# Modus ponens

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Modus ponens ("mode of affirming") is a logical rule of inference based on conditional propositions. Also called affirming the antecedent, modus ponens involves affirming the sufficient condition of the conditional proposition in order to prove the necessary condition. Modus ponens is closely related to modus tollens ("mode of taking").

A modus ponens argument has the following form:

P1: If X, then Y. (i. e. Either not X or Y)
P2: X.
C: Therefore, Y.

For example:

P1: If my friends are coming over tonight, I will bake a cake. (i. e. Either my friends are not coming over tonight, or I will bake a cake.)
P2: My friends are coming over tonight.
C: Therefore, I will bake a cake.

Confusing the directionality of the if-then statement in a modus ponens argument results in the fallacy of affirming the consequent, represented by the following invalid syllogism:

P1: If my friends are coming over tonight, I will bake a cake.
P2: I will bake a cake.
C1: Therefore, my friends are coming over tonight.

As a rule of inference, modus ponens is represented by the following in propositional logic:

$X\rightarrow Y, X\right\$

## Nicod’s postulate and modus ponens

Jean Nicod demonstrated that all of propositional logic can be deduced from a single schema—Nicod’s postulate:
((p | (q | r)) | ((s | (s | s)) | ((u | u) | ((p | q) | (p | q)))))
by applying one inference rule—Nicod’s modus ponens:

1. p
2. (p | (q | r))
3. ∴ r