Psychometricians hypothesize that cognitive abilities such as reading, writing and spelling are measurable. However, they prefer to model item response probabilities than to study the internal structure of cognitive attributes. The theory of conjoint measurement, via its unique capacity to detect the quantitative structure of non-extensive attributes, can be used for the latter purpose. It is shown that conjoint systems analogous to the numerical composition rules advanced by the Rasch, many facet Rasch and two-parameter logistic models can be formulated using well-established axioms, representation theorems and proofs. Examples using empirical and synthetic data are presented. It was concluded that the lack of descriptive, behavioural theory remains the biggest obstacle to the scientific measurement of cognitive abilities.