Logic - Modus Ponens

Modus Ponens is latin for “mode that affirms”

     p→q
     p
    _____
   ∴ q

Modus ponens is the rule of reasoning we all use a hundred times a day without naming it. The truth: it is the engine that turns an “if” into a “therefore”.

The core idea, in plain words

If you know a conditional “if $p$ then $q$ ” is true, and you know $p$ is true, you may conclude $q$ .

\frac{p \to q \qquad p}{\therefore\ q}

It is valid: there is no possible situation where both premises are true and the conclusion false. Check the truth table. The only row with $p$ true and $p \to q$ true is the row where $q$ is true.

$p$	$q$	$p \to q$
T	T	T ← both premises true, $q$ true
T	F	F
F	T	T
F	F	T

An everyday analogy

“If it’s raining, the pavement is wet. It’s raining. So the pavement is wet.” You didn’t need to look outside at the pavement; the conditional plus the fact did the work.

A small worked example

If $n$ is divisible by 4, then $n$ is even. (True for every integer.)
$28$ is divisible by 4.
∴ $28$ is even.

Proofs are chains of exactly this: each step applies a known “if” to a known fact.

A picture in words

Think of $p \to q$ as a one-way door from room $p$ to room $q$ . Modus ponens: you’re standing in room $p$ , so you can walk into $q$ . Nothing about the door lets you walk backwards.

graph LR
  P["p is true"] -- "p → q" --> Q["so q is true"]

The impostor: affirming the consequent

\frac{p \to q \qquad q}{\therefore\ p} \quad \text{(INVALID)}

“If it’s raining, the pavement is wet. The pavement is wet. So it’s raining.” Maybe someone washed their car. In the truth table, row 3 ( $p$ false, $q$ true) has both premises true and the conclusion false. One counter-example row is enough to kill an argument form.

Common misconceptions

“Valid means true.” Validity is about form. “If the moon is cheese, I’m a pilot. The moon is cheese. So I’m a pilot” is perfectly valid, but unsound, because a premise is false. Sound = valid + true premises.
" $p \to q$ means $p$ causes $q$ ." Material implication only rules out “ $p$ true and $q$ false”. It says nothing about causes.
“If $p$ is false, the conditional is false.” No: a conditional with a false antecedent is (vacuously) true. That’s rows 3 and 4.

How it connects

Its mirror image is Modus Tollens: from $p \to q$ and $\neg q$ , conclude $\neg p$ .
Chaining conditionals ( $p \to q$ , $q \to r$ , so $p \to r$ ) is the hypothetical syllogism, i.e. transitivity of implication.
In formal systems, modus ponens is often the only inference rule; everything else is axioms. See Logic.

The truth revealed: “if” plus “it is so” gives “therefore”. And “therefore” never flows backwards.