Logic
NOT: Negation
Цели урока
- Understand the negation operation and its truth table
- Master De Morgan's laws for negating AND/OR
- Learn to simplify expressions with double negation
- Correctly negate conditional statements
"It's not the case that he is smart AND kind" - does that mean he is dumb AND mean? No! De Morgan's laws explain why.
- Legal phrasing: "Not guilty" vs "Innocent"
- Programming: simplifying conditions in if-else
- Mathematical proofs by contradiction
- Critical thinking: how to refute claims
Negation: flipping truth values
**Negation** (¬, NOT) is the simplest logical operation. It flips the truth value of a statement: true becomes false, false becomes true.
**Negation notation:** • In logic: ¬P, ~P, P̄ • In programming: !P, not P • In math: P' (P prime) • In English: "not", "it is not the case that"
Negating a simple statement is easy - just add "not". But negating complex statements requires care. "Not (it is raining AND it is cold)" ≠ "It is not raining AND it is not cold"!
If P = "Today is Monday" and today is indeed Monday, what is ¬¬P?
De Morgan's Laws: negating AND and OR
**De Morgan's Laws** show how to correctly negate complex statements involving AND and OR. This is a key tool for logical transformations.
**Common mistake:** ¬(P ∧ Q) ≠ ¬P ∧ ¬Q ❌ "Not (cat AND dog)" ≠ "Not cat AND not dog" Correct: "Not cat OR not dog" (at least one is false)
Mnemonic: negation "breaks" the parentheses and flips the connective. AND becomes OR, OR becomes AND. Each part also gets negated.
What is the correct negation of "The film is interesting AND short"?
Double negation: minus times minus
**Double negation** is the negation of a negation. In classical logic ¬¬P = P. Two "nots" cancel each other out.
**Interesting fact:** In intuitionistic logic ¬¬P ≠ P! There, "cannot prove it is false" ≠ "can prove it is true". But in classical logic, which we are studying, double negation always cancels out.
Double negation in speech is often used to soften statements: "not bad" is gentler than "good". But logically they mean the same thing.
Simplify: ¬(¬P ∨ ¬Q)
Negating implication: not "if-then", but "and yet"
What is the negation of "if P, then Q"? Many think ¬(P → Q) = (¬P → ¬Q). This is wrong! The correct answer is surprising.
**The word "but":** In English "but" often means P ∧ ¬Q: "Expensive, but quality" = Expensive AND quality "Tried hard, but failed" = Tried AND didn't succeed "But" = "and yet (unexpectedly)"
To refute "if A, then B", a single counterexample suffices: a case where A holds but B does not. That is exactly P ∧ ¬Q.
Negating "if P, then Q" gives "if not P, then not Q"
Negating an implication gives "P, but not Q" (P ∧ ¬Q)
An implication is false only when the condition holds but the consequence doesn't. So its negation = exactly that case: P is true AND Q is false. This is a conjunction, not an implication!
How is the claim "If a person is rich, then they are happy" refuted?
Key Takeaways
- Negation flips the truth value: ¬T = F, ¬F = T
- De Morgan's: ¬(P ∧ Q) = ¬P ∨ ¬Q, ¬(P ∨ Q) = ¬P ∧ ¬Q
- Double negation: ¬¬P = P
- Negating implication: ¬(P → Q) = P ∧ ¬Q (condition holds, consequence doesn't)
What's Next
Now we'll study valid argument forms - templates for error-free reasoning
- Valid Forms — next lesson
- IF-THEN — previous lesson
Вопросы для размышления
- Find an example of double negation in everyday speech. What does it mean?
- How is the claim 'Hard study guarantees a good job' formally negated in predicate logic?
- Why are De Morgan's laws important for programmers?