Propositional logic · Definition

Implication

logicpropositional

Before you start you need to know

Implication is the connective that expresses a condition between two propositions: if the first holds, the second must hold.

Definition

The material implication \(P\Rightarrow Q\) is false exactly when \(P\) is true and \(Q\) is false.

\[P\Rightarrow Q\]

The implication \(P\Rightarrow Q\) expresses a condition: if \(P\) is true, then \(Q\) must be true. \(P\) is called the antecedent and \(Q\) the consequent.

In classical propositional logic, the implication is false only when \(P\) is true and \(Q\) is false. If \(P\) is false, the implication turns out to be true under this material definition.

Guide example: if \(P\) is «it rains» and \(Q\) is «the street gets wet», with \(P\) true and \(Q\) false the implication is false. Typical error: believing that an implication with a false antecedent is false.

Think about it

Which case directly refutes \(P\Rightarrow Q\)?

The case where \(P\) is true and \(Q\) is false.

Exercises 10

  1. P ⇒ Q is false only when...
  2. If P and Q are true, P ⇒ Q is...
  3. If P is false and Q is true, P ⇒ Q is...
  4. Which symbol represents implication?
  5. In the sentence "If it rains, then the street gets wet", P represents...
  6. P⇒Q is false only when...
  7. In P⇒Q, P is called...
  8. The material implication P⇒Q is equivalent to...
  9. Select the only case in which P⇒Q is false.
  10. The contrapositive of P⇒Q is...

What comes next

References

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