Classification of conjectures according to the number of counter-Examples

Question

I ask here if there is in mathematics academic classification of conjectures ( Weak, strong,middle) if it were disproved with counter examples, For example Pólya conjecture,The conjecture fails to hold for most values of $n$ in the region of $906,150,257 ≤ n ≤ 906,488,079$ .This conjecture has weak resistance to have one counter example or at most $2$ counter- examples and probably there are some conjectures which have infinity many counter-example , Now my question here :Is there in mathematics classifications of conjectures according to the numbers of counter examples which occurs ? for example we call it middle if it has only finitely many counter examples and conversaly we call it weak if it has infinity many counter examples and strong if it is proven to be true ?

Answer 1

I don't know if this is what you are going for, but conjectures which are false for a finite number of counterexamples can still be extremely useful/enlightening. For example, if the Riemann Hypothesis has finitely many counterexamples, then

$$\beta=\max\{\Re(z):z\text{ is a counterexample to the Riemann Hypothesis}\}$$

is well defined (and less than $1$). This then implies

$$\pi(x)-\text{li}(x)=O(x^\beta \ln(x))$$

which is already better than the current best such result.