Given your edit, I think what you mean to ask is what kind of conditional has the property that if P is true, then Q is true; but when P is false the conditional itself has no value.
To express this you might use a three-valued logic, with values T/F/U (U for unknown). There are several existing three-valued logics, e.g. by Kleene and Łukasiewicz, but these do not do what you want, because they have the value T when P is false. You want the table:
P Q Con
T T T
T F F
F T -
F F -
This is sometimes referred to as a conditional with 'gappy' truth conditions, i.e. that under some valuations it does not return T or F. One approach to understanding conditionals of this kind appeals to the concept of 'conditional negation' which allows that a proposition can be false if it has a truth value, but it is not required to have a truth value. It plausibly accounts for a number of real usages of conditionals where "if P then Q" is denied by "if P then not Q". If you want more information about this kind of conditional, there is a useful paper by John Cantwell called The Logic of Conditional Negation, Notre Dame Journal of Formal Logic, Volume 49, Number 3, 2008, pp 245-260.