Alfred Tarski was the first mathematician to give a formal definition of what it means for a formula to be ``true'' in a structure. To do this, we need to provide a meaning to terms, and truth-values to the formulas. In doing this, free variables cause a problem : what value are they going to have ? One possible answer is to supply temporary values for the free variables, and define our notions in terms of these temporary values.

Let $\mathcal{A}$ be a structure with signature $\tau$ . Suppose $\I$ is an interpretation, and $\sigma$ is a function that assigns elements of $A$ to variables, we define the function $\val_{\I,\sigma}$ inductively on the construction of terms :

\begin{eqnarray*} \val_{\I,\sigma}(c)& = &\I(c)\qquad c\;\;\text{a constant symbol}\\ \val_{\I,\sigma}(x)& = &\sigma(x)\qquad x\;\;\text{a variable}\\ \val_{\I,\sigma}(f(t_1,...,t_n))& = &\I(f)(\val_{\I,\sigma}(t_1),...,\val_{\I,\sigma}(t_n))\qquad f\text{ an }\; n\text{-ary function symbol} \end{eqnarray*} Now we are set to define satisfaction. Again we have to take care of free variables by assigning temporary values to them via a function $\sigma$ . We define the relation $\mathcal{A},\sigma\models\varphi$ by induction on the construction of formulas :

\begin{eqnarray*} \mathcal{A},\sigma & \models & t_1=t_2\text{ if and only if }\val_{\I,\sigma}(t_1)=\val_{\I,\sigma}(t_2)\\ \mathcal{A},\sigma & \models & R(t_1,...,t_n)\text{ if and only if } (\val_{\I,\sigma}(t_1),...,\val_{\I,\sigma}(t_1))\in\I(R)\\ \mathcal{A},\sigma & \models & \neg\varphi\text{ if and only if }\;\;\mathcal{A},\sigma\not\models\varphi\\ \mathcal{A},\sigma & \models & \varphi\Or\psi \text{ if and only if either } \mathcal{A},\sigma\models\psi \text{ or } \A,\sigma\models\psi \\ \mathcal{A},\sigma & \models &\exists x.\varphi(x)\text{ if and only if for some } \;\;a\in A,\; \mathcal{A},\sigma[x/a]\models \varphi\\ \end{eqnarray*} Here

In case for some $\varphi$ of $L$ , we have $\mathcal{A},\sigma\models\varphi$ , we say that $\mathcal{A}$ models, or is a model of, or satisfies $\varphi$ . If $\varphi$ has the free variables $x_1,...,x_n$ , and $a_1,...,a_n\in A$ , we also write $\mathcal{A}\models\varphi(a_1,...,a_n)$ or $\mathcal{A}\models\varphi(a_1/x_1,...,a_n/x_n)$ instead of $\mathcal{A},\sigma[x_1/a_1]\cdots[x_n/a_n]\models\varphi$ . In case $\varphi$ is a sentence (formula with no free variables), we write $\mathcal{A}\models\varphi$ .