Proof theory · Definition

Modus ponens

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Before you start you need to know

From \(P\Rightarrow Q\) and \(P\), modus ponens allows concluding \(Q\).

Definition

Let \(P\) and \(Q\) be propositions. Modus ponens allows concluding \(Q\) from \(P\Rightarrow Q\) and \(P\).

\[\frac{P\Rightarrow Q,\quad P}{Q}\]

Modus ponens, also called "rule of detachment", allows concluding \(Q\) from \(P\Rightarrow Q\) and \(P\).

The rule uses the affirmation of the antecedent together with the implication. It must not be confused with affirming the consequent: from \(P\Rightarrow Q\) and \(Q\), \(P\) does not follow in general.

Its short form makes it one of the first schemes used to build formal proofs.

Think about it

What is missing to apply modus ponens to \(P\Rightarrow Q\)?

The premise \(P\); with it, \(Q\) can be concluded.

Exercises 11

  1. If P ⇒ Q and P are true, modus ponens allows concluding...
  2. Which is the correct form of modus ponens?
  3. Modus ponens needs as a premise an...
  4. From P ⇒ Q and ¬P, modus ponens allows concluding...
  5. Modus ponens is a rule of...
  6. The scheme of modus ponens is...
  7. If 'if it rains then I carry an umbrella' and it rains, you conclude...
  8. In modus ponens, the second premise asserts...
  9. Select the valid application of modus ponens.
  10. If P⇒Q and P are true, then Q is...
  11. Given \(P\Rightarrow Q\), \(Q\Rightarrow R\) and \(P\), applying modus ponens twice one concludes...

References

  • Frederic P. Schuller, Lectures on the Geometrical Anatomy of Theoretical Physics · Lecture 1, Introduction / Logic of Propositions and Predicates, YouTube, 2015.