A comparison is an operation involving three entities:

Primum and secundum comparationis

The primum and secundum comparationis may be any kind of entity. However, they are objects of the same kind. In its proper interpretation, the common saying according to which one does not compare apples with pears does not actually exclude such a comparison. It makes sense if one compares them in their capacity of being fruits. What the saying mocks is comparing apples with pears as if they were pears. Comparison which is faulty in this respect is not at all rare in science; examples are further below.

The secundum comparationis has occasionally been called ‘standard of comparison’. This is a misleading term. The primum and the secundum comparationis are objects of the same ontological type and have the same methodological status. The secundum comparationis does not represent a standard any more than the primum. Whether we say Linda is taller than Irvin or Irvin is shorter than Linda has nothing to do with their status in a (size) comparison, but rather depends on their position in the information structure of the current discourse.

For the same reason, the terms primum and secundum comparationis are based on their syntactic function in a comparative construction of some language (in the simplest case, the primum comparationis is the syntactic subject). Likewise, the above characterization of the primum comparationis as an “entity to be assessed” refers to such a construction. It is true that in a comparison like Linda is taller than Irvin, we may take Irvin's measure for granted and assess Linda's size with respect to Irvin's size. Likewise in comparative studies, it is possible to characterize one language by contrasting it with other languages. However, equally often two or more objects of a kind are compared with each other in a symmetric way without any intention to characterize one of them in contrast with the others. Instead, the purpose may be to use the tertium comparationis to establish an order of some epistemic interest in the field of such objects.

The tertium comparationis

A basic requirement is that the tertium comparationis be clearly distinct and independent from the objects compared. It makes no sense to compare the primum comparationis with respect to a generic property that only the secundum comparationis possesses. For instance, the elephant is bulkier than the field makes no sense because something must possess the property of being a three-dimensional object in order to be able to be bulky.

Unlike the primum and secundum comparationis, the tertium comparationis is not an entity taken absolutely. The tertium comparationis is more abstract than the primum and secundum comparationis; it is something that applies to the primum and to the secundum comparationis in like fashion. This is to say that the tertium comparationis is a generic property. This is a property which takes on different values (which can be named ‘specific properties’). A generic property for which objects take values by which they can be measured or at least be assessed is a parameter. The values may vary gradually over the parameter. When we say that Linda's hair is longer than Irvin's, we use length as a tertium comparationis, implicitly assign both Linda's and Irvin's hair a value on that parameter and note that the value of Linda's hair is higher than the value of Irvin's hair on that parameter. Length is a parameter and, thus, a generic property, while longness and shortness are specific properties. In such a case, the parameter provides a measure, and the values may be numerical values.

It is also possible to compare two entities with regard to the question whether they bear a certain relation to some third entity. For instance, we may compare Linda and Irvin with respect to their marital status, finding that Linda is married while Irvin is a bachelor. Our criterion for this judgment is the fact that Linda has a spouse while Irvin lacks one. However, the spouse is not the tertium comparationis. The tertium comparationis is the binary property of having or lacking a spouse, where the spouse is, of course, non-specific.

In the nineteenth century, typological comparison of languages used the properties of Indo-European languages as a tertium comparationis and found that all other languages fell short of this standard. Disregarding chauvinistic ideologies orienting this kind of enterprise, the methodological mistake consisted in using a specific property of the secundum comparationis as the tertium comparationis instead of identifying the parameter of which this specific property was but a possible value. In the last resort, it consisted in insufficient abstraction.

Likewise in the twentieth century, people compared pronominal indexing on verbs with case marking on nouns, regarding the former as a kind of case marking. The presupposition or the result of such a comparison is a false understanding of the function of pronominal indexing. What one can fruitfully do is compare these two techniques under the common denominator of providing information on relations between verbs and their dependents – again, something more abstract.