Can only counterexamples disprove conditional statements Consider the statement "If $\sum x_n$ converges conditionally and every $x_n$ contains no digit 2 in its decimal expansion then $\sum x_n$ converges absolutely". It is almost certainly false, containing a digit 2 is irrelevant from converging absolutely. Yet it seems quite difficult/awkward to come up with a counter example. Please don't give one, its besides the point of this question. Is there any logical argument that disproves a conditional statement like this without having to provide an example?
Something to the effect of  "Property $X$ is irrelevant to Property $Y$ thus $X \nRightarrow Y $". Traditionally we're taught that this line of reasoning is cardinal sin so I am anticipating a fat "NO".
 A: The point is: If the claim is false and a counter example exists, then there are often many counter examples or even infinitely many (which applies IMO to your example).
This means that any generic proof will be a superset of (infinitely) many explicit counter examples.  And such "generic" counter examples are usually harder to find than a specific one; that's the reason why you'll typically see specific counterexamples.  The smaller the set is that the generic proof encompasses, the harder it is usually to write and to understand.  Yet trying some generic proof can lead you to (more) specific counter examples, or to more refined versions of the proposition.
Moreover, "conditional convergence" is a rather technical term from inifnite series, which means that re-ordering the sequence under consideration might change its convergence behavior or value.  So I am not shure if you confused "conditional statement" with "conditional convergence"?  "Conditional convergence" basically means "convergent but not absolutely convergent".
