For thinking about numeric types, measurement, truth, social structures ...
This is the mathematical background to Alan Page Fiske's book Structures of Social Life
Fiske argues that all social life is composed of patterns of interaction that are based on four types of scale: categorical, ordinal, interval, and ratio. The scales provide ways of perceiving, and thereby of organizing social interaction.
The four scales are not mere manifestations of a single culture, but are different primary mathematical structures. They are different axiomatically. They are transcultural.
In 1944, Louis Guttman pointed out that all forms of measurement belong to one of four types of scale: categorical, ordinal, interval, and ratio. (Actually, forms of measurement can belong to more than four, but people conflate them into these four.)
The four scales are different primary mathematical structures: equivalence relation, linear ordering, ordered Abelian group, and Archimedean ordered field. They are different axiomatically, but all serve as means of measurement.
Thus you can say this stone weights twice as much as that stone (ratio scale), but you cannot meaningfully say this Fahrenheit temperature (interval scale) is twice that temperature since the Fahrenheit scale has an arbitrary zero. But you can add ten F. degrees to a F. temperature. Similarly, you can say that a captain in the Army is superior (ordinal scale) to a lieutenant but you cannot say by how much he is superior (and indeed, the `how-muchness' is irrelevant). Likewise, you can say that topaz is harder than quartz (Moh's ordinal scale of hardness for minerals) but not how many degrees harder. Finally, you can say that one animal is a cat and another one is a dog.
Much progress in science comes from changing the type of scale used in a measurement: from `it is cold outside' to `it is colder today than yesterday' to `it is 10 F. degrees colder today than yesterday' to `the thermal energy content of this piece of iron is 0.6% less than it was yesterday'.
As for truth: if you are using a categorical scale, you may say that a proposition belongs to the category of truthful propositions or the category of false propositions. If you use such a scale, you are not saying how much truth there is in a proposition, only that it is true, not false. Much logic is based on there being only two categories, true and false; it makes the mathematics simpler. The various fuzzy logics are a formal attempt to add interval or ratio scales to logic.
Or you can say that this first proposition is more credible than that second proposition, and that second proposition is more credible than a third. This is an ordinal scaling. In a court case, a jury may have to judge whether one person's testimony is more credible than another's (ordinal scale) so as eventually to place the defendant in one of the categories `guilty' or `not guilty'.
In artificial intelligence programs, numbers may be used to indicate the quality of the evidence for a proposition. Even though the numbers appear to suggest a familiar ratio scale, as used in measuring weight or density, the computer program often limits operations on the numbers to a more restrictive set of axioms than that used by rational numbers.
Here is a table:
Scales of Measurement ==================== Scale Basic Empirical Permissible Statistics Examples Operations (invariantive) Name of mathematical structure -------------------------------------------------------------------------- Categorical Determination of Number of cases Assign model numbers (or Nominal) equality Mode Specify species of Contingency animal Equivalence correlations relation Ordinal Determination of Median Hardness of minerals greater or less Percentiles Quality of leather, Order correlation lumber, wool Linear ordering (type O) Pleasantness of odor Interval Determination of Mean Temperature equality of Standard deviation (Fahrenheit and intervals or Order correlation Celsius) differences (type I) Calendar dates Product-moment Ordered Abelian correlation group Ratio Determination of Geometric mean Length, weight, density, equality of ratios Coefficient of resistance variation Loudness scale (sones) Archimedean Decibel ordered field transformations(For a little more detail, see Groups and Rings.)
From: S. S. Stevens, 1951, _Mathematics, Measurement, and Psychophysics_, in Handbook of Experimental Psychology_, S. S. Stevens, Ed., NY: Wiley See also: Louis Guttman, 1944, _A Basis for Scaling Qualitative Data_, American Sociological Review 9:139-150 Patrick Suppes, 1957, _Introduction to Logic_, NY: Van Nostrand S. S. Stevens, 1958, _Measurement and Man_, Science 127:383-389 Louis Narens and R. Duncan Luce, 1986, _Measurement: the Theory of Numerical Assignments_, Psychological Bulletin, Vol. 99 No. 2, p. 166-180 Alan Page Fiske, 1991, _Structures of Social Life_, NY: Macmillan
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