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).