Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty type
proposition(-1)-truncated objecth-proposition, mere proposition
proofgeneralized elementprogram
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
logical conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type/path type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
completely presented setdiscrete object/0-truncated objecth-level 2-type/preset/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
modalityclosure operator, (idempotent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels




A system in formal logic is called inconsistent if it admits a proof of a contradiction (that is, usually, a proof of false, or an inhabitant of the empty type).

Accordingly an axiom is called inconsistent or to lead to an inconsistency if adding it to an (implicitly understood) ambient logical system makes that system inconsistent.

In most usual logical systems, it follows that an inconsistent system admits a proof of every proposition, by the rule ex falso quodlibet (which is just the elimination rule for the empty type). For this reason, sometimes (especially in type theory), the adjective “inconsistent” is used to mean a system with this property instead. If we want to distinguish, then a system which admits a proof of every proposition may be called trivial.

As a further complication, a paraconsistent logic is often described as ‘inconsistent but not trivial’. However, many paraconsistent logics (such as dual-intuitionistic logic?) admit ex falso quadlibet and fail to prove \bot; their ‘inconsistency’ is a proof of p¬pp \wedge \neg{p} (or two proofs, one of pp and one of ¬p\neg{p}), which then fails to entail \bot. We thus get three distinct levels of inconsistency: * triviality (proving everything), * proving a statement identified as \bot, * proving pp and ¬p\neg{p} for some statement pp and using an operator identified as negation.


Last revised on September 7, 2018 at 01:58:06. See the history of this page for a list of all contributions to it.