page_title: Theory of conjoint measurement
section_title: Single cancellation or independence axiom
hierarchical_section_title: Theory of conjoint measurement / Measurement and quantification / Theory / Single cancellation or independence axiom
caption_attribution_description: English: This is a graphical representation of the single cancellation axiom of the theory of conjoint measurement.
context_page_description: The theory of conjoint measurement is a general, formal theory of continuous quantity. It was independently discovered by the French economist Gérard Debreu and by the American mathematical psychologist R. Duncan Luce and statistician John Tukey.
  The theory concerns the situation where at least two natural attributes, A and X, non-interactively relate to a third attribute, P. It is not required that A, X or P are known to be quantities. Via specific relations between the levels of P, it can be established that P, A and X are continuous quantities. Hence the theory of conjoint measurement can be used to quantify attributes in empirical circumstances where it is not possible to combine the levels of the attributes using a side-by-side operation or concatenation. The quantification of psychological attributes such as attitudes, cognitive abilities and utility is therefore logically plausible. This means that the scientific measurement of psychological attributes is possible. That is, like physical quantities, a magnitude of a psychological quantity may possibly be expressed as the product of a real number and a unit magnitude.
context_section_description: The single cancellation axiom is as follows. The relation upon P satisfies single cancellation if and only if for all a and b in A, and x in X, (a, x) > (b, x) is implied for every w in X such that (a, w) > (b, w). Similarly, for all x and y in X and a in A, (a, x) > (a, y) is implied for every d in A such that (d, x) > (d, y). What this means is that if any two levels, a, b, are ordered, then this order holds irrespective of each and every level of X. The same holds for any two levels, x and y of X with respect to each and every level of A.
  Single cancellation is so-called because a single common factor of two levels of P cancel out to leave the same ordinal relationship holding on the remaining elements. For example, a cancels out of the inequality (a, x) > (a, y) as it is common to both sides, leaving x > y. Krantz, et al., (1971) originally called this axiom independence, as the ordinal relation between two levels of an attribute is independent of any and all levels of the other attribute. However, given that the term independence causes confusion with statistical concepts of independence, single cancellation is the preferable term. Figure One is a graphical representation of one instance of single cancellation.
  Satisfaction of the single cancellation axiom is necessary, but not sufficient, for the quantification of attributes A and X. It only demonstrates that the levels of A, X and P are ordered. Informally, single cancellation does not sufficiently constrain the order upon the levels of P to quantify A and X. For example, consider the ordered pairs (a, x), (b, x) and (b, y). If single cancellation holds then (a, x) > (b, x) and (b, x) > (b, y). Hence via transitivity (a, x) > (b, y). The relation between these latter two ordered pairs, informally a left-leaning diagonal, is determined by the satisfaction of the single cancellation axiom, as are all the "left leaning diagonal" relations upon P.
caption: Figure One: Graphical representation of the single cancellation axiom. It can be seen that a > b because (a, x) > (b, x), (a, y) > (b, y) and (a, z) > (b, z).
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