1. Idea of the Connective NOT
The connective NOT is used to form the negation of a given statement. It expresses the opposite meaning of the original statement by reversing its truth value. If a statement asserts something, NOT asserts the opposite.
NOT is the simplest logical connective because it deals with only one statement and flips its truth.
1.1. Why NOT is important
Negation is used when describing what does not happen, what fails, or what contradicts a given statement. Many logical arguments depend on understanding the exact opposite of a claim.
2. Formal Meaning of NOT
If P is a statement, then:
\(\text{NOT } P\)
is a new statement that is true exactly when P is false, and false exactly when P is true.
2.1. Truth reversal
NOT transforms truth to falsehood and falsehood to truth. This is the core idea behind negation.
3. Truth Behaviour of NOT
Since NOT applies to a single statement, its truth behaviour is straightforward.
3.1. Intuitive truth patterns
- P true → NOT P is false
- P false → NOT P is true
4. Examples
The following examples show how NOT is used in everyday language and mathematics.
4.1. Everyday examples
- Statement: "The light is on."
NOT form: "The light is not on." - Statement: "The bottle is full."
NOT form: "The bottle is not full."
4.2. Mathematical examples
- Statement: "9 is an even number."
NOT form: "9 is not an even number." - Statement: "x is positive."
NOT form: "x is not positive."
The negation clearly reverses the meaning each time.
5. Notes and Observations
Useful points about the connective NOT:
- NOT creates the logical opposite of any statement.
- The truth value always flips.
- Negations need to be written carefully to avoid ambiguity.
- NOT is frequently used when disproving arguments or forming logical complements.