Logical Approach to Scientific Theories/Laws

I don't think it's fair to say that it's "illogical", in any case. When really the only logical leap made is a relatively minor one: assuming that induction works in general, because it's never failed.

Personally, I think the law of large numbers is the most telling argument in favour of induction:
Just because you don't know something is true or false, does not mean the two are equally likely.
If I claim that I've flipped 10 coins, and the result was 10 heads, it's quite proper for you to be more skeptical than if I claim that it was 4 tails and 6 heads. Probabilistically speaking, of all the outcomes after flipping 10 coins, it takes a greater amount of "faith" to believe in the least likely proposition.
In order for the universe to suddenly change, and make induction invalid, it would have to go against every scientific test to this date - every attempt to predict the future that progressed successfully. The odds of that happening at the exact moment of your test, give that it has never happened before should be considered miniscule, and thus not cause for great concern.

If you like, you could break down the universe into smaller and smaller components of time, see that the same laws have governed each of these forward progressions - and the closer you get, the more it defies probability that in the next tiny unit of time, it'll go against all history and do something new. When you have 10^40 observations to the affirmative, and none to the negative, it's fair to say your next observation is more likely to also be affirmative.

Induction fails all the time:

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

You can use induction to try to predict that its raining, but the case can also be I'm just in the shower or fell in the pool or a car splashed me. This is why induction is such a dangerous tool, especially within science.

But to progress further in this discussion, I'll just give up that induction is necessary in science. There are pretty extreme views regarding the use of induction, such as Popper, who thinks science should never confirm a hypothesis, but should only consist of falsification and corroboration. He thinks, that within science, hypotheses should be bold conjectures which open up tons of possibilities for it to be proven false, and should in fact be an endless falsifying process. Corroboration he would say just means the hypothesis has just not yet been refuted.

The next problem after admitting the use of induction is permitted within strict regulation and more so unavoidable, is which regularities are the regularities to track and make generalizations on. Goodman suggests that in order for induction to be used correctly, it involves the differentiating between law-like generalizations and accidental generalizations. And so the new question is, which regularities in nature should we track and generalize on? So,

All emeralds are green -- confirmed by observations of green emeralds, this also confirms that all emeralds are grue -
where grue is by definition observed before 2012 and green, otherwise blue

So, All emeralds I've observed are green, thus all emeralds are green. So, any emerald I observe in 2012 will be green.

And, All emeralds I've observed are grue, thus all emeralds are grue. So, any emerald I observe in 2012 will be grue, and given grue's definition; Any emerald I observe in 2012 will be blue.

Discuss and keep an open mind as we proceed, because this is where it will get a bit confusing, but there's a deeper point than what it appears to be on the surface.

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.

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.

What you think is induction, is deduction. Yes, it is affirming the consequent under deduction, but under induction it's not and that point was whether we can accept any form of induction in science. You are arguing about what induction is, it's hard for me to even feel the need to justify that I am right, but this is a widely taught argument within the philosophy of science. The structure is not wrong, not the problem, and doesn't contain the discussion.

Your understanding of the rain example is also confused: when it is raining, you get wet - wait until it rains, see if you get wet, you do (this is just affirming the deduction). The induction is when it rains (the phenomena to be explained / hypothesis, imagine it unobservable, since this is where it applies in science, the observable is simply an empirical finding), I am wet (the result) - I am wet (the expected result in accordance with the hypothesis, affirming the phenomena e.g. electrons), that is induction or deductively speaking affirming the consequent, which has naively been mentioned several times, as if thinking I overlooked it.

Yes, it is affirming the consequent under deduction, but under induction it's not and that point was whether we can accept any form of induction in science

The only form I can recall about induction that resembles "affirming the consequence" would be causal inference. Not only is that just one side of the Rubilk's Cube, it is also explicitly required that additional factors be confirmed to establish the exact form of the causal relationship. Wet -> Rain is simply absurd.
Not only are you saying Inductive reasoning IS causal inference, you're also doing it wrong.

The most intuitive forms of inductive reasoning would be generalization and future prediction, which are, yes, arguably reliable. Be reminded, though, that being logical is entirely different from being right.
Philosophers argue there aren't even truths. But then, how can they even be right?

Okay, if you're using non-mathematical induction, the answer is simple - it's not something that can be deduced logically, as Hume and others pointed out.

This is not a problem, because it still works very well.

At worst, there are certain cases which can be identified where the principle works differently. (eg. Newton's laws of motion, at relativistic speeds you need a more complex modeL) You can continue using it for most stuff, just be wary.
At best, it's actually a universal law and you never run into problems with it. (eg. the laws of thermodynamics)