Mathwords logoMathwords

Modus Ponens — Definition, Rule & Examples

Modus Ponens

A logical argument of this form:

If pp then q.q\text{.}
pp is true.
Therefore qq is true.

Example: If it is raining then the grass is wet.
It is raining.
Therefore the grass is wet.

See also

Modus tolens

Key Formula

P⇒Q,P∴QP \Rightarrow Q,\quad P \quad \therefore \quad Q
Where:
  • PP = A proposition (the hypothesis of the conditional)
  • QQ = A proposition (the conclusion of the conditional)
  • ⇒\Rightarrow = The conditional connective, read 'implies'
  • ∴\therefore = The symbol meaning 'therefore'

Example

Problem: Given these two premises, determine what conclusion follows: Premise 1: If a number is divisible by 6, then it is divisible by 3. Premise 2: 18 is divisible by 6.
Step 1: Identify the conditional statement and match it to the form P ⇒ Q.
P:a number is divisible by 6,Q:it is divisible by 3P: \text{a number is divisible by 6},\quad Q: \text{it is divisible by 3}
Step 2: Check whether premise 2 affirms the hypothesis P. Since 18 is divisible by 6, we know P is true for the number 18.
P is true (18 ÷ 6 = 3)P \text{ is true (18 ÷ 6 = 3)}
Step 3: Apply Modus Ponens: because P ⇒ Q is true and P is true, Q must be true.
P⇒Q,P∴QP \Rightarrow Q,\quad P \quad \therefore \quad Q
Answer: Therefore, 18 is divisible by 3.

Why It Matters

Modus Ponens is one of the most fundamental rules of deduction and underpins nearly every logical proof. Whenever you write a chain of reasoning in a geometry proof or an algebraic argument, you are often applying Modus Ponens—using a known theorem (if P then Q) together with a verified condition (P) to reach a conclusion (Q). Understanding this rule helps you construct valid arguments and recognize when someone else's reasoning is logically sound.

Common Mistakes

Mistake: Affirming the consequent: concluding P from 'P ⇒ Q' and Q. For example, arguing 'If it rains the ground is wet; the ground is wet; therefore it rained.'
Correction: Modus Ponens requires you to affirm the hypothesis P, not the conclusion Q. Knowing Q is true does not let you conclude P, because Q could be true for other reasons.

Related Terms