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Show Rationality and the Structure of the Self, Volume II: A Kantian Conception 123 (∀x) (Fxg ~Gxf), etc. (T1) would be equivalent to (T2) (∀x)(Fxg.Gxh.Fxh).7 This proposes to express the pairwise comparison between F and G as a sentential function (1) ((Fxg v Gxf).~(Fxg.Gxf)).~Gxf, an extended quantificational description of what is involved in selecting between two proffered alternatives. (1) is truth-functionally equivalent to (2) Fxg.~Gxf, which, with the application of the asymmetry axiom, becomes (3) Fxg. (T1)'s second conjunct and consequent similarly can be paired down to essentials. This notation is pleasing in certain respects. By replacing Savage's preference relation ">" with a chooser x in the same location, it preserves through several sentential transformations the same symmetrical placement of alternatives found in Savage's notation that made it an intuitively plausible representation of a pairwise comparison. At the same time, it reformulates this relation in identifiably sentential and quantificational terms. And it offers an expanded quantificational interpretation of Savage's "F>G" that unpacks it sententially. However, this proposed notation is very counterintuitive in other respects. It is certainly possible to express the preferred alternative in a particular pairwise comparison as a relational predicate, so that the predicate letter changes accordingly with each such ranking. It is also possible to assign the chooser and nonpreferred alternative to variables related by that predicate, so that the preferred alternative is in effect a property that the chooser and nonpreferred alternative are expressed as having. And it is possible to assign to the preference relation itself no symbolization at all in each of three preference rankings, so that the distinguishing structure of that relation is effectively obscured. This proposal was originally suggested to me by Henry Richardson some years ago (personal written communication, October 2, 1987). 7 © Adrian Piper Research Archive Foundation Berlin |
