Property
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Definition
Quantitative properties, as opposed to qualitative ones, are defined in a way which makes possible to measure them and to operate with the results, i.e. to use mathematical methods. Qualitative properties are defined on a so-called nominal, or categorial scales, which means that the only possible operation is to compare them with respect to identity or non-identity. Every day color terms are an example of qualitative values, they constitute the qualitative property color. An object can be compared to other objects with respect to color only by qualitative means, i.e. by determining whether both objects share their color or not.
Quantitative properties can be defined on three different scales. The first one is the ordinal or comparative scale. An exaple of such a property is the way school-boys may determine which of them is the strongest one, the second strongest etc. The values of such a scale allows to arrange objects according to the corresponding order but they do not allow to give more precise information. They allow us to determine that an object possesses more or less of a given property than another one, or the same amount of it – formally: P(A) > P(B), P(A) = P(B) or P(A) < P(B). Applying this kind of concept yields a higher degree of order, viz. a ranking of the objects with respect to a given property. A linguistic example is the grammatical acceptability of sentences. The highest degree of order is achieved with the help of metrical concepts, which are needed if the difference between the amounts of a given property object a and b possess plays a role. In this case, the values of the property are mapped to the elements of an appropriate set of numbers, i.e. a set of numbers in which the relations between these numbers correspond to the relations between the values of the properties of the objects. In this way, specific operations such as subtraction correspond to specific differences or distances in the properties between the objects – formally: P(A) – P(B) = d. This enables the researcher to establish an arbitrarily fine conceptual grid within his field of study. Concepts which allow determining distances or similarities between objects are called interval-scale concepts. If another feature is added, viz. a fixed point of reference (e.g. an absolute zero) ratio-scaled concepts are obtained, which allow the operation of multiplication and division, formally: P(A) = aP(B) + d. Only the latter scale enables to formulate how many times object A has more than B of a property.
Often, quantitative concepts are introduced indirectly. Quantification can start from established (or potential) quantitative concepts and adding the needed features. One has to make sure that the conceptual scale is chosen properly, i.e. the concepts must be formed according to the mathematical operations which correspond to the properties and relations of the objects. The polysemy of words may serve as a linguistic example of an indirectly introduced quantitative concept. Polysemy is originally a qualitative concept in traditional linguistics which identifies or differentiates words with respect to lexical ambiguity. Taking this as a starting point, a quantitative variant of this concept can easily be created: It may be defined as the number of meanings of a linguistic expression; the values admitted are cardinal numbers in the interval [1,∞), i.e. the smallest possible value is 1 whereas an upper limit cannot be specified. This is a well-defined ratio-scale concept: using basic mathematical operations, differences in polysemy between words can be expressed (e.g. word x has three meanings more than word y) and even the ratio between the polysemy of two words values can be specified (word x has twice as many meanings as word y), since we have a fixed reference point: the minimum polysemy 1. Only by means of concepts on higher scales, i.e. quantitative ones is it possible to pose deeper-reaching questions and even to make corresponding observations. Thus, without our quantitative concept of polysemy no-one could even notice that there is a lawful relation between the number of meanings of a word and its length).