Herbrand structure

A structure that fits a set of predicate logic formulas F is called a Herbrand structure if the following properties are met: ${\displaystyle A=\left(U,I\right)}$

• The universe is the Herbrand universe generated from F , so . ${\displaystyle U=D\left(F\right)}$
• Interpretations I are Herbrand interpretations .

During an interpretation, the function and constant symbols are assigned actual functions and constants. In the Herbrand interpretation, each functional term is assigned an interpretation by itself. This is possible because the Herbrand universe consists precisely of the set of all possible terms with function and constant symbols. A Herbrand structure is therefore a term interpretation .

Example: Be the Herbrand universe . Then the assignment between function symbols and elements from the universe is: ${\displaystyle D\left(F\right)=\{a,f(a),f(f(a)),...\}}$

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

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

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

...

Herbrand structures are used in Herbrand's theorem and are named after Jacques Herbrand .

See also

Herbrand interpretation

literature

• H.-D. Ebbinghaus, J. Flum, W. Thomas: Introduction to mathematical logic , spectrum Akademischer Verlag 1996, ISBN 3-8274-0130-5