how does induction relate to falsifiability, and when does one trump the other?
Inductive Reasoning is the traditional tool of the Empiricism epistemology. Falsifiability or testability is Popper's criterion for demarcating scientific knowledge from everything else.
Perhaps a better formed version of your question would be:
How does classical Empiricism relate to Critical Rationalism and when does one trump the other?
So in your original question, the latter relates to the former most famously in the writings of Popper, first in The Logic of Scientific Discovery, then in Conjectures and Refutations.
Popper suggests his own Epistemology, titled Critical Rationalism, according to which the advancement of our knowledge goes as follows: Using creativity, you come up with a hypothesis (hypothesis here means a falsifiable statement or theory), then, you try to refute it by aiming to find counter examples of deduced statements from said hypothesis. As long as no successful such refutations has been made, your hypothesis remains a Tentative Knowledge. The other kind of knowledge Popper acknowledges of is not scientific. i.e., for Popper, all scientific knowledge is tentative in that it can never be ultimately justified. The only thing we can do is endlessly minimize our error towards an accurate description of reality.
This works, according to Popper, because if "If A then B" then given "Not B" we get "Not A". Now A is the hypothesis at question, and B is a conclusion deduced from A; If we are convinced that the conclusion is false, then by modus tollens, the hypothesis is refuted.
In this sense, every failed serious attempt to refute a hypothesis is a further corroboration of it.
The methodology suggested by the classical Empiricist epistemology, employing induction alone, is to say that if "If A then B" then given "B" we get corroboration of A. Popper attacks this by reminding us that this is not logically valid. "B" could also be explained by infinite other theories other then A. The only logical way around this is to say that A could not be true if "Not B" is true.
According to induction alone, Newtonian physics is true just as anything else is, including the Theory of Relativity. This is Popper's point: Only refutations progress our knowledge, and his canonical example is that of moving from Newton to Einstein. The newer theory replaced the old one (in terms of scientific explanation) only because it stood the tests in which the older one has failed.
It seems clear that in that case the induction should trump the
non-falsifiability of the statement of "I am mortal", since it's a
no-brainer ... But that no-brainer is based on intuition, not on a well defined reason.
Now this is exactly the point of Critical Rationalism. Popper is troubled by the fact that while such Inductive reasoning may seem to go hand in hand with common sense, it is entirely lacking any logical base. Exemplified in this here case by the following:
- Your theory at hand suggests that "if men are mortal then they sometime die."
- Your only evidence to provide here is that all the time men do die.
- Hence, popper would note that all you have is an "If A [is true] then B [is true]" formed statement and then you have provided that indeed B is true.
- His argument would be to stress the fact that from this point onward (as I have described above) A is true is by no logical means deducible. We all know that from "If A then B", we only get that "If not B then not A" (which is exactly the logical basis of his doctrine), but definitely not "If B then A".
- As in, what you have provided, using Induction alone, does not leave you with the logical possibility to claim your "A" ("men are mortal").
So for your last question:
when should one trump the other regarding induction vs falsifiability?
the answer in this context is Critical Rationalism always trumps over Induction.