modus ponens




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
presettype without identity types
completely presented setdiscrete object/0-truncated objecth-level 2-type/set/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




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.