nLab modus ponens





The basis of it all

 Set theory

set theory

Foundational axioms

foundational axioms

Removing axioms

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
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
introduction rule for implicationcounit for hom-tensor adjunctionlambda
elimination rule for implicationunit for hom-tensor adjunctionapplication
cut elimination for implicationone of the zigzag identities for hom-tensor adjunctionbeta reduction
identity elimination for implicationthe other zigzag identity 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 set (into subsingleton)internal hom (into subterminal object)function type (into h-proposition)
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian product (of family of subsingletons)dependent product (of family of subterminal objects)dependent product type (of family of h-propositions)
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijection setobject of isomorphismsequivalence 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




In formal logic, modus ponens is the elimination rule for the logical connective \to (forming conditional statements).

On more general types in type theory this is function application.

Even in Hilbert-style logic where there are almost no rules of inference (besides the many axioms), there is usually a rule of modus ponens. (In particular, modus ponens is the only non-axiom rule of inference in Hilbert's version of the propositional calculus.)


Modus ponens is the rule of inference which says that from the sequents

ψ \vdash \psi


ψϕ \vdash \psi \to \phi

asserting (respectively) the judgement that a proposition ψ\psi is true and the judgement that the conditional statement ψϕ\psi \to \phi is also true, the sequent

ϕ \vdash \phi

(asserting the judgement that the proposition ϕ\phi is true) may be deduced.

Depending on what sort of sequents are allowed in the sequent calculus that one is working with, there may be additional context on either side of the sequents. So rather generally, from

ΓΔ,ψ,Θ \Gamma \vdash \Delta,\; \psi,\; \Theta


ΓΔ,(ψϕ),Θ, \Gamma \vdash \Delta,\; (\psi \to \phi),\; \Theta ,

we may deduce

ΓΔ,ϕ,Θ; \Gamma \vdash \Delta,\; \phi,\; \Theta ;

this rule may be summarized as

ΓΔ,ψ,Θ;ΓΔ,(ψϕ),ΘΓΔ,ϕ,Θ. \frac{ \Gamma \vdash \Delta,\; \psi,\; \Theta ;\;\;\; \Gamma \vdash \Delta,\; (\psi \to \phi),\; \Theta } { \Gamma \vdash \Delta,\; \phi,\; \Theta } .

In linear logic, we must distinguish between the additive conditional \to, whose elimination rule is as above, and the multiplicative conditional \multimap, whose elimination rule is

ΓΔ,ψ,(ψϕ),ΘΓΔ,ϕ,Θ. \frac{ \Gamma \vdash \Delta,\; \psi,\; (\psi \multimap \phi),\; \Theta } { \Gamma \vdash \Delta,\; \phi,\; \Theta } .

In noncommutative logic?, we would further distinguish this last version from

ΓΔ,(ψ opϕ),ψ,ΘΓΔ,ϕ,Θ. \frac{ \Gamma \vdash \Delta,\; (\psi \multimap^{op} \phi),\; \psi,\; \Theta } { \Gamma \vdash \Delta,\; \phi,\; \Theta } .


The converse to modus ponens, which is the introduction rule for conditional statements, is much less commonly admitted directly as a rule of inference, but its validity is typically a theorem, the deduction theorem; see also the discussion at metalanguage.

The categorical semantics of modus ponens in the form

ψ,(ψϕ)ϕ \psi,\; (\psi \to \phi) \;\vdash \; \phi

is the evaluation map of the ambient (locally) cartesian closed category.

Last revised on January 26, 2014 at 21:23:33. See the history of this page for a list of all contributions to it.