Brouwer–Heyting–Kolmogorov interpretation
E459568
The Brouwer–Heyting–Kolmogorov interpretation is a foundational explanation of intuitionistic logic that interprets logical connectives and proofs in terms of explicit constructions and algorithms rather than classical truth values.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Brouwer–Heyting–Kolmogorov interpretation canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T4597342 — 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: Brouwer–Heyting–Kolmogorov interpretation Context triple: [Luitzen Egbertus Jan Brouwer, notableFor, Brouwer–Heyting–Kolmogorov interpretation]
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A.
Elements of Intuitionism
Elements of Intuitionism is a foundational philosophical and logical treatise by Michael Dummett that systematically develops and defends intuitionistic logic and mathematics.
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B.
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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C.
Remarks on the Foundations of Mathematics
Remarks on the Foundations of Mathematics is a posthumously published collection of Ludwig Wittgenstein’s later writings that critically examines the nature of mathematical truth, proof, and practice from a philosophical and language-centered perspective.
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D.
Recherches sur la théorie de la démonstration
Recherches sur la théorie de la démonstration is Jacques Herbrand’s foundational work in mathematical logic, introducing key results in proof theory and what is now known as Herbrand’s theorem.
-
E.
Löb's theorem
Löb's theorem is a fundamental result in mathematical logic that characterizes when a sufficiently strong formal system can prove statements about its own provability, closely refining the insights of Gödel’s incompleteness theorems.
- 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: Brouwer–Heyting–Kolmogorov interpretation Target entity description: The Brouwer–Heyting–Kolmogorov interpretation is a foundational explanation of intuitionistic logic that interprets logical connectives and proofs in terms of explicit constructions and algorithms rather than classical truth values.
-
A.
Elements of Intuitionism
Elements of Intuitionism is a foundational philosophical and logical treatise by Michael Dummett that systematically develops and defends intuitionistic logic and mathematics.
-
B.
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.
-
C.
Remarks on the Foundations of Mathematics
Remarks on the Foundations of Mathematics is a posthumously published collection of Ludwig Wittgenstein’s later writings that critically examines the nature of mathematical truth, proof, and practice from a philosophical and language-centered perspective.
-
D.
Recherches sur la théorie de la démonstration
Recherches sur la théorie de la démonstration is Jacques Herbrand’s foundational work in mathematical logic, introducing key results in proof theory and what is now known as Herbrand’s theorem.
-
E.
Löb's theorem
Löb's theorem is a fundamental result in mathematical logic that characterizes when a sufficiently strong formal system can prove statements about its own provability, closely refining the insights of Gödel’s incompleteness theorems.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
constructive semantics
ⓘ
proof interpretation ⓘ semantics of intuitionistic logic ⓘ |
| aimsTo | explain intuitionistic logic in constructive terms ⓘ |
| appliesTo |
conjunction in intuitionistic logic
ⓘ
disjunction in intuitionistic logic ⓘ existential quantification in intuitionistic logic ⓘ implication in intuitionistic logic ⓘ negation in intuitionistic logic ⓘ universal quantification in intuitionistic logic ⓘ |
| basedOn | intuitionism ⓘ |
| characterizes |
proofs as constructions
ⓘ
truth as existence of a proof ⓘ |
| contrastsWith | classical truth‑value semantics ⓘ |
| describes | meaning of logical connectives in intuitionistic logic ⓘ |
| field |
constructive mathematics
ⓘ
intuitionistic logic NERFINISHED ⓘ mathematical logic ⓘ proof theory NERFINISHED ⓘ |
| historicalContributor |
Andrey Kolmogorov
NERFINISHED
ⓘ
Arend Heyting NERFINISHED ⓘ L. E. J. Brouwer NERFINISHED ⓘ |
| influenced | Curry–Howard correspondence NERFINISHED ⓘ |
| interprets |
A → B as a method transforming any construction of A into a construction of B
ⓘ
A ∧ B as a construction of A and a construction of B ⓘ A ∨ B as a construction of either A or B together with a tag ⓘ ¬A as a method transforming any construction of A into a contradiction ⓘ ∀x A(x) as a method producing for each x a construction of A(x) ⓘ ∃x A(x) as a witness x together with a construction of A(x) ⓘ |
| motivatedBy | Brouwer’s intuitionism NERFINISHED ⓘ |
| namedAfter |
Andrey Kolmogorov
NERFINISHED
ⓘ
Arend Heyting NERFINISHED ⓘ L. E. J. Brouwer NERFINISHED ⓘ |
| provides | operational meaning to intuitionistic proofs ⓘ |
| rejects | law of excluded middle as generally valid ⓘ |
| relatedTo |
constructive type theory
NERFINISHED
ⓘ
proofs‑as‑programs paradigm ⓘ realizability interpretation NERFINISHED ⓘ type theory ⓘ |
| semanticsType |
intensional semantics
ⓘ
proof‑theoretic semantics ⓘ |
| timePeriod | 20th century ⓘ |
| usedIn |
foundations of constructive mathematics
ⓘ
philosophy of mathematics ⓘ theoretical computer science ⓘ |
| usesConcept |
algorithms
ⓘ
explicit constructions ⓘ realizers of proofs ⓘ witnesses for existential statements ⓘ |
How these facts were elicited
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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: Brouwer–Heyting–Kolmogorov interpretation Description of subject: The Brouwer–Heyting–Kolmogorov interpretation is a foundational explanation of intuitionistic logic that interprets logical connectives and proofs in terms of explicit constructions and algorithms rather than classical truth values.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.