🔭 What Is Deductive Reasoning?
Deductive reasoning starts from a general rule (or set of accepted facts) and applies it to a specific case to reach a guaranteed conclusion.
If the premises (starting facts) are true and the argument is valid, the conclusion must be true — no exceptions.
- ⬇️ General → Specific
- ⬇️ Rules → Facts → Conclusion
- ⬇️ Used in math proofs, law, and science
🏺 Syllogisms
The most classic form of deductive reasoning is the syllogism — an argument with two premises and a conclusion.
∴ The Symbol for "Therefore"
The three-dot triangle symbol ∴ means "therefore" and introduces the conclusion of an argument.
✅ Valid vs. ❌ Invalid Arguments
An argument is valid if the conclusion logically follows from the premises — regardless of whether the premises are actually true in the real world.
An argument is sound if it is valid and the premises are actually true.
Valid Valid
The structure is correct — conclusion follows from premises. But it is not sound because the first premise is false.
Invalid Invalid
The conclusion does NOT follow from the premises. Even though all three sentences sound reasonable, the logic is broken.
Valid + True Premises = Sound argument
A sound argument guarantees a true conclusion. Validity alone only guarantees the conclusion follows the structure.
🔑 Law of Detachment (Modus Ponens)
If a conditional p → q is true, and the hypothesis p is true, then the conclusion q must be true.
⚠️ Common Mistake: Affirming the Consequent
You cannot run this in reverse! Knowing "the streets are wet" does NOT mean "it is raining" — the streets could be wet from a fire hydrant.
🔗 Law of Syllogism (Chain Rule)
If p → q and q → r are both true, then p → r is also true. You can chain conditionals together.
Math Example
If a number is divisible by 4, then it is divisible by 2. (p → q)
If a number is divisible by 2, then it is even. (q → r)
∴ If a number is divisible by 4, then it is even. (p → r)
🔄 Law of Contraposition (Modus Tollens)
If p → q is true and q is false, then p must also be false. This is using the contrapositive.
✏️ Practice — Click to Reveal Answers
1. Is this argument valid?
"All fish live in water. Sharks are fish. ∴ Sharks live in water."
Yes, valid — and sound! The structure is a correct syllogism, and both premises are true.
2. Is this argument valid?
"All birds have wings. Penguins have wings. ∴ Penguins are birds."
No, invalid. Having wings doesn't mean you're a bird — bats and butterflies have wings too. The conclusion does not logically follow.
3. Apply the Law of Detachment:
"If a student scores above 90, they get an A. Maria scored 95."
∴ Maria gets an A.
p → q is true. p is true (Maria scored 95 > 90). Therefore q (she gets an A) is true.
4. Apply the Law of Syllogism:
"If it is a rose, then it is a flower. If it is a flower, then it needs sunlight."
∴ If it is a rose, then it needs sunlight. (p → r chained through q)
5. Apply Modus Tollens:
"If the power is on, the light glows. The light is not glowing."
∴ The power is not on.
¬q is given. Since p → q, by contrapositive (¬q → ¬p), we conclude ¬p.
🎯 Key Takeaways
- Deductive reasoning moves from general rules to specific conclusions.
- A syllogism has two premises and one conclusion (∴).
- An argument is valid if the conclusion follows from the premises; sound if also the premises are true.
- Law of Detachment: p → q is true, p is true → q is true.
- Law of Syllogism: (p → q) and (q → r) → (p → r).
- Modus Tollens: p → q is true, q is false → p is false.