Logic

Logic is a set of rules organized into logical systems, also known as formal systems. It underpins Set Theory and Category Theory, both of which add a single primitive to the axioms of Logic. ¹

Logic contains propositions, which may only be true or false. Propositions which are always true are known as tautologies and propositions which are always false are known as contradictions. Each tautology can be turned into a contradiction by adding a "not" and vice versa. ¹

Operators

Material Conditional

The Material Conditional is the → operator, as in X → Y, and is read as "X implies Y" or "if X, then Y". It asks "when could this statement be true?". The truth table for this operator appears as follows: ¹

It so happens that this truth table is the same as the truth table for ¬P ∨ Q: ¹

Thus, X → Y can always be replaced with ¬X ∨ Y