When are conditional statements actually true?
I am getting contradicting answers. Please help.
One resource, a geometry book, says that to prove a conditional statement true, you must show the conclusion is true every time the hypothesis is true.
On the contrary, however, a discrete mathematics book says a conditional statement is true unless the hypothesis is true and the conclusion is false.
These methods for checking the truth of a conditional statement do not produce the same results, however. For example, consider the conditional statement
(1) If today is Saturday, then 5 + 5 = 6.
Under the first method, this (1) is false, because when there is a time when the hypothesis is true (It is Saturday), but the conclusion is false (5 + 5 never equals 6). A counterexample exists, as they would say.
But under the second method, the statement's truth value changes with time. It is true when it is not Saturday since the condition for falsehood, that it is Saturday and 5 + 5 does not equal 6, is not met. But it is false on Saturday, since the condition for falsehood is met.
Which one of these contradicting methods correctly determines the truth of a conditional statement?