98
is the provision of a meaningful interpretation of the ratio of individual properties of
the same kind (Rossi & Crenna, 2013): this can be granted by the empirical additivity of some quantities, and may be derived for quantities which are functions of
additive quantities, but in principle could also be obtained in other ways. This is the
strategy, in particular, that led to the development of the so-called additive conjoint
measurement (Luce & Tukey, 1964), and is arguably also at the core of the Rasch
approach to measurement (Rasch, 1960; see, e.g., Borsboom, 2005: ch. 4; Wilson,
2013). In our structural perspective this path remains in position β, and therefore
could be characterized as a constructive approach for embedding nonphysical properties into the Galilean conception of measurement.
The other path was triggered by the emphasis on the representational role of
measurement, thus with an emphasis on numerical assignment rather than determination (Mari, 1997). By conceiving of numbers in the Euclidean sense of ratios of
quantities, Campbell’s claim that “measurement is the process of assigning numbers to represent qualities” (1920: p. 267, emphasis added) was still conservatively
bound to the algebraic condition that only properties that admit of ratio- or intervallevel representation are measurable. But this was also the starting point for another
interpretation, according to which the important point is not representation by
means of numbers, but representation as such. As discussed in Sect. 4.2.3 this standpoint was developed in particular by Stevens, who accepted measurement as representation by means of informational entities (which he called “numerals”), instead
of numbers only, and introduced a condition of consistency in the assignment that
he called “permissibility” (closely related to what was referred to as “meaningfulness” by Narens, 2002): the relations observed among measured properties must
also apply among the assigned values, and—most importantly—only the informational relations corresponding to empirical relations should be exploited in inference and computation.
24
Such a removal of both experimental conditions on the
process and algebraic conditions on the processed properties is epitomized by
Stevens’ previously quoted assertion that “measurement is the assignment of numerals to objects or events according to rule” (1959). On this basis, representational
theories of measurement (Krantz et al., 1971) provided a mathematization (and in
fact an axiomatization) of Stevens’ standpoint, where the rule of assignment is
required to be a morphism and no mathematical conditions are imposed on properties for their being measurable. Hence, this appears to be a case of position γ, as
depicted in Fig. 4.5.
4.4.2 Towards a different perspective?
The transition from the Galilean (β) position to the representational (γ) position
provided a context for expanding measurability to nonphysical properties, obtained
at the price of removing experimental conditions on processes claimed to be mea24 See Sect. 6.5.1 for an analysis of this condition and of the critiques it has received.
4 Philosophical perspectives on measurement
is the provision of a meaningful interpretation of the ratio of individual properties of
the same kind (Rossi & Crenna, 2013): this can be granted by the empirical additivity of some quantities, and may be derived for quantities which are functions of
additive quantities, but in principle could also be obtained in other ways. This is the
strategy, in particular, that led to the development of the so-called additive conjoint
measurement (Luce & Tukey, 1964), and is arguably also at the core of the Rasch
approach to measurement (Rasch, 1960; see, e.g., Borsboom, 2005: ch. 4; Wilson,
2013). In our structural perspective this path remains in position β, and therefore
could be characterized as a constructive approach for embedding nonphysical properties into the Galilean conception of measurement.
The other path was triggered by the emphasis on the representational role of
measurement, thus with an emphasis on numerical assignment rather than determination (Mari, 1997). By conceiving of numbers in the Euclidean sense of ratios of
quantities, Campbell’s claim that “measurement is the process of assigning numbers to represent qualities” (1920: p. 267, emphasis added) was still conservatively
bound to the algebraic condition that only properties that admit of ratio- or intervallevel representation are measurable. But this was also the starting point for another
interpretation, according to which the important point is not representation by
means of numbers, but representation as such. As discussed in Sect. 4.2.3 this standpoint was developed in particular by Stevens, who accepted measurement as representation by means of informational entities (which he called “numerals”), instead
of numbers only, and introduced a condition of consistency in the assignment that
he called “permissibility” (closely related to what was referred to as “meaningfulness” by Narens, 2002): the relations observed among measured properties must
also apply among the assigned values, and—most importantly—only the informational relations corresponding to empirical relations should be exploited in inference and computation.
24
Such a removal of both experimental conditions on the
process and algebraic conditions on the processed properties is epitomized by
Stevens’ previously quoted assertion that “measurement is the assignment of numerals to objects or events according to rule” (1959). On this basis, representational
theories of measurement (Krantz et al., 1971) provided a mathematization (and in
fact an axiomatization) of Stevens’ standpoint, where the rule of assignment is
required to be a morphism and no mathematical conditions are imposed on properties for their being measurable. Hence, this appears to be a case of position γ, as
depicted in Fig. 4.5.
4.4.2 Towards a different perspective?
The transition from the Galilean (β) position to the representational (γ) position
provided a context for expanding measurability to nonphysical properties, obtained
at the price of removing experimental conditions on processes claimed to be mea24 See Sect. 6.5.1 for an analysis of this condition and of the critiques it has received.
4 Philosophical perspectives on measurement