# Theory of Conjoint Measurement - Measurement and Quantification - Theory - Double Cancellation Axiom

Double Cancellation Axiom

Single cancellation does not determine the order of the "right-leaning diagonal" relations upon P. Even though by transitivity and single cancellation it was established that (a, x) > (b, y), the relationship between (a, y) and (b, x) remains undetermined. It could be that either (b, x) > (a, y) or (a, y) > (b, x) and such ambiguity cannot remain unresolved.

The double cancellation axiom concerns a class of such relations upon P in which the common terms of two antecedent inequalities cancel out to produce a third inequality. Consider the instance of double cancellation graphically represented by Figure Two. The antecedent inequalities of this particular instance of double cancellation are:

and

.

Given that:

is true if and only if ; and

is true if and only if, it follows that:

.

Cancelling the common terms results in:

.

Hence double cancellation can only obtain when A and X are quantities.

Double cancellation is satisfied if and only if the consequent inequality does not contradict the antecedent inequalities. For example, if the consequent inequality above was:

, or alternatively,

,

then double cancellation would be violated (Michell 1988) and it could not be concluded that A and X are quantities.

Double cancellation concerns the behaviour of the "right leaning diagonal" relations on P as these are not logically entailed by single cancellation. (Michell 2009) discovered that when the levels of A and X approach infinity, then the number of right leaning diagonal relations is half of the number of total relations upon P. Hence if A and X are quantities, half of the number of relations upon P are due to ordinal relations upon A and X and half are due to additive relations upon A and X (Michell 2009).

The number of instances of double cancellation is contingent upon the number of levels identified for both A and X. If there are n levels of A and m of X, then the number of instances of double cancellation is n! × m!. Therefore, if n = m = 3, then 3! × 3! = 6 × 6 = 36 instances in total of double cancellation. However, all but 6 of these instances are trivially true if single cancellation is true, and if anyone of these 6 instances is true, then all of them are true. One such instance is that shown in Figure Two. (Michell 1988) calls this a Luce — Tukey instance of double cancellation. If single cancellation has been tested upon a set of data first and is established, then only the Luce — Tukey instances of double cancellation need to be tested. For n levels of A and m of X, the number of Luce — Tukey double cancellation instances is . For example, if n = m = 4, then there are 16 such instances. If n = m = 5 then there are 100. The greater the number of levels in both A and X, the less probable it is that the cancellation axioms are satisfied at random (Arbuckle & Larimer 1976; McClelland 1977) and the more stringent test of quantity the application of conjoint measurement becomes.

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