論理

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The Art of Formal Reasoning

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I. The Premises

Universal Truth

For every proposition, there exists a path through formal reasoning that leads to its truth or falsity. The universality of logic transcends language, culture, and era.

Existential Proof

There exists at least one valid derivation for every well-formed theorem. The act of proof is the act of discovering what was always already true.

Derivability

From axioms to theorems, every step in a proof is a bridge built from pure reason. The turnstile asserts what follows from what we know.

II. The Derivation

1. P → Q Assumption
2. P Assumption
3. Q Modus Ponens (1, 2)
4. Q → R Assumption
5. R Modus Ponens (3, 4)
6. P → R Hypothetical Syllogism (1, 4)
Q.E.D.

III. The Operators

Conjunction
Both must hold true
PQP∧Q
TTT
TFF
FTF
FFF
Disjunction
At least one must hold
PQP∨Q
TTT
TFT
FTT
FFF
Implication
If P then Q
PQP→Q
TTT
TFF
FTT
FFT
¬
Negation
The opposite holds
P¬P
TF
FT

IV. The Conclusion

"Logic is the beginning of wisdom, not the end."

— Adapted from the tradition of formal reasoning

The proof is complete. From premises through derivation, through the careful application of logical operators, we arrive at truth. 論理 — the beauty of reasoning made visible.

∀x(Logic(x)Beauty(x))