Double Negatives
A free Logic lesson from the “Statements and Negation” unit, with a worked example and practice problems including step-by-step solutions.
Negation reverses a truth value. Two consecutive negations of the same whole statement cancel in classical logic. Respect parentheses: negating a conjunction is different from merely negating one of its parts.
What you'll learn
- Simplify statements with two negations
- Explain the idea in plain English before using symbols
- Use examples, non-examples, or counterexamples to check the reasoning
Worked example
Problem. Simplify ¬¬p.
- Each negation reverses the truth value.
- Two reversals restore the original value.
Answer: p
Lesson preview: this page shows 5 practice items. The interactive activity contains 6 practice items and 6 quiz items (multiple choice). Counts include repeated prompts and variants.
Practice problems
1. Simplify ¬¬p.
Choices: p ∧ ¬p · p ∨ ¬p · p · ¬p
Show solution
- Each negation reverses the truth value.
- Two reversals restore the original value.
Answer: p
2. Simplify ¬¬¬p.
Choices: False for every p · ¬p · p · True for every p
Show solution
- Cancel two of the three negations; one remains.
Answer: ¬p
3. If p is false, then ¬¬p is true. Evaluate this claim.
Choices: False · True
Show solution
- For p=false, ¬p=true and ¬¬p=false.
Answer: False
4. Rewrite “It is not true that the lamp is not on.”
Choices: The lamp is off · The lamp may be on or off · No lamp exists · The lamp is on
Show solution
- The outer negation cancels the inner negation of the same claim.
Answer: The lamp is on
5. Simplify ¬¬(p ∧ q).
Choices: p ∨ q · ¬(p ∧ q) · p ∧ q · ¬p ∧ ¬q
Show solution
- The two negations apply to the entire parenthesized claim.
Answer: p ∧ q
Practice this interactively with instant feedback and an AI tutor.
Practice Double Negatives Take the free placement check