# Term and formula assignment

In a previous card, it was described how an \( \mathcal{L} \)-structure and a variable assignment fuction act together to identify a set theoretic interpretation for every symbol of a
language. For terms and formulas, they have an interpretation assigned to them by a *term assignment function* and a *formula assignment function*. Terms get
assigned to elements while (some) formulas get assigned "truth" (also called "satisfaction").

### 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 \( \bar{s} \), 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, [\( \bar{s}(t) = \, ? \, \)]
- if \( t \) is a constant symbol \( c \), [\( \bar{s}(t) = \, ? \, \)]
- if \( t \) is \( ft_1t_2...t_n \) for some function and term symbols, then [\( \bar{s}(t) = f^{ \mathfrak{U} }(\, ? \, ) \) ]

Formulas get a very different mapping.

### 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 \( \bar{s} \) the resulting term assignment function).

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

- \( \phi :\equiv Rt_1t_2...t_n \) and [\( \, ? \, \in \, ? \, \)].
- \( \phi :\equiv (\alpha \lor \beta) \) and [condition 1] or [condition 2].
- \( \phi :\equiv \forall x \; (\alpha) \) and for each element \( a \in A, \; \mathfrak{U} \vDash \alpha[s(x|a)] \; \).
- \( \phi :\lnot (\alpha) \) and \( \mathfrak{U} \nvDash \alpha[s] \).

\( \mathfrak{U} \nvDash \phi \) means it's not the case that \( \mathfrak{U} \vDash \phi \).

There is a discussion about the meaning of \( \nvDash \) on the back side.