On the failure of the weak Beth property

A general method for showing the failure of the weak Beth definability property for certain pairs of logics is discussed. Applications are made to the game logic, various infinitary logics and partially ordered quantifier logics. In (5), Craig essentially shows that the satisfaction relation for second order logic is implicitly but not explicitly definable in second order logic-thereby showing that the analogue of Beth's definability theorem fails for second order logic. (See also Kreisel's review (10).) In this note we shall elaborate on Craig's idea to obtain the failure of the Beth property in various other situations. In so doing we shall obtain some new results as well as give some new proofs of old results. All the logics we deal with are assumed to be such that their formulas are coded by sets in some standard way. In particular, we assume that the variables are indexed by ordinals, that va is coded by (0, a) and that each logic contains a binary relation symbol E such that atomic formulas of the form E(va,Vs) are coded by (l,a,s). The results we obtain actually show that an even weaker property than the usual Beth property fails. Following Friedman (6), we say that a logic L, has the weak Beth property with respect to a logic L2 (Weak Beth (LX,L2)) if for every sentence <p of Lx, of type p U {R}, such that each structure <31t of type p has exactly one expansion <?H, /?) which is a model of tp, there is a formula \p of L2, of type p, such that