Propositional logic · Definition
Negation
Before you start you need to know
Given a proposition \(P\), its negation \(\neg P\) changes its truth value to the opposite. It is one of the basic operations of mathematical logic.
Definition
Given a proposition \(P\), its negation is the proposition \(\neg P\).
Negation changes the truth value of a proposition to its opposite: what is true becomes false and what is false becomes true.
The symbol \(\neg\) is placed in front of the proposition. If \(P\) is the statement it rains, then \(\neg P\) is it does not rain, which is true exactly when \(P\) is false.
Typical error: writing \(\neg(P\land Q)\) as \(\neg P\land\neg Q\). The negation of a conjunction is the disjunction of the negations: \(\neg(P\land Q)\equiv\neg P\lor\neg Q\).
Think about it
If a statement turns out to be false, what happens when it is negated?
Its negation turns out to be true.
Exercises 10
- The negation of a proposition...
- If P is true, then ¬P is...
- If P is false, then ¬P is...
- Which symbol represents negation?
- Double negation ¬¬P is logically equivalent to...
- If P is a proposition, what is ¬P?
- Double negation satisfies...
- If ¬P is true, then P is...
- P and ¬P always have truth values that are...
- Which expression represents the negation of P?
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.