The problem of induction cannot be solved, it is fatally flawed. Induction might be simply characterised as 'more of the same' so we see something, we see it often and it is tempting to conclude we can predict the future. An argument is valid when there is no way (meaning no possible way) that the premises or starting points could be true without the conclusions being true. However, as Hume and others have shown, when we reason inductively there is a way in which our premises can be true and our conclusion false. For example I can say it has rained every Thursday at 3pm for the last 4 weeks and that can be is unquestionably true but if I suggest the conclusion that therefore it will rain next Thursday at 3pm it is obvious this may not be true.
What induction does is engineer a bridge between past and future, but cannot argue that the bridge is reliable. If we think of this in the context of scripture the truth is that one can make a sacred books and obstacle to moral and intellectual; progress, because it can consecrate the ideas of a given epoch and its customs as divinely appointed and hence unchangeable and not possible to improve but that is illogical for the simple reason that one cannot know what you yourself or others might know in the future.
The only way forward is to let everything be discussed and every objection heard because to but think that some particular principle or doctrine should be forbidden to be questioned because it is so certain, that is, because they are certain that it is certain so we decide the question for others, without allowing them to hear what can be said on the contrary side so he who prevents the opinion from being heard assumes infallibility. However unwillingly a person who has a strong opinion may admit the possibility that his opinion may be false, he ought to be moved by the consideration that however true it may be, if it is not fully, frequently, and fearlessly discussed, it will be held as a dead dogma, not a living truth. Where there is a tacit convention that principles are not to be disputed; where the discussion of the greatest questions which can occupy humanity is considered to be closed, we cannot hope to find that generally high scale of mental activity which has made some periods of history so remarkable - in short we must assert our mental freedom and any religion or dogma that forbids it or worse is to be fought. No one can be a great thinker who does not recognize, that as a thinker it is his first duty to follow his intellect to whatever conclusions it may lead.