Term and formula assignment

Term assignment function

Let $\mathcal{L}$ be a language, and let $\mathfrak{U}$ be an $\mathcal{L}$-structure for the language. Let $A$ be the universe of $\mathfrak{U}$ and let $s$ be a variable assignment function.

The term assignment function, denoted $\overline{s}$, is a mapping of terms to elements of $A$. It is defined recursively and is fully determined by the combination of the $\mathcal{L}$-structure and the variable assignment function.

Let $t$ be a term, then:

• if $t$ is a variable symbol, [$\overline{s}(t)=?$]
• if $t$ is a constant symbol $c$, [$\overline{s}(t)=?$]
• if $t$ is $f{t}_1{t}_2...t_n$ for some function and term symbols, then [$\overline{s}(t)=f^{\mathfrak{U}}(?)$ ]

Formula assignment: truth/satisfaction

Let $\mathcal{L}$ be a language, and let $\mathfrak{U}$ be an $\mathcal{L}$-structure for the language. Let $A$ be the universe of $\mathfrak{U}$ and let $s$ be a variable assignment function (and $\overline{s}$ the resulting term assignment function).

Let $\varphi$ be a formula valid in $\mathcal{L}$. We say that $\varphi$ is true in $\mathfrak{U}$, or that $\mathfrak{U}$ satisfies $\varphi$, and we write $\mathfrak{U}\models \varphi$, iff one of the following conditions holds:

1. $\varphi :\equiv R{t}_1{t}_2...t_n$ and [$?\in ?$].
2. $\varphi :\equiv (\alpha \vee \beta )$ and [condition 1] or [condition 2].
3. $\varphi :\equiv \mathrm{\forall }x(\alpha )$ and for each element $a\in A,\mathfrak{U}\models \alpha [s(x|a)]$.
4. $\varphi :\mathrm{\neg }(\alpha )$ and $\mathfrak{U}⊭\alpha [s]$.

$\mathfrak{U}⊭\varphi$ means it's not the case that $\mathfrak{U}\models \varphi$.