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Show Chapter IV. The Utility-Maximizing Model of Rationality: Formal Interpretations 152 (A3) If option A is equivalent to option B and B to C, then A to C; call this the Transitivity of Equivalent Options Axiom. (A3) establishes the unit consistency among comparable outcomes of whatever the unit quantity by which all of them are compared. (A4) If FG = HI, HI = JK, then FG = JK; call this the Transitivity of Equal Intervals Axiom. (A4) establishes the interval consistency among the intervals of measurement relative to which each outcome, conceived as some multiple of their unit quantity, is calibrated. Together axioms (A2) - (A4) ensure that, independent of the particular numerical values assigned to alternative outcomes, all such comparable outcomes can be ranked relative to one another as multiples of some unit quantity along a cardinal scale that assigns some numerical value to each. There are eight axioms in all. (A5) and (A6) stipulate unit quantity uniqueness, (A7) stipulates ordinal continuity, (A8) the ratio between two numbers a and b given integers m, n such that na>b and mb>a. Together they demonstrate how a value a might be correlated with a real number u(F) such that the value interval between F and G can be represented by the numerical expression u(F) - (u)G. I focus here on the reasoning behind (A2)-(A4) just summarized. 2.2. Consistency and Intensionality From the outset, Ramsey's exposition relies on intuitive and unexplicated notions of preference and ranking. Axioms (A1) - (A8) articulate these intuitive notions with formal precision. But they do not function as foundations that these intuitive notions presuppose. The relationship is rather the reverse: The intuitive notions of preference and ranking provide the foundations that the formal axioms presuppose, and on the basis of which those formal axioms are interpreted. That is, we need to assume and accept the intuitive notions in order to make sense of the formalization. This means that implicit from the very beginning in the conception of utilitymaximization and partial belief Ramsey develops formally is a more primitive, clearly intentional proto-concept of utility-maximization as basic action - (U), in fact, as defined in Chapter III, Section 1 above - that gives the formalized axioms meaning. Because this more primitive conception of utility-maximization is implicitly intentional, Ramsey's value axioms, in so far as they are valid, are implicitly intentional as well. The point in Ramsey's exposition at which intensionality is abandoned for the flexibility, precision and objectivity of extensional notation is the point at which these axioms cease to pertain to preference as we ordinarily understand that concept. © Adrian Piper Research Archive Foundation Berlin |
