How are the logical connectives not, and, or, and if-then used in formal logic?

In formal logic, specifically within the propositional calculus, logical connectives are operators used to form new propositions from existing ones. These are known as truth-functional operators because the truth value of the resulting proposition is determined solely by the truth values of its arguments.1

The four primary connectives function as follows:

• Negation ("not," symbol $~$): A monadic operator that takes one argument. The proposition $~p$ is true if $p$ is false, and false if $p$ is true.1
• Conjunction ("and," symbol $·$): A dyadic operator. The conjunction $p·q$ is true only when both $p$ and $q$ are true; otherwise, it is false.1
• Disjunction ("or," symbol $\vee$): A dyadic operator. The disjunction $p\vee q$ is false only when both $p$ and $q$ are false; it is true if at least one of the arguments is true.1
• Conditional ("if-then," symbol $\supset$): A dyadic operator representing material implication. The statement $p\supset q$ is false only when the antecedent ($p$) is true and the consequent ($q$) is false; in all other cases, it is true.1