How can one validly infer a universal statement from any number of existential statements?

Inductivist methodology supposed that one can somehow move from a series of singular existential statements to a universal statement.

Those explorations, in turn, become universal statements on the impact politics has had on environment, sports, religion, race, and government in the South.

Thus, a cryptic work about jostlingfor power and mysterious relationships is transformed into a recognizable, universal statement of shared need.

Does the invalidity of all universal statements invalidate itself?

The mothers' desperate search for their sons makes a universal statement.

In its initial formulation, it made universal statements meaningless, and this was seen as a problem for science.

The reformulated principle is still a universal statement.

A realistic scientific theory will consist of a complex of universal statements rather than a single statement like 'All swans are white'.

For example, a universal affirmative statement would become a universal negative statement.