Abstract

We introduce a natural class of quantifiersTh containing all monadic type quantifiers, all quantifiers for linear orders, quantifiers for isomorphism, Ramsey type quantifiers, and plenty more, showing that no sublogic ofL ωω (Th) or countably compact regular sublogic ofL ∞ω (Th), properly extendingL ωω , satisfies the uniform reduction property for quotients. As a consequence, none of these logics satisfies eitherΔ-interpolation or Beth's definability theorem when closed under relativizations. We also show the failure of both properties for any sublogic ofL ∞ω (Th) in which Chang's quantifier or some cardinality quantifierQ α, with α≧1, is definable