Imagine a researcher is exploring a dataset and runs 1000 different regressions and he finds one interesting relationship among them.
Now imagine another researcher with the same data runs just 1 regression, and it turns out it's the same one that the other researcher took 1000 regressions to find. Researcher 2 does not know researcher 1.
Should researcher 1 make different inferences than researcher 2? Why? For example, should researcher 1 perform multiple comparisons correction, but researcher 2 should not?
If researcher 2 showed you his single regression first, what inferences would you make? If after that researcher 1 showed you his results, should you change your inference? If so, why should it matter?
PS 1: If talking about hypothetical researchers makes the problem abstract, think about this: imagine you ran just one regression for your paper, using the best method available. Then another researcher explored 1000 different regressions with the same data, until he found the exact same regression you ran. Should you two make different inferences? Is the evidence the same for both cases or not? Should you change your inference if you knew the other researcher results? How should the public assess the evidence of the two studies?
PS 2: please try to be specific and to provide a mathematical/theoretical justification, if possible!