propositions as projections



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


Linear algebra

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts




In intuitionistic type theory the perspective of propositions as types identifies mere propositions with (speaking in categorical semantics) monomorphisms into the given context type. If one instead considers linear type theory then it makes sense to consider those propositions, in this sense, which come from monomorphisms that are split monomorphisms, hence those for which there is a projection from the context to the subobject that represents the proposition. For these one might speak of “propositions as projections”.

For instance in a category of Hilbert spaces, regarded as semantics for bare multiplicative linear type theory, then projections correspond to (closed) linear subspaces. These are just the propositions in the corresponding quantum logic. If we regard a proposition as its corresponding projection, then it is natural to consider it also as being the application of this projection, in that the truth of the proposition is thought of as the image of applying this projection. In terms of quantum physics this relation between propositions (about a quantum mechanical system) and the projection onto the corresponding subspace of the space of quantum states is what is referred to as the “collapse of the wave function”.


  • Kurt Engesser, Dov Gabbay, Daniel Lehmann, A New Approach to Quantum Logic, vol. 8 of Studies in Logic, College Publications, London, UK, 2007. (pdf)

Last revised on May 13, 2021 at 16:09:35. See the history of this page for a list of all contributions to it.