Yes, without the rule that true conclusion makes the whole implication true we would not be able to say general facts whose converse is false, both in mathematics and in real life.

An example from Wikipedia:

The converse of

the intermediate value theorem is false: roughly speaking, if for all

*a*,

*b* and every

*u* between f(a) and f(b) there exists a

*c* such that f(c) = u, it does not follow that f is continuous.

Still another example: If f : A -> B and g : B -> C are injections, then so is the composition g o f, but not conversely (if g o f is injective, then so is f, but not necessarily g).

Other concepts of implications have been considered in philosophy and logic. See

this Wikipedia section, for example. You can probably find a lot of information in the Stanford Encyclopedia of Philosophy.