Logical Approach to Scientific Theories/Laws

Induction fails all the time:

If it's raining outside, then I'll get wet
I'm wet
It's raining

That's not a failure of induction, and the scientific result there would simply be "when it is raining, you get wet." Testing the hypothesis: you wait until it rains, and see if you get wet. If you do get wet, while it is raining, it's corroboration.

As for using "it's wet" to predict "it's raining" - yes, that's a valid prediction. It's not always going to be right, but some amount of the time, it will work. And, as a matter of weather prediction, it'll give you a better rate of success than guessing at random.

I'm still not sure why you're calling this induction, though. Induction is the use of chains of logical steps.
Given
A -> B
B -> C
C -> D
Then
A -> D
This is induction. It has nothing to do with
Given
A -> B
B
Then
A
Which like I mentioned before is a fairly well known logical fallacy.

If you can provide an example of actual induction where
A -> B is true
B -> C is true
A -> C is false
I will be quite impressed.