The interpretation that Arrow's Condition I, independence of irrelevant alternatives, prohibits the use of individuals' intensities of preference in the construction of social choices is not precise. Rather, it is the social-welfare function (as defined in Chapter 4) which demands both individual and social orderings, and thereby prohibits cardinal utility inputs. Condition I, as Arrow wrote it, redundantly requires individual orderings, but goes further and demands that, even given the ordinal data from individual orderings, the social choice over any two alternatives not be influenced by individuals' preferences involving any third alternatives. This is explicit in Arrow (1963/1951, 59, emphasis added):
It is required that the social ordering be formed from individual orderings and that the social decision between two alternatives be independent of the desires of individuals involving any alternatives other than the given two … These conditions taken together serve to exclude interpersonal comparison of social utility either by some form of direct measurement or by comparison with other alternative social states.
Arrow's is a strong independence condition. Slight weakenings of it allow the Borda count or the Young–Kemeny rule as possible social welfare functions and further weakenings permit further voting procedures.
Barry and Hardin (1982, 217–218) agree that Arrow's IIA is a powerful condition. “Part of its power is that one cannot easily intuit what it means or why it matters … Perhaps because of its subtlety, condition I is apparently the condition that is most readily taken for granted in the proof of Arrow's and related theorems.”