Model (in logic)

An interpretation of a formal language satisfying certain axioms (cf. Axiom). The basic formal language is the first-order language $ L _ \Omega $ of a given signature $ \Omega $ including predicate symbols $ R _ {i} $, $ i \in I $, function symbols $ f _ {j} $, $ j \in J $, and constants $ c _ {k} $, $ k \in K $. A model of the language $ L _ \Omega $ is an algebraic system of signature $ \Omega $.

Let $ \Sigma $ be a set of closed formulas in $ L _ \Omega $. A model for $ \Sigma $ is a model for $ L _ \Omega $ in which all formulas from $ \Sigma $ are true. A set $ \Sigma $ is called consistent if it has at least one model. The class of all models of $ \Sigma $ is denoted by $ \mathop{\rm Mod} \Sigma $. Consistency of a set $ \Sigma $ means that $ \mathop{\rm Mod} \Sigma \neq \emptyset $.

A class $ {\mathcal K} $ of models of a language $ L _ \Omega $ is called axiomatizable if there is a set $ \Sigma $ of closed formulas of $ L _ \Omega $ such that $ {\mathcal K} = \mathop{\rm Mod} \Sigma $. The set $ T ( {\mathcal K} ) $ of all closed formulas of $ L _ \Omega $ that are true in each model of a given class $ {\mathcal K} $ of models of $ L _ \Omega $ is called the elementary theory of $ {\mathcal K} $. Thus, a class $ {\mathcal K} $ of models of $ L _ \Omega $ is axiomatizable if and only if $ {\mathcal K} = \mathop{\rm Mod} T ( {\mathcal K} ) $. If a class $ {\mathcal K} $ consists of models isomorphic to a given model, then its elementary theory is called the elementary theory of this model.

Let $ \mathbf A $ be a model of $ L _ \Omega $ having universe $ A $. One may associate to each element $ a \in A $ a constant $ c _ {a} $ and consider the first-order language $ L _ {\Omega A } $ of signature $ \Omega A $ which is obtained from $ \Omega $ by adding the constants $ c _ {a} $, $ a \in A $. $ L _ {\Omega A } $ is called the diagram language of the model $ \mathbf A $. The set $ O ( \mathbf A ) $ of all closed formulas of $ L _ {\Omega A } $ which are true in $ \mathbf A $ on replacing each constant $ c _ {a} $ by the corresponding element $ a \in A $ is called the description (or elementary diagram) of $ \mathbf A $. The set $ D ( \mathbf A ) $ of those formulas from $ O ( \mathbf A ) $ which are atomic or negations of atomic formulas is called the diagram of $ A $.

Along with models of first-order languages, models of other types (infinitary logic, intuitionistic logic, many-sorted logic, second-order logic, many-valued logic, and modal logic) have also been considered.

For references see Model theory.

Comments

English usage prefers the word "structure" where Russian speaks of a "model of a language" or an "algebraic system" ; "model" is reserved for structures satisfying a given theory (set of closed formulas).