\(\land\) is associative: \[ A \land (B \land C) = (A \land B) \land C \]
\(\lor\) is associative: \[ A \lor (B \lor C) = (A \lor B) \lor C \]
\(\land\) is commutative: \[ A \land B = B \land A \]
\(\lor\) is commutative: \[ A \lor B = B \lor A \]
\(\land\) is idempotent: \[ A \land A = A \]
\(\lor\) is idempotent: \[ A \lor A = A \]
\(\land\) has identity \(\top\): \[ A \land \top = A \]
\(\lor\) has identity \(\bot\): \[ A \lor \bot = A \]
\(\land\) has annihilator \(\bot\): \[ A \land \bot = \bot \]
\(\lor\) has annihilator \(\top\): \[ A \lor \top = \top \]
\(\land\) distributes over \(\lor\): \[ A \land (B \lor C) = (A \land B) \lor (A \land C) \]
\(\lor\) distributes over \(\land\): \[ A \lor (B \land C) = (A \lor B) \land (A \lor C) \]
\(\square\) distributes over \(\land\): \[ \square (A \land B) = \square A \land \square B \] \[ \square \top = \top \]
\(\lozenge\) distributes over \(\lor\): \[ \lozenge (A \lor B) = \lozenge A \lor \lozenge B \] \[ \lozenge \bot = \bot \]
\[ A \land (A \lor B) = A \]
\[ A \lor (A \land B) = A \]
\[ A \land \neg A = \bot \]
\[ A \lor \neg A = \top \]
\[ \frac{}{A = A} \]
\[ \frac{A = B}{C[x / A] = C[x / B]} \]