What is the difference between Herbrand Logic and Relational Logic or Predicate Logic? I am learning a course from Stanford University, and it introduces the notion of Herbrand Logic. However in Wikipedia I cannot find a definition specifically for "Herbrand Logic", only for Herbrand Base or Herbrand Theorem, concepts that also exist in relational locic / predicate logic ?
 A: "Herbrand Logic = First-order syntax + Herbrand semantics".
Essentially, in Herbrand semantics a model M is a set of ground terms, which are exactly those ground terms that it satisfies. In other words, a model is always some subset of the Herbrand base.
This is very different from the Tarski semantics, where a model may contain a whole bunch of elements that cannot be written down as ground terms, but they can take part in the interpretation. In particular, the quantifiers range over such elements as well.
A: See here for Relational Logic.
Is first-order predicate logic without function symbols, i.e. tehre are only $n$-ary relation constants.
The semantics is based on Herbrand semantics.
Due to the lack of function symbols, the interpretation has only a finite number of objcets; thus, we can use truth assignments, as in propositional logic, and use them to evaluate formulas.
A: Consider a first order language L.
In first order logic: A formula F follows from a set of formulas T if every first order structure for L that satisfies T also satisfies F.
In Herbrand logic: A formula F follows from a set of formulas T if every first order Herbrand structure for L that satisfies T also satisfies F.
