Fine's Theorem on First-Order Complete Modal Logics
  Fine's influential Canonicity Theorem states that if a modal logic is determined by a first-order definable class of Kripke frames, then it is valid in its canonical frames. This article reviews the background and context of this result, and the history of its impact on further research. It then develops a new characterisation of when a logic is canonically valid, providing a precise point of distinction with the property of first-order completeness. The ultimate point is that the construction of the canonical frame of a modal algebra does not commute with the ultrapower construction. 
Robert Goldblatt
math.LO