Herbrand Base in First Order Logic

Question

I've just studied the Least Herbrand Model, that is the intersection of every possible Herbrand Base.

So, i guess that it's possible to have an Herbrand Base with more elements than needed . Is it correct ? If no, i dont understand the need to introduce the Least Herbrand model...

Answer 1

See Minimal models:

The relation $⊆$ (subset) is a partial ordering for subsets of $\text {atoms}(P)$ of a program $P$, and therefore for Herbrand interpretations and Herbrand models. So we can talk of minimal Herbrand models.

A Herbrand model is minimal if no proper subset of it is also a model.

Note that minimal does not imply unique, in general. A set of formulas might have several minimal Herbrand models.

For example, $\{ a ∨ b \}$ has the following Herbrand models: $\{ a, b \}, \{ a \}, \{ b \}$.

The models $\{ a \}$ and $\{ b \}$ are both minimal.

The usual terminology for any partial ordering is that least means "the least" (i.e. unique minimal). So sometimes we speak of the least Herbrand model, which is the unique minimal model, if it exists.