nLab
adjoint logic

Contents

Context

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
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive 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, (idemponent) 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

semantics

Modalities, Closure and Reflection

Duality

Contents

Idea

Adjoint logic or adjoint type theory is formal logic or type theory which natively expresses adjunctions of modal operators, adjoint modalities.

References

  • Nick Benton, Philip Wadler, Linear logic, monads and the lambda calculus, In IEEE Symposium on Logic in Computer Science. IEEE Computer Society Press, 1996.

  • Jason Reed, A judgemental deconstruction of modal logic, 2009, (pdf)

  • Klaas Pruiksma, William Chargin, Frank Pfenning, and Jason Reed, Adjoint Logic, 2018, (pdf)

A framework for (homotopy-)type theoretic adjoint logic (modal type theory) is discussed, in various stages of generality, in

Review includes

Last revised on January 7, 2019 at 16:10:30. See the history of this page for a list of all contributions to it.