ABSTRACT
The “empirical meaningfulness” analysis in theory of measurement imposes a priori restrictions on statements involving a given set of quantities, by striking down as “empirically meaningless” those of logically possible statements whose truth value (true or false) is not invariant under mutual substitutions of “admissible” measurements of the quantities involved. However, any logically unambiguous statement that is “empirically meaningless” by this invariance criterion can be equivalently reformulated to become, by the same criterion, “empirically meaningful.” This is achieved by explicating in the statement all its measurement-dependent constants, whose numerical values covary with choices of measurements within a specified class. Provided that the basic, nonderivable laws of a given area (such as mechanics or psychophysics) can be formulated in some specific measurements (such as mass in grams or in absolute threshold units), a simple algorithm described in this chapter determines the set of measurement-dependent constants that ensure the invariance of these basic laws under any specified class of transformations of these measurements: the choice of this class, and thereby of the measurement-dependent constants, is subject to no substantive constraints. The only context in which the invariance considerations may be restrictive is that of deciding whether a given statement is logically derivable from a given list of basic laws: if it is, then one should be able to make it invariant under the same class of transformations with the aid of the same set of measurement-dependent constants. Dimensional analysis in physics, for instance, can determine that a statement is not derivable from a given set of physical laws (such as the gravitation law and the second law of motion) by demonstrating that it cannot be made dimensionally homogeneous (invariant under scaling transformations) if one only utilizes the dimensional constants that have been explicated in these basic laws themselves, when presenting 114them in a dimensionally homogeneous form. Outside the context of derivability, however, the requirement of dimensional homogeneity does not restrict the class of possible laws of physics, as their dimensional homogeneity can always be achieved by an appropriate choice of dimensional constants.
