QUEST 01 / V.II

LOGIC.QUEST

Every journey begins with a premise.
What follows is a chain of consequence,
traced step by step until the summit is earned.

Discipline Formal Logic
Method Quest, Stepwise
Reward Q.E.D.
Style Swiss / Stacked
01 Propositional Logic / Modus Ponens

If the path is open, the traveler will pass.
The path is open.

P → Q  ·  P  ⊢  ?

  1. i. Assume P → Q.
  2. ii. Observe P holds.
  3. iii. By Modus Ponens, derive Q.

∴ The traveler will pass.

02 Propositional Logic / Modus Tollens

If the lantern burns, the cavern is lit.
The cavern is not lit.

P → Q  ·  ¬Q  ⊢  ?

  1. i. Assume P → Q.
  2. ii. Observe ¬Q.
  3. iii. By Modus Tollens, derive ¬P.

∴ The lantern does not burn.

03 Predicate Logic / Universal Instantiation

Every cartographer carries a compass.
Mira is a cartographer.

∀x (C(x) → K(x))  ·  C(m)  ⊢  ?

  1. i. Universal premise: ∀x (C(x) → K(x)).
  2. ii. Instantiate at m: C(m) → K(m).
  3. iii. With C(m), derive K(m).

∴ Mira carries a compass.

04 Predicate Logic / Existential Generalization

The cartographer Mira has crossed the river.
What does the world know now?

R(m)  ⊢  ?

  1. i. Particular fact: R(m).
  2. ii. By Existential Generalization, derive ∃x R(x).

∴ Someone has crossed the river.

05 Modal Logic / Necessity & Possibility

It is necessary that the sun rise on a clear morning.
The morning is clear.

□(C → S)  ·  C  ⊢  ?

  1. i. In every accessible world, C → S.
  2. ii. Our world is accessible & C holds here.
  3. iii. So S holds here.

∴ In every clear morning, the sun rises.

FINAL ASCENT

A Multi-Step Proof

From three premises, derive the conclusion in seven steps. Each step rises into view as you climb. The final line earns the glow.

P1∀x (T(x) → (W(x) ∨ M(x)))
P2∀x (W(x) → ¬M(x))
P3T(a) ∧ M(a)
Goal¬W(a)
  1. 01 T(a) ∧ M(a) Premise (P3)
  2. 02 M(a) ∧-Elim from 01
  3. 03 ∀x (W(x) → ¬M(x)) Premise (P2)
  4. 04 W(a) → ¬M(a) ∀-Elim from 03
  5. 05 Assume W(a) For Reductio
  6. 06 ¬M(a) Modus Ponens 04, 05
  7. 07 M(a) ∧ ¬M(a) → ⊥ Contradiction with 02
  8. 08 ∴ ¬W(a) Reductio Ad Absurdum
Quod Erat Demonstrandum

Q.E.D.

The summit is reached. The premises rest. The proof endures.