論理 — a daily practice of logical thinking
The foundation of deductive reasoning
If P, then Q. P is true. Therefore, Q must be true. The simplest and most powerful rule of inference — the engine that drives all logical deduction forward, one certain step at a time.
P → Q, P ⊢ Q
Seeing truth from the other side
If not Q, then not P. The logical mirror of any conditional statement. Sometimes the clearest path to understanding is to consider what happens when the conclusion fails — the contrapositive reveals what was hidden in the original.
(P → Q) ≡ (¬Q → ¬P)
Truth through contradiction
Assume the opposite of what you wish to prove. Follow the assumption to its logical conclusion. When it collapses into contradiction, the original proposition stands revealed as true. Beauty in the breakdown.
¬P → ⊥ ⊢ P
The duality of AND and OR
Not (A and B) is the same as (not A) or (not B). Not (A or B) is the same as (not A) and (not B). These twin laws reveal the deep symmetry between conjunction and disjunction — two sides of the same logical coin.
¬(A ∧ B) ≡ ¬A ∨ ¬B
The architecture of argument
All A are B. All B are C. Therefore, all A are C. Aristotle's gift to reason — the formal structure of valid argument, where premises link together like roots drawing nourishment upward into inevitable conclusion.
∀x(A(x) → B(x)), ∀x(B(x) → C(x)) ⊢ ∀x(A(x) → C(x))
Foundational axioms from which all logic grows