Carnap’s chapters and the short paper by Stevens cover pretty much the same ground from slightly different perspectives. I’m asking you to read both partly in hopes that the two different views of the same ideas will help make them clearer and partly because Stevens covers a bit of material that appears in Carnap only by way of his discussion in ch. 7, which we will not read.

Both Carnap and Stevens are concerned with measurement. Stevens quotes an unnamed writer as saying that this is “the assignment of numerals to things so as to represent facts and conventions about them” (p. 680). Much of science is a product of an interaction of fact and convention. Measurement stands out as a feature of our thinking has served for millennia as a paradigm example of this interaction, and the account of measurement that Carnap and Stevens describe tries to provide a clear way of distinguishing its factual and conventional components. Much of the notes below are designed to help you negotiate the difference between Carnap and Stevens and the technical aspects of their discussions, but the final paragraph suggests some broader issues and I hope our discussion will focus on those.

The vocabulary that Carnap and Stevens use is somewhat different. The following table gives the correspondence between a few of the basic ones in case there is any doubt how they match up.

The correspondence between classificatory concepts and nominal scales may be the least obvious, but Stevens intends nominal “scales” to be numerical labelings of either individual objects or collections of objects, and collections of objects is just what classificatory concepts provide. Moreover, a classification of objects is associated with an equivalence relation (Carnap defines this idea on p. 55) that holds between any two objects that fall in the same class, and this equivalence relation can be understood as the “determination of equality” that Stevens gives as the operation required for a nominal scale.

Later Stevens distinguished a fifth sort of scale that he called “absolute.” It is absolute in the sense that it does not depend on a unit of measure, and a quantity associated with such a scale is “dimensionless” in the sense that it is just a number and not so many feet or pounds or whatever. Probability has been suggested as an example of this sort of quantity.

The ideas discussed in the third and fourth columns of the table in Stevens are somewhat less important for our purposes, but it is still worth thinking about them. He explains the fourth column with the example of the median and mean (i.e., average), and that example is enough for our purposes. The idea of “group” that appears in the third column is a kind of mathematical structure. The term is used in this context simply because changes in units of measurement and the other changes indicated by the formulas in that column can be combined repeatedly to form a series of changes and are reversible by other changes of the same sort. A “one-to-one substitution” replaces numbers in a way that never replaces two different numbers by the same one but may not preserve the order of the numbers assigned while a “monotonic increasing function” will also preserve the order (so its graph will always move upwards).

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Most of the discussion in Carnap and Stevens is devoted to discussing the distinctions among concepts or scales and the significance of the specific differences among them, so you may need to make a special effort to stand back and think about the broader significance of these ideas. First, think about Carnap’s view that moving down the list towards quantitative concepts is good when it is possible. Why might someone object to that, and what could be said on the other side? Also, think about the distinction between factual and conventional components of our knowledge more generally. What does it mean to say something is conventional as opposed to factual? (That’s not an easy question to answer.) The third and fourth columns in Stevens are associated with a general claim that a statement is factually significant only to the extent that it remains invariant when we change conventional factors (e.g., change a unit of measure). Try to think of other examples of this point. Is it something you would agree with? If it is right, does it mean we should strive to avoid conventional factors in making scientific claims?