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Show Rationality and the Structure of the Self, Volume II: A Kantian Conception 151 8. The Variable Term Calculus: Subsentential Predication So far I have suggested some notational revisions to Savage's system designed to enable us to represent the language of preference within the familiar constraints of the Boolean connectives. Essentially these have amounted to embedding and expanding within the place conventionally held by variable terms some familiar operations of sentential logic on the variables to which an n-adic predicate ordinarily is ascribed; this is why I describe these proposed revisions as constituting a variable term calculus. In order for the variable term calculus to represent an ordinal ranking in an intuitively acceptable way within the constraints imposed by the Boolean connectives, certain further notational revisions, familiar from predicate logic, need to be introduced. Let A be a two-place predicate that denotes the "above" relation. Then (1) Pw(Axy) states that w ranks (or prefers) x above y. Notice first that (1) avoids begging the questions raised in Section 2 against Savage's assumptions about the numerical commensurability of x, y and z. I may rank veggies above rice without being committed to any sense in which veggies are more than beans (other than the unhelpful sense in which they perhaps mean more to me). A noncommittal stance toward numerical commensurability is a virtue in an ordinal ranking of alternatives. (1) introduces the possibility of conceiving not only variable terms but predicate letters - and, if needed, quantifiers as well - as subsentential constituents that can be nested within other predicates that govern entire sentences, such that the scope of the outermost is the entire sentence whereas the scope of one enclosed within the brackets is the variable term(s) enclosed within sub-brackets to the right. Call the outermost governing predicates. In this discussion, P would be a governing predicate. A, like any predicate letter whose scope is a variable term or relation among some but not all variable terms in the sentence, would exemplify what I shall call a subsentential predicate. The same constraints on linguistic interpretation mentioned in Section 4, above, apply here. So, for example, (2) (Axy) is not a sentential proposition but rather a constituent of one that says merely "…x above y…." And similarly, the interpretation of (2) will depend on the context and intentional operator that modify it. (1) demonstrates the interpretation of (2) in a sentence asserting a preference ranking. In a sentence © Adrian Piper Research Archive Foundation Berlin |
