Worked Examples
Building and verifying logical arguments.
1Worked Example
Problem
Prove R from premises: (1) P → Q, (2) Q → R, (3) P
1
We have P → Q and P. Apply Modus Ponens to get Q.
2
Now we have Q → R and Q. Apply Modus Ponens again to get R.
3
Formal proof:
1. P → Q (Premise)
2. Q → R (Premise)
3. P (Premise)
4. Q (MP: 1, 3)
5. R (MP: 2, 4) ✓2Worked Example
Problem
Is this valid? "All cats are mammals. Felix is a cat. Therefore, Felix is a mammal."
1
Let C(x) = x is a cat, M(x) = x is a mammal.
2
Premise 1: ∀x(C(x) → M(x)) — All cats are mammals.
3
Premise 2: C(Felix) — Felix is a cat.
4
By Universal Instantiation: C(Felix) → M(Felix)
5
By Modus Ponens: M(Felix) ✓ Valid!
3Worked Example
Problem
Identify the fallacy: "If you studied, you passed. You passed. So you studied."
1
Structure: P → Q, Q, therefore P
2
This is Affirming the Consequent — INVALID!
3
Q (passing) could be true for other reasons: easy test, cheating, luck, etc.
⚠
Watch Out!
Affirming the Consequent looks like Modus Ponens but isn't! MP needs P, not Q.