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Show Rationality and the Structure of the Self, Volume II: A Kantian Conception 127 quantificational notation, in conjunction with the elements of the variable calculus just sketched, in order to fashion a logical analogue of (T). Let a triadic preference relation P for pairwise comparisons be defined as follows. Given variables w, x, y and z, let w be a chooser and x, y and z be any alternatives - actions, states, events, gambles, compound lotteries, plans, prospects or discrete objects - between which that chooser decides, such that (P) Pw(x.~y).9 (P) is a sentential function - call it a strict preference- [or P-] function - that states that a conscious and intentional chooser w strictly prefers alternative x to alternative y, i.e. that she selects x and rejects y. (P) expresses the implicit rejection of the not-preferred alternative involved in all strict preference rankings (if this seems too strong, try convincing the lover you have just jilted that your preference for someone else does not imply your rejection of him, and see how he takes it). So one advantage of (P) is that on an intuitive linguistic level, it captures better what goes on when a chooser makes a pairwise comparison - "this one, not that one" - than the intuitive linguistic reading of Savage's "more than" relation. A disadvantage is that it replaces Savage's asymmetric n-place connective ">" with a rigidly two-place combination ".~". This replaces a simple, streamlined, and aesthetically pleasing function with a clunkier and less graceful one that (as we have seen in Section 2, above) makes the move from (T') to (O) clumsier. But perhaps there will be compensations. Similarly, let (I1) Pw(x v y) be an indifference [or I-] function defined in terms of P that states that w prefers either x or y, i.e. either one is acceptable. Now were I to follow Savage's lead, I would define the indifference relation strictly in terms of (P) rather as (I2) ~Pw(x.~y).~Pw(y.~x). But Savage's definition (2. (I)), and (I2) even more precisely, express, if anything specific to preference, something closer to revulsion for both x and y rather than true indifference between them. It is the difference between Fenrong Liu and Dick de Jongh arrived at a similar interpretation of strict preference independently. They proposed it in their "Optimality, Belief and Preference," delivered to the Models of Preference Change Workshop, Freie Universität Berlin, 15 September 2006; and published in Proceedings of the Workshop on Rationality and Knowledge, Ed. Sergei Artemov and Rohit Parikh, Eds (ESSLLI, 2006). I discuss their proposal in Section 9, below. 9 © Adrian Piper Research Archive Foundation Berlin |
