In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, for a language

Formal language

A formal language is a set of words—that is, finite strings of letters, symbols, or tokens that are defined in the language. The set from which these letters are taken is the alphabet over which the language is defined. A formal language is often defined by means of a formal grammar...

  , define the Herbrand universe to be the set of ground terms of .

A structure

Structure (mathematical logic)

In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations which are defined on it....

  for is a Herbrand structure if the domain

Domain of discourse

In the formal sciences, the domain of discourse, also called the universe of discourse , is the set of entities over which certain variables of interest in some formal treatment may range...

 of is the Herbrand universe of This fixes the domain of , and so each Herbrand structure can be identified with its interpretation, leading to the alternative nomenclature of Herbrand interpretation

Herbrand interpretation

In mathematical logic, a Herbrand interpretation is 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...

.

A Herbrand model of a theory  is a Herbrand structure which is a model of

See also

• Herbrand base
• Herbrand's theorem

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