Propositional form X is logically equivalent to propositional form Y (i.e., X and Y are logically equivalent [to one another]) iff (1) it is impossible for X to be true while Y is false and (2) it is impossible for Y to be true while X is false. In other words, (1) X logically implies Y and (2) Y logically implies X.
Example from everyday life: 'x is taller than y' is logically equivalent to 'y is shorter than x'
Example from categorical logic: 'Some S are not P' is logically equivalent to 'Some nonP are not nonS' (this is contraposition of an 'O' proposition)
Example from propositional logic: 'p ﬤ q' is logically equivalent to 'p ﬤ (p • q)'
Example from predicate logic: '~(Ǝx)Øx' is logically equivalent to '(x)~Øx'
