Herbrand semantics
E822886
Herbrand semantics is a formal framework in logic and automated theorem proving that interprets first-order formulas over the Herbrand universe of ground terms to define truth and satisfiability.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Herbrand semantics canonical | 3 |
How this entity was disambiguated
This entity first appeared as the object of triple T9809623 — 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 semantics Context triple: [Herbrand universe, usedIn, Herbrand semantics]
-
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 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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C.
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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D.
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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E.
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.
- 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 semantics Target entity description: Herbrand semantics is a formal framework in logic and automated theorem proving that interprets first-order formulas over the Herbrand universe of ground terms to define truth and satisfiability.
-
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 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.
-
C.
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.
-
D.
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.
-
E.
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.
- F. None of above. chosen
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
framework in automated theorem proving
ⓘ
semantics in mathematical logic ⓘ |
| advantage |
avoids arbitrary non-denumerable domains by using syntactic terms as domain elements
ⓘ
reduces semantic reasoning to combinatorics over ground atoms ⓘ |
| appliesTo | first-order logic ⓘ |
| assumes |
fixed function symbols and constants of the underlying language
ⓘ
standard syntactic formation rules of first-order logic ⓘ |
| basedOn |
Herbrand base
ⓘ
Herbrand universe ⓘ |
| characteristicFeature |
restricts interpretations to ground terms built from the language’s function symbols and constants
ⓘ
treats predicate symbols as the only non-fixed part of interpretations ⓘ |
| clarifies | relationship between syntactic derivations and semantic models in first-order logic ⓘ |
| concerns | truth of ground atoms in Herbrand interpretations ⓘ |
| defines |
satisfiability for first-order formulas over the Herbrand universe
ⓘ
truth for first-order formulas over the Herbrand universe ⓘ |
| field |
mathematical logic
ⓘ
theoretical computer science ⓘ |
| formalObject |
Herbrand structure
ⓘ
linked to:
Herbrand interpretation
|
| historicalContext | developed in the context of early 20th-century proof theory ⓘ |
| implies | a formula is satisfiable iff it has a Herbrand model (under suitable conditions) ⓘ |
| interprets | first-order formulas ⓘ |
| namedAfter | Jacques Herbrand ⓘ |
| relatedConcept |
Herbrand interpretation
ⓘ
Herbrand model ⓘ
linked to:
Herbrand interpretation
Herbrand’s theorem ⓘ
linked to:
Herbrand's theorem
ground instantiation of clauses ⓘ least Herbrand model ⓘ |
| relatedTo |
Tarskian semantics
ⓘ
model-theoretic semantics ⓘ |
| roleIn |
connecting syntactic derivability with semantic satisfiability
ⓘ
formalizing semantics of Horn clause programs ⓘ providing a basis for completeness proofs in first-order logic ⓘ |
| supports |
construction of countermodels via ground instances
ⓘ
formal analysis of logic programs ⓘ |
| typicalUse |
analyzing satisfiability of clause sets via ground instances
ⓘ
reasoning about sets of clauses ⓘ |
| usedBy |
logic programming systems such as Prolog (at the semantic level)
ⓘ
resolution-based theorem provers ⓘ |
| usedIn |
automated theorem proving
ⓘ
logic programming ⓘ model theory for logic programming languages ⓘ proof theory ⓘ |
| usesDomain |
Herbrand universe
ⓘ
set of ground terms ⓘ |
| varies | interpretations of predicate symbols ⓘ |
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 semantics Description of subject: Herbrand semantics is a formal framework in logic and automated theorem proving that interprets first-order formulas over the Herbrand universe of ground terms to define truth and satisfiability.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.