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.
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 ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.