In first-order logic, a Herbrand structure ${\displaystyle S}$ is a structure over a vocabulary ${\displaystyle \sigma }$ (also sometimes called a signature) that is defined solely by the syntactical properties of ${\displaystyle \sigma }$. The idea is to take the symbol strings of terms as their values, e.g. the denotation of a constant symbol ${\displaystyle c}$ is just "${\displaystyle c}$" (the symbol). It is named after Jacques Herbrand.

Herbrand structures play an important role in the foundations of logic programming.

The Herbrand universe serves as the universe in a Herbrand structure.

Let ${\displaystyle L^{\sigma }}$, be a first-order language with the vocabulary

then the Herbrand universe ${\displaystyle H}$ of ${\displaystyle L^{\sigma }}$ (or of ${\displaystyle \sigma }$) is

${\displaystyle H=\{c,f(c),g(c),f(f(c)),f(g(c)),g(f(c)),g(g(c)),\dots \}}$

The relation symbols are not relevant for a Herbrand universe since formulas involving only relations do not correspond to elements of the universe.

A Herbrand structure interprets terms on top of a Herbrand universe.