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A recursively enumerable Kripke complete first-order logic not complete with respect to a first-order definable class of frames
It is well-known that every quanti ed modal logic complete with respect to a first-order defi nable class of Kripke frames is recursively enumerable. Numerous examples are also known of natural quanti ed modal logics complete with respect to a class of frames de ned by an essentially second-order condition which are not recursively enumerable. It is not, however, known if these examples are instances of a pattern, i.e., whether every recursively enumerable, Kripke complete quanti ed modal logic can be characterized by a first-order defi nable class of frames. While the question remains open for normal logics, we show that, in the context of quasi-normal logics, this is not so, by exhibiting an example of a recursively enumerable, Kripke complete quasi-normal logic that is not complete with respect to any first-order defi nable class of (pointed) frames.
