Gettiered by degrees

Consider a standard Gettier case. A cutout of a sheep in a field hides a sheep behind it. At that distance, the cutout looks just like a sheep. You have a justified true belief that there is a sheep, but you don’t know it (or so the story goes).

Now imagine that cutout is to some degree transparent, so some of the whiteness you see is in fact from the sheep, and some from the cutout. Consider the continuum of cases as the cutout goes from fully opaque to full transparent. Perhaps it fades from opaque to transparent as you’re looking—all without you knowing that it is fading. When it’s fully or nearly opaque, you are Gettiered and don’t know there is a sheep. When it’s fully or nearly fully transparent, you know there is a sheep.

Supposing that knowledge has a distinctive value over and beyond the value of justified true belief, it seems plausible to think that this value increases monotonically with the transparency of the cutout. If the cutout is becoming more and more transparent before your eyes, you are gaining epistemic value, without noticing you are doing so.

It’s an interesting question: What kind of a function is there from cutout-transparency to value? Is it continuous, or is there a transparency threshold for knowledge at which it jumps discontinuously? If it is continuous, is it linear?

I have to confess that these kinds of questions seem a bit silly, and this gives some ammunitition to the thought that knowledge does not have a distinctive value.