論理 — the shape of logical thought
A proposition is a declarative statement that is either true or false — never both, never neither. It is the atomic unit of logical reasoning: a single, indivisible claim about the world.
"The sky is blue" is a proposition. "Is the sky blue?" is not. Propositions are the bubbles from which all logical structures are built.
Negation flips the truth value of a proposition. If p is true, then ¬p is false — and vice versa. It is the simplest logical operation: a mirror that turns every truth into its opposite.
Click to negate
Conjunction — logical AND — is true only when both operands are true. Think of it as two bubbles merging: only when both are present does the combined form appear.
p ∧ q requires both p and q to hold.
Disjunction — logical OR — is true when at least one operand is true. The union of two bubbles glows: as long as either is present, truth persists.
p ∨ q is false only when both p and q are false.
Implication states that if p is true, then q must also be true. It is false only when the antecedent is true but the consequent is false.
Think of it as a promise: p → q — "if p, then q."
A truth table systematically enumerates every possible combination of truth values for a set of variables, showing the resulting value of a compound expression.
Click any input bubble to toggle its value and watch the output recalculate.
Two propositions are logically equivalent when they have the same truth value in every possible scenario. They may look different on the surface, but underneath they are the same shape.
¬(p ∧ q) is equivalent to ¬p ∨ ¬q — this is De Morgan's Law.