🧠 What Is Logic?
Logic is the study of correct reasoning. It gives us tools to figure out whether an argument is valid — whether the conclusion really follows from the evidence — regardless of whether the topic is math, science, law, or everyday life.
The most basic building block of logic is the statement.
📋 What Is a Statement?
A statement (also called a proposition) is a sentence that is either definitely true or definitely false — not both, not neither.
✅ Statements
"The Earth orbits the Sun." (True)
"A triangle has four sides." (False)
"5 + 3 = 8." (True)
"Whales are fish." (False)
❌ Not Statements
"Close the door!" (command — not true/false)
"Is it raining?" (question — not true/false)
"Wow, that's amazing!" (exclamation — not true/false)
"She is tall." (relative/vague — can't judge without context)
💡 Quick Test
Ask yourself: "Can I say this is definitely true or definitely false?" If yes, it's a statement. If it's a question, command, opinion, or too vague to judge, it is not a logical statement.
⚖️ Truth Values
Every statement has exactly one truth value: either True (T) or False (F). There is no middle ground in classical logic.
Assigning Truth Values
"Paris is the capital of France." → TRUE
"The moon is made of cheese." → FALSE
"2 × 6 = 11." → FALSE
"Sharks are a type of fish." → TRUE
In logic, we often use variables like p, q, and r to stand for any statement, just like algebra uses x and y for unknown numbers.
- Let p = "It is raining."
- Let q = "The ground is wet."
🔄 Negation: Flipping the Truth
The negation of a statement is its opposite. If a statement is true, its negation is false — and vice versa.
We write the negation of statement p as ¬p (read: "not p").
Examples of Negation
p: "The cat is sleeping." (True)
¬p: "The cat is not sleeping." (False)
p: "7 is an even number." (False)
¬p: "7 is not an even number." (True)
| p | ¬p (NOT p) |
|---|---|
| T | F |
| F | T |
⚠️ Watch Out!
Negation is not always just adding "not."
p: "All birds can fly." → ¬p: "Some birds cannot fly." (not "No birds can fly!")
📂 Open vs. Closed Statements
Closed Statement
Has a definite, fixed truth value.
"The sum of angles in a triangle is 180°." (Always True)
Open Statement
Contains a variable; truth value depends on what the variable represents.
"x + 5 = 12" — True if x = 7, False otherwise.
In logic, we work mostly with closed statements that have definite truth values. Open statements become closed once you substitute a specific value.
✏️ Practice — Click to Reveal Answers
1. Is "Stop making so much noise!" a logical statement?
No. It is a command. It cannot be judged as true or false.
2. What is the truth value of "The Pacific Ocean is the largest ocean on Earth"?
True. The Pacific Ocean covers more area than any other ocean.
3. Write the negation of: "All squares are rectangles."
¬p: "Not all squares are rectangles." (equivalently: "Some squares are not rectangles.")
Note: "No squares are rectangles" would be too strong — that's not the direct negation.
4. If p is FALSE, what is ¬p?
¬p is TRUE. Negation always flips the truth value.
5. Is "x > 10" a statement? Explain.
It is an open statement — it has a variable (x), so its truth value changes depending on what x is. It becomes a closed statement (with a definite truth value) once we substitute a specific number.
🎯 Key Takeaways
- A statement (proposition) is a sentence that is either true or false — not both, not neither.
- Questions, commands, and exclamations are not logical statements.
- Every statement has a truth value: T or F.
- The negation (¬p) of a statement flips its truth value.
- An open statement contains a variable; it becomes closed when the variable is replaced.
- Logic variables like p, q, r stand in for any statement.