1. Idea of Negation
The negation of a statement is a new statement that expresses the opposite meaning. Negation is used to reverse the truth value: if a statement is true, its negation is false, and if the statement is false, its negation is true.
Negation allows us to understand what it means for a claim to fail. In reasoning, negations are essential when disproving statements or forming logical complements.
1.1. Why negation is useful
Many arguments rely on knowing not only when something is true, but also when it is not true. Negation provides a precise way to express the "opposite" of any given statement.
2. Formal Definition
The negation of a statement \(P\) is a new statement, written as not P (or \(\neg P\)), which is true exactly when \(P\) is false, and false when \(P\) is true.
2.1. Truth reversal
If:
- \(P\) is true → \(\neg P\) is false
- \(P\) is false → \(\neg P\) is true
3. Negation of Simple Statements
For simple declarative statements, negation usually begins with words like "It is not the case that…" or a direct opposite form.
3.1. Examples
- Statement: "7 is an even number."
Negation: "7 is not an even number." - Statement: "The sun rises in the east."
Negation: "The sun does not rise in the east."
4. Negation of Quantified Statements
Statements involving "for all" or "there exists" require special care when negating. The quantifiers switch roles during negation.
4.1. Universal statements
Negation switches:
"For all x, P(x)" → "There exists an x such that P(x) is false."
Example:
- Statement: "Every natural number is positive."
- Negation: "There exists a natural number that is not positive."
4.2. Existential statements
Negation switches:
"There exists an x such that P(x)" → "For all x, P(x) is false."
Example:
- Statement: "There exists a number whose square is -1."
- Negation: "For all numbers, their square is not -1."
5. Negation of Compound Statements
Negating compound statements requires using logical patterns. Instead of memorizing truth tables, it is easier to understand the meaning of each connector.
5.1. Negation of AND (Conjunction)
Statement: "P and Q"
Negation: "P is false or Q is false"
Meaning: both conditions cannot be true together.
5.2. Negation of OR (Disjunction)
Statement: "P or Q"
Negation: "P is false and Q is false"
Meaning: neither condition holds.
5.3. Negation of Implication (If P then Q)
Statement: "If P, then Q"
Negation: "P is true and Q is false"
Meaning: for the implication to fail, the first part must happen while the second part does not.
5.4. Negation of Bi-conditional (P if and only if Q)
Statement: "P if and only if Q"
Negation: "P and Q have different truth values"
Meaning: one is true, the other is false.
6. Worked Examples
The following examples show how negations are formed in common situations.
6.1. Example 1: Simple sentence
Statement: "10 is greater than 20."
Negation: "10 is not greater than 20."
6.2. Example 2: Universal statement
Statement: "Every even number is divisible by 4."
Negation: "There exists an even number that is not divisible by 4."
6.3. Example 3: Compound statement
Statement: "A number is positive and even."
Negation: "The number is not positive or it is not even."
7. Notes and Observations
Some important points to remember:
- Negation reverses the truth of a statement.
- Negating quantified statements requires switching the quantifiers.
- Compound statements follow specific patterns during negation.
- Adding "not" blindly is not always correct; structure must be considered.