≡Negation of Propositions
What is it?
Negation of propositions is the process of turning a sentence into its correct opposite form, while keeping the logical meaning.
👉 It is NOT “creating any random opposite sentence”
👉 It is making the sentence express the exact logical opposite
Main idea
To negate a sentence means:
👉 “This is NOT true”
But carefully:
The negation must be logically equivalent to the opposite
It cannot change the meaning incorrectly
Basic examples
1. Universal quantifier (ALL)
Sentence:
“All students study”
Correct negation:
👉 “Not all students study”
or
👉 “There is at least one student who does not study”
Idea: just 1 exception breaks the “all”
2. Existential quantifier (SOME / THERE EXISTS)
Sentence:
“Some student passed”
Correct negation:
👉 “No student passed”
or
👉 “There is no student who passed”
Here you completely remove existence
Most important rules
ALL → negates to EXISTS NOT
“All A are B”
➡️ “There exists A that is not B”
EXISTS → negates to NONE
“There exists A that is B”
➡️ “No A is B”
SIMPLE SENTENCES
Example:
“John studies”
Negation:
👉 “John does not study”
Common exam traps
Mistake 1: wrong negation of “all”
“All students study” ❌
“All students do not study” (WRONG)
✔️ Correct:
“Not all students study”
Mistake 2: confusing quantity
“Some student passed” ❌
“Some student did not pass” (WRONG as negation)
✔️ Correct:
“No student passed”
How to think fast
🔄 Simple rule:
ALL → becomes “THERE EXISTS AT LEAST ONE WHO DOES NOT”
SOME → becomes “NONE”
SIMPLE → just add “NOT”
More examples
Example 1
“All cars are red”
➡️ “There exists a car that is not red”
Example 2
“There exists a student approved”
➡️ “No student is approved”
Example 3
“Maria likes math”
➡️ “Maria does not like math”
Did you know?
