We promised to do an examination of Arrow's R notation to resolve the differences between my friend, Ben at Oxford, and myself. Ben contended that R was only a "representation device." (See Comments.) After delving into this subject I would both agree and disagree. Arrow says on p. 12 of “Social Choice and Individual Values”: “Preference and indifference are relations between alternatives. Instead of working with two relations, it will be slightly more convenient to use a single relation. ‘preferred or indifferent.’ The statement ‘x is preferred or indifferent to y’ will be symbolized by xRy. The letter R, by itself, will be the name of the relation and will stand for a knowledge of all pairs such that xRy.” [emphasis added]
So R is both a representation device (when it stands alone) and a logical relation when it stands between two letters representing alternatives. A relation of the form aPbPcIdPf... (where a,b,c... stand for alternatives; P stands for preference and I stands for indifference) makes perfect sense since the logical relationships are clear. However, a relation of the form aRbRcRdRf... makes no sense since one must know the truth values of aRb and bRa, aRc and cRa, aRd and dRa etc. etc.
We have assumed that Arrow’s intent was to maintain a 1-1 relationship between P and I, on the one hand, and R on the other so that individual voters would submit their ballots in terms of P and I. These ballots could then be translated to terms of R as long as one knew both xRy and yRx. The dichotomy between the two notations is that one only need know xPy, yPx or xIy since they are all mutually exclusive. If you know that xPy is true, for example, you need not know the truth values of xIy or yPx. However, you do need to know the truth values for both xRy and yRx in order to maintain the 1-1 relationship between R and {P,I}.
We think that it is more transparent and less confusing to use the P and I notation instead of the R notation . Arrow’s use of the R notation because it is, according to him, “slightly more convenient,” turns out to be more cumbersome and more confusing. The same proofs could be done using P and I instead of R. The in-depth analysis of this conumdrum continues here.
