Existential vs Preserved by Extensions, and Other Syntax/Semantics Correspondences

We look at various preservation theorems of classical logic (first of all, / Los - Tarski theorem) which fail in the context of finite models, and examine the extent of their failure. In some cases, even if the old correspondence between syntactical and semantical properties does not hold, one can find a new semantical counterpart for the syntactical property, and vice versa - a new syntactical counterpart for the semantical property. 1 Introduction It is well known that famous theorems about first-order logic fail in the case when only finite structures are allowed (Gurevich [1984]). A more careful examination shows that it is wrong to lump all these failing theorems together. On one side we have theorems like completeness or compactness where the failure is sort of hopeless. On the other side there are theorems like that of / Los - Tarski, which we prefer to formulate in the following form: Theorem 1.1 (/Los - Tarski) A first order formula is preserved by extensions iff it is equiv..