Textbook
The negation $\neg P$ is true when $P$ is false and false when $P$ is true. In classical logic, double negation restores the original value: $\neg\neg P\Leftrightarrow P$.
For a real number $x$, the negation of $x>3$ is $x\le3$, not $x<3$. Equality belongs to the negation because the original condition excluded it. Negating a compound assertion requires respecting its full scope; parentheses show which expression the symbol $\neg$ changes.
In classical two-valued logic, a proposition and its negation cannot both be true or both be false. For a numerical condition, the two must cover the declared domain without overlap. Replacing $x>3$ with $x<3$ fails this test: $x=3$ belongs to neither condition. The domain matters because negation is assessed within that domain.
Definitions and further reading: Mathematics for Computer Science.
In plain language
Negation changes true to false and false to true, so “not older than three” includes exactly three. Admissible cases fall into two baskets: those passing the original check and those failing it. A boundary value stranded between the baskets signals a sorting mistake rather than a special kind of negation. The model assumes a definite truth value; “not measured yet” does not automatically mean mathematically false.
Example
The example uses $P(x):x>3$. The original predicate is false at 2 and 3 and true at 4. Negation must reverse all three answers.
Negation: Reversing a Truth Value
Original condition
Only 4 passes the threshold.
Step 1 of 3: Original condition
Read all steps
Original condition
Only 4 passes the threshold.
- State
- 2: falsecurrent3: falsecurrent4: truecurrent
Negation
Now 2 and the boundary value 3 pass.
- State
- 2: truecurrent3: truecurrent4: falsecurrent
Double negation
Switching again restores the original answers.
- State
- 2: falsecurrent3: falsecurrent4: truecurrent
After the first reversal, the successful values 2 and 3 must be exactly those rejected by the original condition. Checking 3 separately matters: values far from the boundary would not reveal the incorrect inequality sign. The second reversal restores the complete original pattern, not just the number of successful checks. Matching the number of ones is insufficient if they belong to different inputs.
Related concepts: Implication.