Model (in logic)

An interpretation of a formal language satisfying certain axioms (cf. Axiom). The basic formal language is the first-order language $L_{\mathrm{\Omega }}$ of a given signature $\mathrm{\Omega }$ 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 $L_{\mathrm{\Omega }}$ is an algebraic system of signature $\mathrm{\Omega }$.

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

A class $\mathcal{K}$ of models of a language $L_{\mathrm{\Omega }}$ is called axiomatizable if there is a set $\mathrm{\Sigma }$ of closed formulas of $L_{\mathrm{\Omega }}$ such that $\mathcal{K}=\mathrm{Mod}\mathrm{\Sigma }$. The set $T(\mathcal{K})$ of all closed formulas of $L_{\mathrm{\Omega }}$ that are true in each model of a given class $\mathcal{K}$ of models of $L_{\mathrm{\Omega }}$ is called the elementary theory of $\mathcal{K}$. Thus, a class $\mathcal{K}$ of models of $L_{\mathrm{\Omega }}$ is axiomatizable if and only if $\mathcal{K}=\mathrm{Mod}T(\mathcal{K})$. 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 $L_{\mathrm{\Omega }}$ having universe $A$. One may associate to each element $a\in A$ a constant $c_a$ and consider the first-order language $L_{\mathrm{\Omega }A}$ of signature $\mathrm{\Omega }A$ which is obtained from $\mathrm{\Omega }$ by adding the constants $c_a$, $a\in A$. $L_{\mathrm{\Omega }A}$ is called the diagram language of the model $\mathbf{A}$. The set $O(\mathbf{A})$ of all closed formulas of $L_{\mathrm{\Omega }A}$ 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[edit]

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).