satisfaction relation

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satisfaction relation

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 $\mathcal{I}$ is an interpretation, and $\sigma$ is a function that assigns elements of $A$ to variables, we define the function $\operatorname{Val}_{{\mathcal{I},\sigma}}$ inductively on the construction of terms :

$\displaystyle\operatorname{Val}_{{\mathcal{I},\sigma}}(c)$	$\displaystyle=$	$\displaystyle\mathcal{I}(c)\qquad c\;\;\text{a constant symbol}$
$\displaystyle\operatorname{Val}_{{\mathcal{I},\sigma}}(x)$	$\displaystyle=$	$\displaystyle\sigma(x)\qquad x\;\;\text{a variable}$
$\displaystyle\operatorname{Val}_{{\mathcal{I},\sigma}}(f(t_{1},...,t_{n}))$	$\displaystyle=$	$\displaystyle\mathcal{I}(f)(\operatorname{Val}_{{\mathcal{I},\sigma}}(t_{1}),.% ..,\operatorname{Val}_{{\mathcal{I},\sigma}}(t_{n}))\qquad f\text{ an }\;n% \text{-ary function symbol}$

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 :

$\displaystyle\mathcal{A},\sigma$	$\displaystyle\models$	$\displaystyle t_{1}=t_{2}\text{ if and only if }\operatorname{Val}_{{\mathcal{I},\sigma}}(t_{1})=\operatorname{Val}_{{% \mathcal{I},\sigma}}(t_{2})$
$\displaystyle\mathcal{A},\sigma$	$\displaystyle\models$	$\displaystyle R(t_{1},...,t_{n})\text{ if and only if }(\operatorname{Val}_{{% \mathcal{I},\sigma}}(t_{1}),...,\operatorname{Val}_{{\mathcal{I},\sigma}}(t_{1% }))\in\mathcal{I}(R)$
$\displaystyle\mathcal{A},\sigma$	$\displaystyle\models$	$\displaystyle\neg\varphi\text{ if and only if }\;\;\mathcal{A},\sigma\not\models\varphi$
$\displaystyle\mathcal{A},\sigma$	$\displaystyle\models$	$\displaystyle\varphi\vee\psi\text{ if and only if either }\mathcal{A},\sigma% \models\psi\text{ or }\mathfrak{A},\sigma\models\psi$
$\displaystyle\mathcal{A},\sigma$	$\displaystyle\models$	$\displaystyle\exists x.\varphi(x)\text{ if and only if for some }\;\;a\in A,\;% \mathcal{A},\sigma[x/a]\models\varphi$

Here

\sigma[x/a](y)\begin{cases}a&\text{ if }\;\;x=y\\ \sigma(y)&\text{ else.}\end{cases}

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 $\mathcal{A}\models\varphi$ .

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satisfaction, satisfy

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Definition

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bourbaki

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