論理
P → Q
P
∴ Q
If the premise holds and the implication is valid, the conclusion follows with certainty. The simplest engine of deductive reasoning, floating here in infinite digital space.
P → Q ≡ ¬Q → ¬P
Every implication carries its shadow. To deny the consequent is to deny the antecedent. Logic mirrors itself in negation, a symmetry buried inside every conditional statement.
¬(P ∧ Q) ≡ ¬P ∨ ¬Q
¬(P ∨ Q) ≡ ¬P ∧ ¬Q
Negation distributes across conjunction and disjunction by swapping the operator. The boundary between "and" and "or" dissolves under the sign of negation.
P ∨ ¬P
Every proposition is either true or false. There is no third state, no twilight zone between affirmation and denial. Classical logic demands a binary universe.
Every logically valid formula is provable. The syntactic and semantic worlds converge at the horizon. Truth and proof meet where the grid lines vanish.
Any consistent formal system powerful enough to express arithmetic contains statements that are true but unprovable within that system. The grid extends beyond what we can see.