Herbrand base
E822884
The Herbrand base is the set of all ground (variable-free) atomic formulas that can be formed from the predicate and constant symbols of a first-order language, serving as the foundational domain for Herbrand semantics and automated theorem proving.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Herbrand base canonical | 4 |
How this entity was disambiguated
This entity first appeared as the object of triple T9809578 — 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 base Context triple: [Herbrand's theorem, usesConcept, Herbrand base]
-
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 function
The Herbrand function is a numerical tool in local class field theory that measures the ramification filtration of Galois groups, playing a key role in understanding how ramification behaves in extensions of local fields.
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C.
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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D.
Herbrand interpretation
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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E.
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.
- 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 base Target entity description: The Herbrand base is the set of all ground (variable-free) atomic formulas that can be formed from the predicate and constant symbols of a first-order language, serving as the foundational domain for Herbrand semantics and automated theorem proving.
-
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 function
The Herbrand function is a numerical tool in local class field theory that measures the ramification filtration of Galois groups, playing a key role in understanding how ramification behaves in extensions of local fields.
-
C.
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.
-
D.
Herbrand interpretation
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.
-
E.
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.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
concept in automated theorem proving
ⓘ
concept in mathematical logic ⓘ concept in model theory ⓘ |
| appearsIn |
Skolemized form of first-order theories
ⓘ
first-order clause logic ⓘ |
| assumes | fixed first-order signature ⓘ |
| componentOf |
Herbrand structure
ⓘ
linked to:
Herbrand interpretation
|
| constructedFrom |
Herbrand universe
ⓘ
predicate symbols applied to ground terms ⓘ |
| contextOfUse |
Horn clause logic
ⓘ
classical first-order logic ⓘ logic programming semantics ⓘ |
| contrastedWith |
non-ground atoms containing variables
ⓘ
set of all formulas of the language ⓘ |
| definedOver | first-order language ⓘ |
| dependsOn |
constant symbols of the language
ⓘ
function symbols of the language ⓘ predicate symbols of the language ⓘ |
| enables | reduction of first-order satisfiability to propositional satisfiability over ground instances ⓘ |
| field |
automated reasoning
ⓘ
first-order logic ⓘ mathematical logic ⓘ |
| formalDefinition | set of all ground atomic formulas over the signature of a first-order language ⓘ |
| hasElementType | ground atomic formula ⓘ |
| hasProperty |
contains all ground atomic formulas that can be formed from the language symbols
ⓘ
contains only variable-free formulas ⓘ is countable when the language signature is countable ⓘ is finite when the language has only finitely many constants and no function symbols of positive arity ⓘ is generally infinite when the language has function symbols of positive arity ⓘ is uniquely determined by the language signature ⓘ |
| namedAfter | Jacques Herbrand ⓘ |
| relatedConcept |
Herbrand interpretation
ⓘ
Herbrand model ⓘ Herbrand theorem ⓘ
linked to:
Herbrand's theorem
Herbrand universe ⓘ ground atom ⓘ |
| roleInProofTheory |
basis for ground instantiations of clauses
ⓘ
underlies resolution-based theorem proving ⓘ |
| roleInSemantics |
provides canonical set of atoms for defining models of clauses
ⓘ
serves as domain of discourse for Herbrand interpretations ⓘ |
| typicalNotation |
B_L
ⓘ
HB ⓘ |
| usedBy |
SLD-resolution in logic programming
ⓘ
resolution calculus ⓘ tableaux methods ⓘ |
| usedIn |
Herbrand semantics
ⓘ
automated theorem proving ⓘ logic programming ⓘ model checking for first-order theories ⓘ |
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 base Description of subject: The Herbrand base is the set of all ground (variable-free) atomic formulas that can be formed from the predicate and constant symbols of a first-order language, serving as the foundational domain for Herbrand semantics and automated theorem proving.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.