I'm struggling to understand truth functionality.
I know that a connective is truth-functional if the truth value of a compound statement formed with that connective is completely determined by the truth values of the input statements.
Truth functional: "and"
- "I love The Beatles and I hate today's music" is true, as is its converse "I hate today's music and I love The Beatles". Since the truth value of the compound is the same in both variations, the truth value is completely determined by the connective "and", and so the connective "and" is truth-functional.
Non-truth functional: "because"
- "The sky is blue because sunlight is scattered through the atmosphere." This statement is true, but the converse (i.e. "Sunlight is scattered through the atmosphere because the sky is blue") is not. Since the positions of the input statements affect the truth value of the compound, the connective "because" is not truth-functional.
But I don't understand how to apply this definition to conditionals. In particular, why are the following conditionals truth-functional:
- If x = 5, then x + 5 = 10.
- If the lines are parallel, then the lines do not have a point in common.
...whereas these are not:
- If the match is struck, then it would light.
- If Shakespeare didn't write Hamlet, then someone else would have.
Is it because in 1) and 2) the antecedent and consequent can be switched without affecting the truth value of implication, whereas this is not the case in 3) and 4)?
I'm not sure how we'd even switch the antecedent and consequent in 3) and 4), since these statements are written in the subjunctive. Does "would" remain in the consequent of the converse as well? So, for example, would the converse of 4) be "If someone else wrote Hamlet, Shakespeare would not have"?
I've seen this post, but it's still not clear to me why 1) and 2) are truth-functional, while 3) and 4) are not.