MEASUREMENT OF MEMBERSHIP FUNCTIONS AND THEIR ACQUISITION

Three basic views of the representation of membership functions are reviewed, together with fundamental measurement of linguistic terms of linguistic variables. The conclusion is that such measurements are either on an 'ordinal' or an 'interval' scale based on whether the appropriate axioms are validated by the empirical data, with an allowance for stochastic variation. The conjoint measurement is introduced for the case of multi-dimensional linguistic variables whose linguistic terms are compositions of two or finitely many more component linguistic terms of distinct (independent) linguistic variables. It is shown that any composition of distinct (independent) component linguistic terms of component linguistic variables by any t-norm or s-norm or any finite convext linear combination preserves the monotonic weak order property of the components in the composite. Once the scale properties of the measurement values of particular terms are validated, composition procedures may be applied to the experimental data to obtain compound membership functions of fuzzy sets induced by meaningful representations of compositions of linguistic terms of linguistic variables. Finally, four methods of membership acquisition and construction are reviewed from the perspective of fundamental measurement.