On the possible scientific laws

Abstract In a foundational paper, Luce (1959) observes that the scale type of the dependent and independent variables in a possible psychophysical or other scientific law determines the general form of the law through the solution of a certain functional equation. Luce (1964) extends the earlier results to the situation where there may be more than one independent variable. In a subsequent paper, Osborne (1970) extends Luce's early results to other scale types. Osborne (1970) discusses the possible psychophysical laws in the situation where the independent variables are independent ratio, interval, log-interval or ordinal scales and the dependent variable is a ratio, interval, log-interval or ordinal scale. In this paper we generalize Osborne's results in the cases that involve ordinal scales and in the case where the independent variables are independent ratio scales and the dependent variable is a log-interval scale and present new, more detailed proofs of his theorems. In addition, for the cases where the independent variables are ratio scales and the dependent variable is a log-interval or ordinal scale, we correct small errors in Osborne's results.