Do Inconsistency Indices Measure Inconsistency of Preferences?

In this manuscript, we question the capacity of well-known inconsistency indices to measure, in a cardinal sense, the inconsistency of preferences. We argue that, at present, axiomatic properties of inconsistency indices only guarantee the good behaviour of an index when it comes to rank matrices according to their inconsistency, and the measurement of the intensity of preferences is not guaranteed. We propose to adapt the concept of measurability, established by Multiple Attribute Value Theory, to define measurable inconsistency indices. We conclude by presenting some methods to obtain measurable inconsistency indices.