Every journey begins with a premise.
If it rains, the ground is wet. It rains.
What can you conclude?
P → Q (If it rains, the ground is wet)
P (It rains)
Modus Ponens
∴ The ground is wet.
If a shape is a square, then it has four equal sides. This shape does not have four equal sides.
What follows?
P → Q (Square → four equal sides)
¬Q (Not four equal sides)
Modus Tollens
∴ This shape is not a square.
Either the butler or the gardener committed the crime. The gardener has a confirmed alibi.
Who is guilty?
P ∨ Q (Butler or gardener)
¬Q (Not the gardener)
Disjunctive Syllogism
∴ The butler committed the crime.
If you study logic, you think clearly. If you think clearly, you make better decisions. If you make better decisions, you live well.
You study logic. What can you ultimately conclude?
P → Q (Study → think clearly)
Q → R (Think clearly → better decisions)
R → S (Better decisions → live well)
P (You study logic)
Hypothetical Syllogism + Modus Ponens
∴ You live well.
All humans are mortal. Socrates is a human.
What must be true?
∀x(Human(x) → Mortal(x)) (All humans are mortal)
Human(Socrates) (Socrates is a human)
Universal Instantiation + Modus Ponens
∴ Socrates is mortal.
Assume √2 is rational. Then √2 = a/b where a and b share no common factor. Squaring: 2 = a²/b², so a² = 2b². Thus a² is even, so a is even. Let a = 2k. Then 4k² = 2b², so b² = 2k², meaning b is also even. But then a and b share factor 2 — contradicting our assumption.
What does this prove?
Assume ¬P (√2 is rational)
¬P → (Q ∧ ¬Q) (Leads to contradiction)
Reductio ad Absurdum
∴ √2 is irrational.
Prove: If logic is a quest, then the journey itself is the reward.