💬 What Is a Conditional Statement?
A conditional statement (also called an if-then statement) links two statements together with a cause-and-effect relationship.
Form: If p, then q — written as p → q
- p is called the hypothesis (the condition / the "if" part)
- q is called the conclusion (the result / the "then" part)
Examples
If it rains, then the ground gets wet.
p = "It rains" | q = "The ground gets wet"
If a number is divisible by 4, then it is even.
Always true — this is a mathematical conditional.
If you study hard, then you will pass the test.
📊 When Is p → q True?
This is the trickiest truth table in basic logic. The conditional is false only when the hypothesis is true but the conclusion is false. In all other cases, it is true.
| p (If…) | q (…then) | p → q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
💡 Why is F → T true?
Think of a promise: "If you clean your room, I'll give you ice cream." If you never clean your room, the promise was never broken — regardless of whether you got ice cream or not. A false hypothesis makes the whole conditional vacuously true.
Applying the Truth Table
"If a shape is a square, then it has four sides." — T → T = TRUE ✅
"If you are in Tokyo, then you are in France." — T → F = FALSE ❌
"If 2 + 2 = 5, then the moon is made of cheese." — F → F = TRUE ✅ (vacuously)
🔃 The Four Related Conditionals
From any conditional p → q, we can form three related statements by swapping or negating:
Original (Conditional): p → q
"If it is a dog, then it is a mammal."
Converse: q → p (swap p and q)
"If it is a mammal, then it is a dog." — FALSE (cats are mammals too!)
The converse is not necessarily equivalent to the original.
Inverse: ¬p → ¬q (negate both)
"If it is not a dog, then it is not a mammal." — FALSE
The inverse is not necessarily equivalent to the original either.
Contrapositive: ¬q → ¬p (swap AND negate)
"If it is not a mammal, then it is not a dog." — TRUE
The contrapositive is always logically equivalent to the original. If the original is true, the contrapositive is true — and vice versa.
Key Equivalent Pairs:
- Original p → q ≡ Contrapositive ¬q → ¬p (always equivalent)
- Converse q → p ≡ Inverse ¬p → ¬q (these two are equivalent to each other, but not to the original)
↔️ Biconditional: If and Only If p ↔ q
A biconditional says both directions are true at the same time — "p if and only if q" means p → q AND q → p.
It is true when both statements have the same truth value.
| p | q | p ↔ q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
Example
"A shape is a square if and only if it has four equal sides and four right angles."
If it's a square → it has those properties. If it has those properties → it's a square. Both directions hold.
✏️ Practice — Click to Reveal Answers
1. Identify the hypothesis and conclusion: "If it is snowing, then school is cancelled."
Hypothesis (p): "It is snowing."
Conclusion (q): "School is cancelled."
2. Write the contrapositive of: "If a polygon is a triangle, then it has exactly 3 sides."
Contrapositive: "If a polygon does not have exactly 3 sides, then it is not a triangle."
This is TRUE and logically equivalent to the original.
3. p → q where p = TRUE and q = FALSE. What is the truth value?
FALSE. A conditional is only false when the hypothesis is true and the conclusion is false.
4. Are the converse and the inverse logically equivalent to each other?
Yes! The converse (q → p) and the inverse (¬p → ¬q) always have the same truth value — they are logically equivalent to each other, though not to the original conditional.
5. "A number is even if and only if it is divisible by 2." Is this a true biconditional?
Yes! Both directions hold:
• If a number is even → it is divisible by 2. ✓
• If a number is divisible by 2 → it is even. ✓
So p ↔ q is TRUE.
🎯 Key Takeaways
- A conditional p → q is false only when p is true and q is false.
- Hypothesis = the "if" part; Conclusion = the "then" part.
- Contrapositive (¬q → ¬p) is always equivalent to the original.
- Converse (q → p) and Inverse (¬p → ¬q) are equivalent to each other but not to the original.
- A biconditional (p ↔ q) is true when both statements share the same truth value.