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Show Chapter III. The Concept of a Genuine Preference 152 asserting a desire, (2) will be spelled out as the desire for x above y, or for x to be above y, or that x be above y. Or a sentence may incorporate (2) similarly as the object of intending x above y, or intending x to be above y, or that x be above y. Or (2) may express an agent's perception of x as being above y, or her believing x to be above y; and so on. Within this stipulation, more fine-grained semantic ambiguities are resolvable with the provision of additional context. Correspondingly, (3) (∃y)(∀x)Axy would merely mention "… a(n existing) y such that all xs above it…" rather than asserting sententially that there is such a one. Thus I might believe all xs to be above a y, or intend any x to be above an existing y. Or I might rank any x above an existing y, and express it thus: (4) Pw[(∃y)(∀x)Axy] It is to be hoped that the general idea is clear: it is to do for subsentential constituents with predicate and quantificational logic what I earlier suggested we do with sentential logic, with the same rules and restrictions, plus those peculiar to quantificational inference. One benefit of this approach is that it permits a restatement in quantificational terms of Savage's original conception of ordinality (O) (Section 2, above) that captures what we need from the original: (O') {[Pw(x.~y).Pw(y.~z).Pw(x.~z)]. [Pw(x.~y).~Pw(y.~x)]} (∃z) [Pw(Axy.Ayz)] (Ordinality) (O') says that if w's preferences among x, y, and z are transitive and asymmetric, then w ranks x above y and y above an existing z; i.e. that w's ordering of x, y, and z has a lowest-ranked member and so constitutes a well-ordered triad. (O') enables us to answer the objections raised to Savage's conception of a simple ordering raised in Section 2 by avoiding any suggestion as to the selection criteria on which pairwise comparisons are based. As predicted, this notation sacrifices the streamlined elegance of Savage's measurable and uniform rendering. But as promised, it also avoids begging the question as to what those selection criteria are. A second benefit of introducing subsentential predication into the variable term calculus is that it allows us to symbolize a noncyclical solution to Gertrude's choice problem as described in Section 2. Recall that Gertrude preferred chocolate ice cream to vanilla for its sweetness, vanilla to coffee for © Adrian Piper Research Archive Foundation Berlin |
