Inductive logic has been a methodology of great contention in recent decades. Karl Popper, the famous philosopher of science in the 20th century wrote, in his book "Conjectures and Refutations" that particular facts cannot establish a general claim. Popper also writes that no finite number of particular facts can establish a general claim as that claim would apply in a infinity of circumstances. Furthermore, he thinks that scientific facts are exact, while observations often contain a degree of error. In this case, it would be erroneous for exact claims to come from inexact observations. Although such sound attacks have been made, I do not think all hope is lost for inductive reasoning. Inductive reasoning is useful when we are unable to verify an infinite number of situations and yet require general claims. Inductive logic can often offer a probabilistic estimation of general claims.
I would first like to point out that not all scientific facts are exact insofar as these facts do not make an absolute claim. In scientific investigations, theories are often proposed and accepted based on what is probable and improbable, not what is exact or inexact. It should also be noted that in medicine, drugs are tested on a sample group of test subjects and the recuperative effects are extended based on that result. For example, if a group of 50 test subjects took penicillin and recovered from a bacterial infection. Claims are often made that penicillin can therefore be considered an effective antibacterial drug. Such is the role of inductive logic. In such a scenario, we would be fair to apply inductive logic to assert the proposition that penicillin is an effective antibacterial drug.
Wouldn't it be a little too risky though? After all, I'm offering penicillin as an efficacious drug despite the fact that this has only been tested on 50 people. Is it absurd to put the lives of billions of people at risk simply because I have tested it on a sample group of 50? Personally, I do not think so. In reality, it is very difficult to test every single individual before making an absolute claim. It should also be noted that the principles of falsification aid us in overcoming this problem. Should a patient's bacterial infection worsen or not get any better after consuming penicillin, we are able to use that example to falsify the claim that penicillin works efficaciously for all people. Attempts should then be made to understand the underlying problems (allergy reaction, unknown bacterial strain, etc) and new information regarding this issue should thus be made accessible to everyone.
The difference between deductive logic and inductive logic is that deductive logic attempts to make a claim for the particular by appealing to the general, while inductive logic attempts to make a claim for the general by appealing to the specific. An example of deductive logic would be "All living creatures are mortal. A fly is a living creature. Hence, a fly is mortal." An example of inductive logic would be "I walked into the country and noticed two bearded men. Hence, all men in the country are bearded." It should be noted that deductive logic is very often correct as it's conclusions follow necessarily from the premises while inductive logic is not necessarily correct as it's conclusions merely follow what is probable.
However, I would also like to point out that deductive logic is imperfect in the sense that it also employs a form of inductive logic. Notice my first premise in the earlier paragraph "All living creatures are mortal"? That is also a form of inductive reasoning as I cannot demonstrate with absolute certainty that all living things are mortal. It could be possible to falsify my premise by giving an example of an immortal living thing. What I can offer, is the functional claim that all living things are "probably mortal" based on inductive reasoning. This is where things get tricky as we have no idea whether inductive logic gives us a confirmed hypothesis or hasty generalization.
A problem with inductive logic is also that we are unable to determine when inductive references constitute a decent generalization. Suppose I had encountered 20 ducks on my way back home, and all of them happened to be brown. Inductive logic tells me that all ducks are brown. But is this claim necessarily true? I think it doesn't require much thought to know that the claim "all ducks are brown" is a fallacious one. How then, should we determine when inductive reasoning is appropriate and when it is inappropriate? I think part of it comes back to probabilities and falsification. The claim that all ducks are brown can be falsified by showing a white duck, a black duck and a multicolored duck. In addition, the claim "all ducks are brown" is functionally probabilistic insofar as claims to absolute certainty is concerned. As long as we remember that "penicillin is an efficacious antibacterial drug" is a functionally certain claim just like "all ducks are brown".
In essence, while inductive logic may not be perfect, I think there is little to fret about in the face of proper tools like falsification which allow us to separate the weed from the chaff. We have developed many modes of testing to empirically test inductive claims. It should also be stated that inductive observations are theory free, which is a plus point when it comes to objective conjectures and conclusions. Facts can also be reliably transferred via inductive reasoning and then be trenchantly verified by falsification and experiments.