nLab contrapositive

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

logicset theory (internal logic of)category theorytype theory
propositionsetobjecttype
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
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
truesingletonterminal object/(-2)-truncated objecth-level 0-type/unit type
falseempty setinitial objectempty type
proposition, truth valuesubsingletonsubterminal object/(-1)-truncated objecth-proposition, mere proposition
logical conjunctioncartesian productproductproduct type
disjunctiondisjoint union (support of)coproduct ((-1)-truncation of)sum type (bracket type of)
implicationfunction setinternal homfunction type
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian productdependent productdependent product type
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalenceset of bijectionsobject of isomorphisms/adjoint equivalencesequivalence type
support setsupport object/(-1)-truncationpropositional truncation/bracket type
n-image of morphism into terminal object/n-truncationn-truncation modality
equalitydiagonal function/diagonal subset/diagonal relationpath space objectidentity type/path type
completely presented setsetdiscrete object/0-truncated objecth-level 2-type/set/h-set
setset with equivalence relationinternal 0-groupoidBishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
equivalence class/quotient setquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
set of truth valuessubobject classifiertype of propositions
domain of discourseuniverseobject classifiertype universe
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

semantics

(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 15:19:33. See the history of this page for a list of all contributions to it.