There are two popular yet nonequivalent definitions of conditional statements in the popular and respected literature on the topic, but only one of them is correct.
Under the first definition, a conditional statement is true if and only if the conclusion is true every time the hypothesis is true.
Under the second definition, a conditional statement is true if and only if it is not the case that: the hypothesis is true and the conclusion is false.
Only the first definition is correct.
Explains what really constitutes a conditional statement, despite references in numerous reputable works to an incorrect definition.
A conditional statement is not necessarily truth-functional. It's truth is not necessarily determined by merely a truth table. A conditional statement is a modal claim. It is a necessary claim. A conditional statement is equivalent to it is necessary that: the hypothesis is false or the conclusion is true. It is equivalent to every time the hypothesis is true, the conclusion is also true.
Be on the lookout! Many reputable sources do not make the necessary distinction between what is actually a conditional statement, and what is not actually a conditional statement! These sources are ambiguous!
Among the tainted works include:
1. "Geometry" (2007) by Ron Larson, Laurie Boswell, Timothy D. Kanold, and Lee Stiff, published by McDougal Littell. The flaw seems to be contained to only Pages 94-95, where truth tables are discussed, and not the rest of the book.
2. "Discrete Mathematics and Its Applications, Sixth Edition" (2007) by Kenneth H. Rosen, published by McGraw-Hill. Definition 5 on Page 6, and the examples that follow, are the worst examples of the problem I have come upon. Watch out for this! It taints the entire book, although Rosen is usually able to circumvent any problems through the use of universal quantifications throughout the book.
3. "Language, Proof and Logic" (2008) by Jon Barwise and John Etchemendy, published by the CSLI (Center for the Study of Language and Information) Publications. Although clear warnings are given, they do seem to be incomplete.