I'm not aware of any widely understood set of formal criteria that a purported measure of dispersion has to meet<sup>&dagger;</sup>&mdash;perhaps it's more that *dispersion* is a vague, pre-quantitative, notion that has motivated & can apply to various precisely defined measures. The assertion that "a measure of dispersion can, in the true sense, be regarded as the proper measure of dispersion if the measure is based on the deviations between all pairs of data" is intriguing; but its author doesn't provide any rationale (or any reference). This criterion would rule out the inter-quartile range, but not in fact the standard deviation: for a sample ${x_1, \ldots, x_n}$, the sum of squared deviations from the mean is equal to the mean of squared pairwise differences,

$$\sum_{i=1}^n{\left(x_i- \frac{\sum_{i =1}^n x_i}{n}\right)^2} = \frac{\sum_{i=1}^n\sum_{j = i + 1}^n(x_i - x_j)^2}{n}$$

(see https://stats.stackexchange.com/a/225758/17230).

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&dagger; @NickCox made me aware in short order that Bickel & Lehmann (1976), *Ann. Statist.*, **4**, 6, propose such a set of criteria. A measure of dispersion must (1) be location-invariant, (2) be scale-equivariant, & (3), given distributions $F_X$ symmetric about $\mu_X$ & $F_Y$ symmetric about $\mu_Y$ where $|X-\mu_X|$ stochastically dominates $|Y-\mu_Y|$, be greater for $F_X$ than for $F_Y$. (Of course standard deviation meets these criteria.)