Not too long ago I asked a question related to the material conditional that ended up proving just how limited my understanding of the material conditional actually was.
In the meantime, I found a fantastic webpage that hosts the best explanation of the material conditional I have encountered thus far. The page in question: http://philosophy.hku.hk/think/sl/ifthen.php
Now, I have some trouble understanding certain parts. I have to make it clear though; I'd appreciate it if we could leave out any references to the technical nature of the conditional when explaining the conditional. So no "that property of the conditional is true because if you check with the conditional, you'll see that..." Specifically, the author explains the material conditional through a method that tries to preserve some of the logical properties of "if..., then...". As such, try to stick to a similar method without involving things that I'd simply have to assume to be (logically) true (e.g. the one where the teacher says "If you get an A, you can go home early.", but you don't get an A and don't get to go home early, so the teacher didn't lie - even though the teacher might just as well have been lying, but we didn't find out because it didn't happen so we have to assume he was right for the sake of the explanation, thereby making the explanation quite worthless) My first issue arises in this part of the explanation where the author begins motivating two of four truth values of the implication:
First of all, one thing that we accept is that "if φ, then φ" is always true for any statement φ. So to preserve this fact, we need to ensure that the truth-value of "(φ → φ)" is always T whether φ is T or F. Since we are assuming that "→" is truth-functional, this implies that "(φ → ψ)" has the truth-value T whenever φ and ψ have the same truth-value, whatever that is.
(1) Why is "if φ, then φ" always true for any statement φ, regardless of the truth of φ? Is this because φ is simply assumed to be true when one says "if φ...", irrespective of the actual truth of φ? For instance, is "If 1=2, then 1=2" true because, in the case that 1=2, it is necessary that 1=2? Or, 1=2 only if 1=2? Am I overthinking this, or is there some logical structure behind all of this?
(2) According to the author, from (1) it follows that the assumption that "→" is truth-functional implies that "(φ → ψ)" has the truth-value T whenever φ and ψ have the same truth-value, be it F or T. I can sort of intuitively feel (1) is true, but I don't see how he's able to extend this to two different statements φ and ψ. Maybe the reason I don't understand this is because I don't fully grasp (1)? Either way it seems odd that you'd be able to say something specific like that about the nature of the material conditional by simply observing that it's a truth-functional"? I mean, how does (1) lead him to conclude that "Since we are assuming that "→" is truth-functional, this implies that "(φ → ψ)" has the truth-value T whenever φ and ψ have the same truth-value, whatever that is."?
Down the line, he gets to this part:
To fill in the third row of the truth-table, a different kind of argument is needed. This time we consider the properties that we do not want "→" to possess. In particular, consider this sequent:
(P→Q) ⊧ (Q→P)
Surely we do not want this argument to be valid.
(3) I think I have an idea as to why we don't want this argument to be valid; something to the effect of not wanting "If x is an apple, x is a fruit" to be logically equivalent to "If x is a fruit, x is an apple", correct? Of course, some additional background here would be appreciated.