nLab modus ponens

Idea

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.)

Statements

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

$$\vdash \psi$$

and

$$\vdash \psi \to \phi$$

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

$$\vdash \phi$$

(asserting the judgement that the proposition $\phi$ 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

$$\Gamma \vdash \Delta,\; \psi,\; \Theta$$

and

$$\Gamma \vdash \Delta,\; (\psi \to \phi),\; \Theta ,$$

we may deduce

$$\Gamma \vdash \Delta,\; \phi,\; \Theta ;$$

this rule may be summarized as

$$\frac{ \Gamma \vdash \Delta,\; \psi,\; \Theta ;\;\;\; \Gamma \vdash \Delta,\; (\psi \to \phi),\; \Theta } { \Gamma \vdash \Delta,\; \phi,\; \Theta } .$$

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

$$\frac{ \Gamma \vdash \Delta,\; \psi,\; (\psi \multimap \phi),\; \Theta } { \Gamma \vdash \Delta,\; \phi,\; \Theta } .$$

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

$$\frac{ \Gamma \vdash \Delta,\; (\psi \multimap^{op} \phi),\; \psi,\; \Theta } { \Gamma \vdash \Delta,\; \phi,\; \Theta } .$$

Properties

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

$$\psi,\; (\psi \to \phi) \;\vdash \; \phi$$

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