Direct and indirect scaling of membership functions of probability phrases

Abstract A crucial issue in the empirical measurement of membership functions is whether the degree of fuzziness is invariant under different scaling procedures. In this paper a direct and an indirect procedure, magnitude estimation and graded pair-comparison, are compared in the context of establishing membership functions for probability phrases such as probable, rather likely, very unlikely , and so forth. Analyses at the level of individual respondents indicate that: (a) membership functions are stable over time; (b) functions for each phrase differ substantially over people; (c) the two procedures yield similarly shaped functions for a given person-phrase combination; (d) the functions from the two procedures differ systematically, in that those obtained directly dominate, or indicate greater fuzziness than do those obtained indirectly; and (e) where the two differ the indirectly obtained function may be the more accurate one. A secondary purpose of the paper is to evaluate the effects of the modifiers very and rather . Very has a general intensifying effect that is described by Zadeh's concentration model for 7 subjects and by a shift model for no one. The effects of rather are unsystematic and not described by any available model.