Herbrand expansion
E238236
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Herbrand expansion canonical | 3 |
How this entity was disambiguated
This entity first appeared as the object of triple T2139571 — 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 expansion Context triple: [Jacques Herbrand, knownFor, Herbrand expansion]
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A.
Knuth–Bendix completion algorithm
The Knuth–Bendix completion algorithm is a procedure in term rewriting and automated theorem proving that transforms a set of equations into a confluent rewriting system, enabling decision of word problems in algebraic structures.
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B.
completeness theorem for first-order logic
The completeness theorem for first-order logic is a fundamental result in mathematical logic, proved by Kurt Gödel, which states that every logically valid first-order formula is provable from the axioms of first-order logic.
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C.
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.
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D.
Entscheidungsproblem
The Entscheidungsproblem is a foundational decision problem in mathematical logic that asks whether there exists a general algorithm to determine the truth or falsity of any given first-order logical statement.
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E.
Church–Rosser property
The Church–Rosser property is a confluence property of rewriting systems stating that if an expression can be reduced in different ways, all reduction paths can be further reduced to a common equivalent form.
- 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 expansion Target entity description: 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.
-
A.
Knuth–Bendix completion algorithm
The Knuth–Bendix completion algorithm is a procedure in term rewriting and automated theorem proving that transforms a set of equations into a confluent rewriting system, enabling decision of word problems in algebraic structures.
-
B.
completeness theorem for first-order logic
The completeness theorem for first-order logic is a fundamental result in mathematical logic, proved by Kurt Gödel, which states that every logically valid first-order formula is provable from the axioms of first-order logic.
-
C.
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.
-
D.
Entscheidungsproblem
The Entscheidungsproblem is a foundational decision problem in mathematical logic that asks whether there exists a general algorithm to determine the truth or falsity of any given first-order logical statement.
-
E.
Church–Rosser property
The Church–Rosser property is a confluence property of rewriting systems stating that if an expression can be reduced in different ways, all reduction paths can be further reduced to a common equivalent form.
- F. None of above. chosen
Statements (42)
| Predicate | Object |
|---|---|
| instanceOf |
method in mathematical logic
ⓘ
technique in proof theory ⓘ |
| appliesTo |
Skolemized formulas
ⓘ
prenex form formulas ⓘ |
| assumes | classical first-order logic semantics ⓘ |
| basedOn | syntactic generation of ground instances ⓘ |
| canBe |
countably infinite for general first-order theories
ⓘ
finite for some formulas ⓘ |
| closelyRelatedTo |
Herbrand disjunction
ⓘ
Herbrand interpretation ⓘ |
| componentOf | Herbrand’s approach to first-order validity ⓘ |
| contrastsWith | semantic methods using structures directly ⓘ |
| field |
mathematical logic
ⓘ
proof theory ⓘ |
| goal |
eliminate quantifiers by instantiation
ⓘ
reduce first-order validity to propositional validity ⓘ |
| historicalContext | introduced in early 20th century ⓘ |
| implies | existence of a finite unsatisfiable subset for unsatisfiable formulas ⓘ |
| inputType | first-order formula ⓘ |
| namedAfter | Jacques Herbrand ⓘ |
| outputType | propositional combination of ground instances ⓘ |
| property |
logically equivalent to original first-order formula
ⓘ
often infinite expansion ⓘ |
| relatedTo |
Herbrand base
ⓘ
Herbrand's theorem ⓘ
surface form:
Herbrand theorem
Herbrand universe ⓘ Skolemization ⓘ first-order logic ⓘ propositional logic ⓘ |
| requires | enumeration of Herbrand universe terms ⓘ |
| step |
construct Herbrand universe from function and constant symbols
ⓘ
form disjunctions and conjunctions of ground instances ⓘ generate ground terms from Herbrand universe ⓘ instantiate quantified variables with ground terms ⓘ |
| usedFor |
reducing first-order entailment to propositional entailment
ⓘ
showing completeness of certain proof calculi ⓘ |
| usedIn |
automated theorem proving
ⓘ
resolution-based proof methods ⓘ satisfiability testing for first-order formulas ⓘ |
| uses |
Herbrand universe
ⓘ
ground terms ⓘ instantiation of quantified variables ⓘ |
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 expansion Description of subject: 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.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.