Suppose we removed the $$ connective from propositional logic. From a semantic point of

view this would not be a significant loss because any formula of the form $$A is equivalent to

A $->.$ In natural deduction, however, we would have to modify the set of rules to account

for the loss of $.$ Interestingly, we would not have to add much to the inference system

because some of the basic natural deduction rules involving $$ can be simulated by the other

basic rules