As it became clear at the end of the last section a formal semantic presupposes an interpretation function mapping some language to some abstract structure . Traditionally this has been called Model Theory. I would like to call it Classical Model Theory because the usual application context of this kind of theory was the language of first order logic (predicate logic) and the construction of a model for this language. Introducing the logical inference concept in parallel with the logical satisfaction concept yields the following conceptual symmetry (cf. Chang 1973 [52]:33) for some set of expressions :

SYNTAX | SEMANTICS |

( is a theorem) | ( is valid) |

is consistent | has a model |

( is deducible from ) | ( is a consequence from ) |

Assuming as a model and as a sentence of the language , then it is common to circumscribe the expression

as follows:

holds in |

satisfies |

is satisfied in |

is a model of |

Examples of theorems in (classical) model theory are the following ones :

Gödel's Completeness Theorem: Given any sentence , is a theorem of if and only if is valid.

Extended Completeness Theorem: Let be any set of sentences, then is consistent if and only if has a model.

Compatness Theorem: A set of sentences has a model if and only if every finite subset of has a model.

Gerd Doeben-Henisch 2010-03-03