Herbrand interpretation
E238813
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.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Herbrand interpretation canonical | 3 |
| Herbrand model | 1 |
| Herbrand models | 1 |
| Herbrand structure | 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.
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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.
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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.
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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.
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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 interpretation
self-linksurface differs
ⓘ
surface form:
Herbrand model
Herbrand semantics ⓘ |
| roleIn |
Herbrand's theorem
ⓘ
surface form:
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
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Herbrand interpretation Description of subject: 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.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Herbrand structure
this entity surface form:
Herbrand models
this entity surface form:
Herbrand model