User:Gregbard/Sandbox/Logical properties

The following properties apply to disjunction:

associativity: $$a \lor (b \lor c) \equiv (a \lor b) \lor c $$

commutativity: $$a \lor b \equiv b \lor a $$

idempotency: $$a \lor a \equiv a $$

monotonic: ($$a \rightarrow b) \rightarrow ((c \lor a) \rightarrow (c \lor b))$$

truth-preserving: The interpretation under which all variables are assigned a truth value of 'true' produces a truth value of 'true' as a result of disjunction.

falsehood-preserving: The interpretation under which all variables are assigned a truth value of 'false' produces a truth value of 'false' as a result of disjunction.

transitivity: ($$a \rightarrow b) \rightarrow ((b \rightarrow c) \rightarrow (a \rightarrow c))$$

distributivity: ($$a \rightarrow (a \rightarrow (b \rightarrow c)) \rightarrow ((a \rightarrow b) ) \rightarrow (a \rightarrow c))$$

commutativity: ($$a \rightarrow (b \rightarrow c)) \rightarrow (b \rightarrow (a \rightarrow c))$$

idempotency: $$a \rightarrow a$$

idempotency: $$a \equiv a$$

commutativity: $$(a \equiv b) \equiv (b \equiv a)$$

monotonic: ($$a \equiv b) \rightarrow ((a \lor c) \equiv (b \lor c))$$ ($$a \equiv b) \rightarrow ((a \land c) \equiv (b \land c))$$