Counterexample in the context of Rigor

Testability is a primary aspect of science and the scientific method. There are two components to testability:

1. Falsifiability or defeasibility, which means that counterexamples to the hypothesis are logically possible.
2. The practical feasibility of observing a reproducible series of such counterexamples if they do exist.

In short, a hypothesis is testable if there is a possibility of deciding whether it is true or false based on experimentation by anyone. This allows anyone to decide whether a theory can be supported or refuted by data. However, the interpretation of experimental data may be also inconclusive or uncertain. Karl Popper introduced the concept that scientific knowledge had the property of falsifiability as published in The Logic of Scientific Discovery.

In reasoning and argument mapping, a counterargument is an objection to an objection. A counterargument can be used to rebut an objection to a premise, a main contention or a lemma. Synonyms of counterargument may include rebuttal, reply, counterstatement, counterreason, comeback and response. An attempt to rebut an argument may involve generating a counterargument, or finding a counterexample.

In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit.

If a series converges, the individual terms of the series must approach zero. Thus any series in which the individual terms do not approach zero diverges. However, convergence is a stronger condition: not all series whose terms approach zero converge. A counterexample is the harmonic series