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If P is true, then Q is true. P being false or unknown says nothing about Q.

Instead of a truth table like this:

P : Q

0 : 0 = 1

0 : 1 = 1

1 : 0 = 0

1 : 1 = 1

You have a truth table like this:

P : Q

0 : 0 = ?

0 : 1 = ?

1 : 0 = 0

1 : 1 = 1

Or perhaps a truth table like this:

1 : 0 = 0

1 : 1 = 1

0
1

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.

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  • I look at it as conditional branches where P being equal to true calls for a specific action. If P is true, then I will set Q to true whether it was or not because it must be true if P = true. If P is false, I will do nothing and leave Q alone because if P is false, that says nothing about Q. If P is false, all bets are off regarding not only the status of Q, except any relationship between P and Q is null and void, and therefore, the final evaluation of any branch of logic starting with P being equal to false ends that branch of logic and is irrelevant to to any truth table involving P. – Abercrombie Dorfen Oct 9 '20 at 6:42

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