Easy example of a herbrand structure

Question

Can someone give me an easy example of a Herbrand structure?

I can't really visualise the difference between a Herbrand and a normal structure.

Answer 1

Example

Consider the very simple FOL formula : $R(c)$.

The domain of the Herbrand structure is :

the set of all ground terms [i.e. closed terms] of the language.

In the above case, we have only the individual constant $c$ as gorund term. Thus, the domain is $H = \{ c \}$.

With it, we define the Herbrand interpretation :

an interpretation in which all constants and function symbols are assigned very simple meanings. Specifically, every constant is interpreted as itself, and every function symbol is interpreted as the function that applies it. The interpretation also defines predicate symbols as denoting a subset of the relevant Herbrand base, effectively specifying which ground atoms are true in the interpretation. This allows the symbols in a set of clauses to be interpreted in a purely syntactic way, separated from any real instantiation.

Again, we have a very simple Herbrand inerpretation $H_S$ :

$H_S = (H, R^H)$,

where $H$ is the domain defiend above and $R^H$ is the subset of $H$ interpreting the relation symbol $R$.

Obviously, $R^H = \{ c \}$.