A proposition is a declarative statement that can be evaluated as either true or false, but not both simultaneously.

The symbol "¬" is used to denote the negation operator, which changes the truth value of a proposition to its opposite.

The truth table for conjunction shows that the result is true only when both operands are true.

The disjunction operator "∨" is also known as the inclusive OR, which means that the result is true if either or both operands are true.

The truth table for implication shows that the result is false only when the antecedent is true and the consequent is false.

The biconditional operator "↔" is equivalent to the compound proposition (P → Q) ∧ (Q → P), which means that both the implication from P to Q and the implication from Q to P must be true.

A tautology is a compound proposition that is always true, regardless of the truth values of its component propositions.

(P ∨ ¬P) is a tautology because it is always true, regardless of the truth value of P.

A contradiction is a compound proposition that is always false, regardless of the truth values of its component propositions.

(P ∧ ¬P) is a contradiction because it is always false, regardless of the truth value of P.

A contingency is a compound proposition that is neither a tautology nor a contradiction.

(P → Q) → ¬Q is a contingency because its truth value depends on the truth values of P and Q.

The distributive law in propositional logic states that (P ∧ (Q ∨ R)) = (P ∧ Q) ∨ (P ∧ R).

The associative law in propositional logic states that (P ∧ (Q ∧ R)) = (P ∧ Q) ∧ R.

De Morgan's laws in propositional logic state that ¬(P ∨ Q) = ¬P ∨ ¬Q and ¬(P ∧ Q) = ¬P ∧ ¬Q.