*Scheiblechner (1995) proposes a probabilistic axiomatization of
measurement called ISOP (isotonic ordinal probabilistic models) that
replaces Rasch's (1980) specific objectivity assumptions with two
interesting ordinal assumptions. Special cases of Scheiblechner's model
include standard unidimensional factor analysis models in which the
loadings are held constant, and the Rasch model for binary item
responses. Closely related are the doubly-monotone item response models
of Mokken (1971; see also Mokken and Lewis, 1982; Sijtsma, 1988;
Molenaar, 1991; and Sijtsma and Junker, 1995). More generally, strictly
unidimensional latent variable models have been considered in some
detail by Holland and Rosenbaum (1986), Ellis and van den Wollenberg
(1993), and Junker (1990, 1993).
*
The purpose of this note is to provide connections with current
research in foundations and nonparametric latent variable and item
response modeling that are missing from Scheiblechner's (1995) paper,
and to point out some related forthcoming work by Hemker et
al. (1995a,b), Ellis and Junker (1995) and Junker and Ellis (1995).
We also discuss counterexamples to three major theorems in the paper.