Definition:Model (Predicate Logic)

Definition
Let $\mathcal P$ be the predicate symbols of a vocabulary of predicate calculus.

A model for predicate logic of type $\mathcal P$ is a system $\mathcal M$ consisting of:
 * A non-empty set $M$ called the universe of the model $\mathcal M$;
 * A function which assigns an $n$-ary relation $q^{\mathcal M}$ to each $n$-ary predicate symbol $q$ of $\mathcal P$.

Only the universe set $M$ is required to be non-empty. A unary relation $p^{\mathcal M}$ may be any subset of $M$ at all, which includes either empty or not.

Such a model is called a model for predicate calculus or a model for predicate logic.

Examples

 * The sentence $\exists x: p \left({x}\right)$ is true in a model $\mathcal M$ iff $p^\mathcal M$ is a non-empty subset of $M$.


 * The sentence $\forall x: p \left({x}\right)$ is true in a model $\mathcal M$ iff $p^\mathcal M = M$.


 * A propositional symbol $q$ will be true in a model $\mathcal M$ iff $q^\mathcal M = M^0$, that is, $q^\mathcal M$ contains the empty sequence, or $\left({}\right) \in q^\mathcal M$.


 * Suppose $\mathcal P$ consists of one unary predicate symbol $p$.

Then a model $\mathcal M$ of type $\mathcal P$ consists of:
 * A nonempty set $M$
 * One subset $p^\mathcal M$ (empty or not) of $M$.

If $M$ has $n$ elements, then from Cardinality of Power Set there are $2^n$ different models of type $\mathcal P$ with universe $M$, one for each subset $p^\mathcal M$ of $M$.

Given an infinite set $M$, there are then infinitely many different models of type $\mathcal P$ with universe $M$.


 * Suppose $\mathcal P$ consists of two unary predicate symbols $p$ and $q$.

Then a model $\mathcal M$ of type $\mathcal P$ consists of:
 * A nonempty set $M$
 * Two subsets $p^\mathcal M$ and $q^\mathcal M$ of $M$.

If $M$ has $n$ elements, then there are $\left({2^n}\right)^2$ different models of type $\mathcal P$ with universe $M$.


 * Suppose $\mathcal P$ consists of one binary predicate symbol $p$.

Then a model $\mathcal M$ of type $\mathcal P$ consists of:
 * A nonempty set $M$
 * One subset $p^\mathcal M$ of $M \times M$.

If $M$ has $n$ elements, then there are $2^{\left({n^2}\right)}$ different models of type $\mathcal P$ with universe $M$.