Abstract model theory is a field that arose in the seventies after Lindström's model-theoretic characterisation of first-order logic. Logics started to be studied and compared in terms of their model theoretic properties -in opposition to extended model theory, which studies logics in terms of their expressive power. This optic gave rise to the search of new logics capable of exhibiting a certain combination of model-theoretic properties. So, to the generalizations of infinitary logics Lkl, Lindström himself (following the path started by Mostowsky in the late fifties) added extensions of first-order logic by generalised quantifiers, corresponding to an arbitrary class of models associated with an abstract sentence -i.e. without a specific syntax.

Description of the contents of the tutorial: This tutorial tries to give a general overview of the developments of abstract model theory, from the beginnings till its actual scenario, as far as classification of logics is concerned. After an introduction to the main concepts of the field, we describe the extensions of first-order logic and state Lindström's theorem. Secondly, we describe back-and-forth systems, which are a main tool in the field, particularly in the proofs of Lindström's and interpolation theorems.

We then present the generalisations of these concepts through, among others, Barwise's, Caicedo's, Lindström's and Shelah-Väänänen's results for Lindström's and interpolation theorems in several logics.

We finish with a schematic presentation of compactness and the relation between generalised interpolation and definability theorems in abstract logics, as studied by Makowsky, Shelah and Stavi.

Main reference

- J. Barwise and S. Feferman (ed.), Model-theoretic logics, Springer-Verlag, New York, 1985.

Lecture Notes will be available here by January 31th 2005.