Definition 3.4 A statement using scales is called meaningful “if its truth or falsity is unchanged whenever any scale is replaced by another acceptable scale” (for instance by changing units) (Roberts 1979); otherwise, it is called meaningless. For example, consider the statement: “I am twice as tall as the Obelisk of Theodosius in Istanbul” (around 20 m high!); this statement is clearly false for all scales of height, i.e., it is false whether we use inches, feet, meters, or any other unit. Thus, the statement is meaningful because its falsity is (in this case) independent of the scales used. Note that meaningfulness is not the same as truth. It has to do with the “compatibility” of a statement with a particular measurement scale.
From the perspective of admissible scale transformations, the Definition 3.4 (of meaningfulness) can be modified as:
Definition 3.5 “A statement using scales is called meaningful if its truth or falsity is unchanged when all scales in the statement are transformed by admissible transformations” (Marcus-Roberts and Roberts 1987); otherwise, it is called meaningless.
In general, the meaningfulness of a statement can be formally demonstrated through an (analytical) proof that its truth/falsity is invariant under whatever admissible transformation. We point out that a specific example of the invariance of the statement under one of the possible admissible transformations does not represent a general proof of meaningfulness (one swallow doesn’t make summer!). On the other hand, the meaninglessness of a statement can be formally demonstrated through either a general (analytical) proof or a single counter-example, which shows the non-invariance of the statement under a specific admissible scale transformation (see also the scheme in Fig. 3.2).
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