I want to understand contrapositive clearly. I'll start by saying: "If it is sunny, then there is light". The statement is true. But now consider the contrapositive: " If there is no light then it is not sunny". The contrapositive is false because you could create light with a flashlight.
What is wrong here?
A usefull distinction to be made here is the distinction between : sufficient condition and necessary condition.
The sentence "Sunny --> Light" says that
the fact "it is sunny" is a SUFFICIENT condition in order " there is light" to be true. ( Saying that "X is sufficient for Y" does not exclude the possibility of obtaining Y without X. If I am a pianist, I am a musician. This is perfectily true, and it does not deny the possibility of being a musician without being a pianist).
One cannot object to this that " one can make light with a flashlight".
Such an objection amounts to saying " the sun is not a necessary condition in order " there is light" to be true".
Such an objection is directed at an assertion that was NOT contained in the original sentence.
Both the original sentence and its contrapositive say exactly the same thing under various points of view, namely
(1) sun is sufficient for light
(2) light is necessary for sun
(3) absence of light is sufficient for absence of sun
(4) absence of sun is necessary for absence of light.
NOTE : Saying " X --> Y " is equivalent to
(1) X is sufficient for Y
or, equivalently
(2) Y is necessary for X
The statement "if $p$ then $q$" has truth value true if it's true or false otherwise. The contrapositive is "if not $q$ then not $p$". The two statements are equivalent, i.e. always the same truth value. This is unrelated to the question of what that common truth value is. The point is the logically possible universes in which "If it's sunny there is light" are true are precisely those in which "if there's no light it's not sunny" are true.
You may have confused the contrapositive with the converse, whose truth value can be different. Confusing "if $p$ then $q$" with its converse, "if $q$ then $p$", is a fallacy called affirming the consequent. Confusing "if not $q$ then if not $p$" with its converse, "if not $p$ then if not $q$", is a fallacy called denying the antecedent.
Read your correct contrapositive again:
If there is no light then it is not sunny.
That's true - it's equivalent to the first statement. You could create light with a flashlght, but then there would be light.
What
you could create light with a flashlight.
tells you is that the original true statement is not "if and only if". The converse
If there is light then it is sunny
is false.
Your conclusion about the flashlight has nothing to do with the contrapositive. You may was well say that the original statement is false because your eyes are closed. It has nothing to do with the claim.
The hypothesis of the contrapositive is that there is no light. So not only is it not sunny, but also you don’t have a flashlight, and there is no fire burning, and your car headlights aren’t on, etc etc.
Note also that even if there were light (say you have a flashlight), then the hypothesis of the contrapositive is false, so the contrapositive is true because it would be a statement of the form “F $\implies$ something” which is always true.
