Herbrand interpretation

In mathematical logic, a Herbrand signature interpretation of a first-order logic language is one - interpretation in which the universe is the Herbrand universe over , that is, the set of all terms without variables, and interprets each term "by itself" becomes. A Herbrand interpretation can thus be fully described by specifying the interpretation of the relational symbols. ${\displaystyle {\mathcal {S}}}$ ${\displaystyle {\mathcal {S}}}$ ${\displaystyle {\mathcal {I}}=(U,I)}$ ${\displaystyle U}$ ${\displaystyle {\mathcal {S}}}$

Formally, every function symbol is interpreted by the function . The amount of simple statements called Herbrand base to . The interpretation of the relation symbols is now fully specified by a subset of the Herbrand basis, with each -digit relation symbol being interpreted by the relation . ${\displaystyle f\in {\mathcal {S}}}$ ${\displaystyle f^{I}\colon U\to U;\ t\mapsto f(t)}$ ${\displaystyle HB}$ ${\displaystyle {\mathcal {S}}}$ ${\displaystyle {\mathcal {A}}\subseteq HB}$ ${\displaystyle n}$ ${\displaystyle R\in {\mathcal {S}}}$ ${\displaystyle R^{I}=\{(t_{1},\dotsc ,t_{n})\in U^{n}\mid R(t_{1},\dotsc ,t_{n})\in {\mathcal {A}}\}}$

example

Contain the signature only the constant symbol and the function symbol . The associated Herbrand universe is . Then the assignment between function symbols and elements from the universe is: ${\displaystyle S}$ ${\displaystyle a}$ ${\displaystyle f}$ ${\displaystyle D\left({\mathcal {S}}\right)=\{a,f(a),f(f(a)),...\}}$

${\displaystyle f^{I}(a)=f(a)}$

${\displaystyle f^{I}(f(a))=f(f(a))}$

${\displaystyle f^{I}(f(f(a)))=f(f(f(a)))}$

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