The $n$-adic first-order undefinability of the Geach formula.
Ray E. Jennings, D. K. Johnston, Peter Κ. Schotch · Notre Dame Journal of Formal Logic · 1981
By adopting natural generalizations of relational frame and relational model we showed in [2] that the deontic law D, Dp -» Op, is not universally first-order definable.In effect, we showed there that for n > 2, there is no n-adic first-order sentence p such that for every n-ary frame F,F ^D iff F ^(3 in the first-order sense.The notion of n-ary frame and model employed there may be summarized as follows: F -(U,R) is an n-ary relational frame iff U is a nonempty set and R is an n-ary relation on U. Valuations on F are classical for PC formulas.For modal formulas,We say that a is valid on F or F is a frame for a(F 1= a) iff F(a) = U for every valuation V on F. That this is the correct generalization of frame and model is argued at some length in [3].Corresponding to the modal notion of a frame is the first-order notion of a model.If F is an n-ary frame and a* is a sentence in the first-order theory of a single n-adic predicate, then we say that F is a first-order model for O!*(F I 3 ot*) iff F(a*) = 1 for every assignment of individual variables to objects in F. Taking these notions of frame and first-order model we arrive at the notion of n-adic first-order definability.If there is an n-adic first-order sentence a* such that for every n-ary frame F, F t= a* (in the first-order sense) iff F t= a, we say that a. is n-adically first-order definable.If a is n-adically firstorder definable for every n, then we say that a is universally first-order definable.