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Show Rationality and the Structure of the Self, Volume II: A Kantian Conception 155 Roughly speaking, then, a constraint is a linguistic formula that functions logically in much the same way as does a predicate in quantificational logic, i.e. it is ascribed to a variable and satisfies the law of non-contradiction for predicates (x)~(Fx . ~Fx) - or, as De Jongh and Liu express it, "either the constraint clearly is true of the alternative or it is not." Hence for purposes of this exposition, we can think of their concept of a constraint as a certain kind of predicate. De Jongh and Liu are interested in the way in which the imposition of constraint predicates engenders a preference ordering among all the alternatives. Hence their approach treats constraint predicates as giving rise to a preference ordering among alternatives, but also presupposes a strict ordering among those predicates themselves. Later in the discussion, they then go on to examine the way in which introduction of the belief operator in doxastic logic offers new ways of thinking about preference change; it must be emphasized that this is their primary concern. But my interest here is confined to De Jongh and Liu's conceptualization of the relation between preference alternatives and the predicates that are argued to order them. De Jongh and Liu define a constraint sequence as a finite, strictly ordered sequence of constraints C1, C2, … Cn, each of which is predicated of exactly one free variable x, such that (1) C1x > C2x … > Cnx and, for example, C1 . ~C2 … . ~C m is preferable to ~C1 . C2 … . Cm; and C1 . C2 . C3 . ~C4 . ~C5 is preferable to C1 . C2 . ~C3 . C4 . C5. They then define a strict preference for x over y Pref (x,y), given a constraint sequence C with n members, as follows: (2) Pref1(x,y) =df. C1x . ~C1y (3) Prefk+1(x,y) =df. Prefk(x,y) v [(C1x ≡ C1y) . … . (Ckx ≡ C ky)23 . Ck+1x . ~Ck+1y], k<n (4) Pref(x,y) =df. Prefn(x,y). Similar in logical structure to my Pw(x.~y), (2) intuitively defines preference for x over y with respect to the first C in n as the case in which that first and lexically prior constraint predicate C1 holds true of x and not of y. On that basis, (3) then defines preference for x over y with respect to subsequent constraints Ck+1 in n as the case in which either x is preferred to y with respect De Jongh and Liu substitute Eqk(x,y) for (C 1x ≡ C1y) . … . (Ckx ≡ Cky) for brevity. I restore the original sentence in order to expose the structure of their definition. I also translate their notation for the Boolean connectives into mine for purposes of comparison. 23 © Adrian Piper Research Archive Foundation Berlin |
