Herbrand interpretation

E238813

model-theoretic construct semantic notion in logic

A Herbrand interpretation is a foundational model-theoretic construct in logic and automated theorem proving that interprets formulas over the Herbrand universe built from a theory’s own function symbols and constants.

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All labels observed (4)

Label	Occurrences
Herbrand interpretation canonical	5
Herbrand model	3
Herbrand structure	3
Herbrand models	1

How this entity was disambiguated

This entity first appeared as the object of triple T2139598 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Herbrand interpretation
Context triple: [Jacques Herbrand, developedConcept, Herbrand interpretation]

A. Herbrand universe
The Herbrand universe is a fundamental concept in mathematical logic and automated theorem proving, consisting of all ground (variable-free) terms that can be built from the function symbols and constants of a given first-order language.
B. Herbrand's theorem
Herbrand's theorem is a fundamental result in mathematical logic and proof theory that characterizes the validity of first-order formulas via finite sets of ground instances, forming a basis for automated theorem proving.
C. Herbrand expansion
Herbrand expansion is a method in mathematical logic that transforms first-order formulas into equivalent (often infinite) propositional combinations by systematically instantiating quantified variables with terms from the Herbrand universe.
D. Herbrand disjunction
Herbrand disjunction is a logical formula formed as a finite disjunction of ground instances of a first-order formula, central to Herbrand’s theorem in proof theory and automated reasoning.
E. Hilbert’s program
Hilbert’s program was an influential early-20th-century initiative in the foundations of mathematics that sought to formalize all of mathematics and prove its consistency using finitistic methods.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Herbrand interpretation
Target entity description: A Herbrand interpretation is a foundational model-theoretic construct in logic and automated theorem proving that interprets formulas over the Herbrand universe built from a theory’s own function symbols and constants.

A. Herbrand universe
The Herbrand universe is a fundamental concept in mathematical logic and automated theorem proving, consisting of all ground (variable-free) terms that can be built from the function symbols and constants of a given first-order language.
B. Herbrand's theorem
Herbrand's theorem is a fundamental result in mathematical logic and proof theory that characterizes the validity of first-order formulas via finite sets of ground instances, forming a basis for automated theorem proving.
C. Herbrand expansion
Herbrand expansion is a method in mathematical logic that transforms first-order formulas into equivalent (often infinite) propositional combinations by systematically instantiating quantified variables with terms from the Herbrand universe.
D. Herbrand disjunction
Herbrand disjunction is a logical formula formed as a finite disjunction of ground instances of a first-order formula, central to Herbrand’s theorem in proof theory and automated reasoning.
E. Hilbert’s program
Hilbert’s program was an influential early-20th-century initiative in the foundations of mathematics that sought to formalize all of mathematics and prove its consistency using finitistic methods.
F. None of above. chosen

Statements (36)

Predicate	Object
instanceOf	model-theoretic construct ⓘ semantic notion in logic ⓘ
appliesTo	Skolemized formulas ⓘ clause sets ⓘ
associatesTruthValuesWith	ground atomic formulas ⓘ
assumes	standard syntactic signature of a theory ⓘ
basedOn	Herbrand base ⓘ Herbrand universe ⓘ
characterizedAs	interpretation determined by truth values of ground atoms ⓘ interpretation with canonical domain of ground terms ⓘ
constrains	interpretation of constants to be themselves as ground terms ⓘ interpretation of function symbols to be term-forming operations ⓘ
contrastedWith	arbitrary first-order structures ⓘ
domainOfInterpretation	Herbrand universe ⓘ
enables	reduction of satisfiability to ground instances ⓘ search procedures over ground clauses ⓘ
formalizedIn	model theory ⓘ
interprets	ground atoms ⓘ ground terms ⓘ
interpretsOver	constant symbols of a theory ⓘ function symbols of a theory ⓘ
namedAfter	Jacques Herbrand ⓘ
relatedTo	Herbrand model ⓘ linked to: Herbrand interpretation Herbrand semantics ⓘ
roleIn	Herbrand’s theorem ⓘ linked to: Herbrand's theorem completeness results for first-order logic ⓘ
truthAssignmentType	two-valued truth assignment ⓘ
usedBy	Prolog ⓘ logic programming semantics ⓘ resolution calculus ⓘ tableaux methods ⓘ
usedFor	decidability investigations of fragments of first-order logic ⓘ soundness and completeness proofs of automated deduction calculi ⓘ
usedIn	automated theorem proving ⓘ first-order logic ⓘ proof theory ⓘ

How these facts were elicited