How do the logical connectives not, and, or, and if-then work?

In the propositional calculus, logical connectives are used to form complex propositions from simpler ones. These connectives are considered "truth-functional," meaning the truth value of the resulting proposition depends entirely on the truth values of its component parts.12

The standard connectives function as follows:

• Negation ("not," ~): A monadic operator that reverses the truth value of a proposition. If a proposition p is true, ~p is false; if p is false, ~p is true.2
• Conjunction ("and," & or ·): A dyadic operator that is true only when both component propositions are true. If either or both are false, the conjunction is false.2
• Disjunction ("or," ∨): A dyadic operator that is true if at least one of the component propositions is true. It is false only when both components are false.2
• Conditional ("if-then," ⊃): A dyadic operator that represents material implication. The statement p ⊃ q is false only when the antecedent (p) is true and the consequent (q) is false; in all other cases, it is considered true.2

These dependencies are systematically represented using truth tables.12