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


(0,1)(0,1)-Category theory

In propositional logic, the contrapositive rule states that it is valid to derive ¬Q¬P\neg{Q} \to \neg{P} from PQP \to Q (where ¬\neg is negation and \to is implication). In symbols:

PQ¬Q¬PCP \frac {P \to Q} {\neg{Q} \to \neg{P}} \;CP

The combination of this rule, followed by modus ponens (the elimination rule for implication) was traditionally called modus tollens:

PQ¬Q¬PCP¬Q¬P \frac {\displaystyle\frac{P \to Q} {\neg{Q} \to \neg{P}} \;CP \;\;\; \neg{Q}} {\neg{P}} \; \to\mathcal{E}

The contrapositive rule is valid in intuitionistic logic, not just in classical logic; however, the reverse rule is valid only in classical logic.

Another intuitionistically valid rule, this one reversible, is

P¬QQ¬P \frac {P \to \neg{Q}} {Q \to \neg{P}}

as both statements are equivalent to the negation of PQP \wedge Q (where \wedge is conjunction). However, the superficially similar

¬PQ¬QP \frac {\neg{P} \to Q} {\neg{Q} \to P}

is again valid only classically.

Last revised on April 23, 2017 at 11:19:33. See the history of this page for a list of all contributions to it.