Harmonious logic: Craig's interpolation theorem and its descendants

model theory S. Feferman and J. Barwise (eds.) (1985), ModelTheoretic Logics. Abstract model theory deals with properties of model-theoretic logics L, specified by an abstract syntaxi.e. a set of “sentences” satisfying suitable closure conditionsand “satisfaction” relation M |= φ for φ a sentence of L. With each such L is associated its collection of Elementary Classes, ECL, and from that its collection of Projective Classes, PCL. L ⊆ L* if ECL ⊆ ECL*.model theory deals with properties of model-theoretic logics L, specified by an abstract syntaxi.e. a set of “sentences” satisfying suitable closure conditionsand “satisfaction” relation M |= φ for φ a sentence of L. With each such L is associated its collection of Elementary Classes, ECL, and from that its collection of Projective Classes, PCL. L ⊆ L* if ECL ⊆ ECL*.