Negation of Compound Statements and ALL / SOME

We know that the negation of a true statement will be false, and the negation of a false statement will be true. But what happens when we try to negate a compound statement?

Negating a Conjunction (and) and a Disjunction (or):

The negation of a conjunction (or disjunction) could be as simple as placing the word “not” in front of the entire sentence.

Conjunction:
p ∧ q – “Snoopy wears goggles and scarves.”
∼(p ∧ q) – “It is not the case that Snoopy wears goggles and scarves.”

While by our negation we know that Snoopy does not wear BOTH goggles and scarves, we cannot say for sure that he does not wear ONE of these items. We can only state that he does not wear goggles or he does not wear scarves. (∼p ∨ ∼q)

Disjunction:
p ∨ q – “I will paint the room blue or green.”
∼(p ∨ q) – “It is not the case that I will paint the room blue or green.”

If I am not painting the room blue or green, then I am not painting EITHER color. So it can be said that “I am not painting the room blue” and “I am not painting the room green”. (∼p ∧ ∼q)

DeMorgan’s Laws: (negating AND and OR)
(The statements shown are logically equivalent.)

∼(p ∧ q) ↔ (∼p ∨ ∼q)
∼(p ∨ q) ↔ (∼p ∧ ∼q)

Notice that the negation symbol is distributed across the parentheses and the symbols are changed from AND to OR (or vice versa).

Negating a Conditional (if … then):

Remember: When working with a conditional, the statement is only FALSE when the hypothesis (“if”) is TRUE and the conclusion (“then”) is FALSE.

p → q – “If 9 + 3 = 12, then 9 is a prime number.” is a FALSE statement.
∼(p → q) – “It is not the case that if 9 + 3 = 12, then 9 is a prime number.” is TRUE.
p ∧ ∼q – “9 + 3 = 12 and 9 is not a prime number.” is a TRUE statement.

Negate a Conditional: (negating IF … THEN)

∼(p → q) ↔ (p ∧ ∼q)
Notice that the statement is re-written as a conjunction and only the second condition is negated.

Negating a Biconditional (if and only if):

Remember: When working with a biconditional, the statement is TRUE only when both conditions have the same truth value.

p ↔ q – “A triangle has only 3 sides if and only if a square has only 4 sides.”
… is logically equivalent to …
(p → q) ∧ (q → p) – “If it is a triangle then it has only 3 sides and if it is a square then it has only 4 sides.”

To negate a biconditional, we will negate its logically equivalent statement by using DeMorgan’s Laws and Conditional Negation.

∼(p ↔ q)
∼[(p → q) ∧ (q → p)]
[∼(p → q) ∨ ∼(q → p)]
[(p ∧ ∼q) ∨ (q ∧ ∼p)]

Negate a Biconditional: (negating IF AND ONLY IF):

∼(p ↔ q) ↔ [(p ∧ ∼q) ∨ (q ∧ ∼p)]

Negating ALL and SOME:

ALL	SOME
Consider: “ALL students are opera singers.” (Meaning that there are NO students who are not opera singers.)	Consider: “Some rectangles are squares.” (Meaning that there exists at least one rectangle that is a square.)
Negation: “It is not true that ALL students are opera singers.” “SOME students are not opera singers.”	Negation: “It is not true that some rectangles are squares.” “NO rectangles are squares.”