Quining Gettier Cases

What is the epistemological relevance of Gettier counterexamples to the definition of “knowledge” as justified true belief (JTB)? Any purported instance of knowledge is either a Gettier case or not. If not, and it is not an instance of JTB, it is no counterexample. If it is not and is also a case of JTB, the belief is both justified and true. And if it is a Gettier case, then the belief is, of necessity, both justified and true. In no Gettier case can a belief be unjustified or untrue. To take a standard example, suppose I’m driving by a field and I see, from a great distance, some piles of shaven wool which I take to be sheep. Let’s also suppose that there are sheep in the field, though I never see them. In this instance some of my beliefs—such as the belief that those wooly things over there are sheep—are false, and I think this is what, unconsciously, makes it seem plausible that more than JTB is required for knowledge, for if we try and act on these belief or make inferences based on them we are liable to go wrong. But since these beliefs are false, they are not JTBs, and hence the scenario is no counterexample to the thesis that knowledge is JTB.[1] Yet my general belief to the effect that there are sheep in the field is both justified and true. I might go wrong if I try and make inferences based on the first, particular belief, yet that is false, and so not a JTB. As long as I confine my inferences[2] to the second, general belief, I will not go wrong, for it entails nothing about which things are sheep, where exactly the sheep are, how many there are, etc. It requires only that there are sheep in the field, and as that is true, it cannot entail any false proposition. For those, such as myself, who view epistemic practices as our means of ensuring (or trying to ensure) that our beliefs are true, Gettier cases should pose no problem, for in no such case can a belief be false, or entail false propositions. So my question is, if knowledge is more than JTB, why should we care whether we have it?

[1] My general belief that there are sheep in the field is true. We can question whether it is justified: The only evidence I had concerning the existence of sheep in the field was also evidence that those wooly things over there are sheep. Sheep really do exist in the field, yet nothing justifies me in believing those particular animals are in the field. In this case we might say we have evidence for the truth of a general belief without having evidence for its truthmakers.

[2] Of course I don’t mean to imply that in this scenario I realize that the first belief is false, as that would ascribe inconsistent beliefs to the hypothetical me. I mean only that inferences based on the second belief result in knowledge, while inferences based on the first do not.