quine]] puts it,

<blockquote>the general adoption of class variables of quantification ushers in a theory whose laws were not in general expressible in the antecedent levels of logic. The price paid for this increased power is [[ontological]]: objects of a special and abstract kind, [[viz.]] classes, are now presupposed. Formally it is precisely in allowing [[Quantification (logic)|quantification]] over class variables α, β, etc., that we assume a range of values for these variables to refer to. To be assumed as an entity is to be assumed as a value of a variable. (''Methods of Logic'', 1950, p. 228)</blockquote>

Another statement about individuals that appears “ontologically innocent” is the well-known [[Geach–Kaplan sentence]]: ''Some critics admire only one another.''