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.
6 comments:
According to the (unfashionable) no-false-lemmas account, Chisholm's original example is a Gettier case because the subject's justified true belief that there is a sheep in the field is based on a false belief that that thing over yonder is a sheep. In your scenario, there is a continuum of cases in which "that thing over yonder is a sheep" goes from false to true, because there is a continuum of cases in which the referent of "that thing over yonder" goes from cutout to sheep. So, on this account, gettieredness-by-degrees is a symptom of reference-by-degrees.
That's an interesting diagnosis! And then, maybe, you have truth-by-degrees in the lemma.
Maybe one can avoid the no-false-lemmas move here by supposing that we have a direct non-inferential perceptually based belief that there is a sheep there. (Though I feel like you would more naturally end up non-inferentially having the belief that *this* is a sheep.)
BTW, here's a fun variant. It's around sunset, and the sun is low on the horizon, and on the side of the barn you see a shadow as of a sheep. You conclude there is a sheep. But as before there is a cutout of a sheep between the sheep and the barn. You now vary the opacity of the cutout. When the cutout is fully transparent, you know about the sheep.
But now I am not quite sure what to say about the case where the cutout is fully opaque. Is the shadow on the barn the shadow of the sheep or of the cutout? It's the sheep that's doing all the blocking of the light. The cutout isn't actually blocking any light--only counterfactually (if the sheep weren't there, the cutout would block the light). So it seems you're seeing the shadow of the sheep. But then you should know. However, it doesn't quite feel right to say you know.
Maybe the thing to say is this. The cutout is blocking the shadow of the sheep, and what you have on the wall isn't a shadow of the sheep. But it's also not the shadow of the cutout. It's a shadow, but not a shadow of either the sheep or the cutout. I suppose Sorensen has a solution to this in his shadows book, but I haven't read it.
I confess I haven't read Sorensen's book either, but I press on.
I suspect that the "shadow of" relation is inessential to your example. Suppose I didn't have that concept, I just had the concept of shadows of various shapes, and I knew everything that optics can tell me about their origins. Then you still have your puzzle, without me having to wonder whether the shadow is the shadow of a sheep. In the case where the cutout is fully opaque (and assuming that I am unaware of the cutout), I form a true belief that there is a sheep there, but it's a belief that I wouldn't form if I were better informed, so arguably(*) it's not knowledge, and then you can twiddle the opacity of the cutout to build your slippery slope.
(*) "arguably" = "I haven't made my mind up about this yet".
Good! I think your observation shows that the reference-by-degrees is not the only basis of Gettier-by-degrees. Actually, I now think it can't be the basis in all cases like the original version. Suppose that my metaphysics is one on which I believe neither in sheep nor in particles, only the wavefunction of the universe. When I say "There is a sheep in the field", I mean something like: "The wavefunction of the universe is arranged sheep-in-the-field-ishly." But now there is nothing for me to have reference-by-degrees-to.
And opacity is not the only way of generating these cases, too. I imagine (without knowing much about the relevant biophysics) that my eyes work something like this. Light increases the energy level of the sensors in my retina, which increases the probability of the sensors emitting an electrical pulse in the optic nerve. But the background energy level is non-zero, so there is a non-zero background probability of the sensors emitting an electrical pulse without any light input. Thus, my apparent perception of a sheep in an ordinary field (with no cutouts or sheepdogs or anything like that) is always caused by a probabilistic process based on the total energy level of the rods and cones. This total energy level is the sum of a background energy level and a light-caused input. We can never say that my perception is wholly caused by the light--the background energy level always increases the probability of the electrical pulse, and there is no fact of the matter as to "which part" of the energy level caused the pulse. So now imagine that we increase the background energy level of those rods and cones that would normally be involved in seeing a sheep so that the proportion of the energy coming from the background is bigger and bigger. Eventually, it will be very likely that I would have a perception as of a sheep whether or not the sheep was there, and "arguably" then I don't know. But in the the ordinary case, where the background level is low, I don't know. And maybe there is never a fact of the matter as to whether it is the sheep or the background energy in the sensors that has caused the perception.
I think I agree with everything in your second paragraph, but I need to hear more about the first.
Here's one way it could go: you distribute credences over possible wave functions by doing a lot of hairy matrix arithmetic and that's all -- no credences assigned to lemmas along the way. In that case -- arguably! -- you don't have the sort of cognitive structure that is susceptible to gettiering.
Here's another way it could go: you do reason step by step, and en route to your conclusion that "The WFOTU is arranged sheep-in-the-fieldishly" you believe something that stands to "That thing over yonder is a sheep" as your conclusion stands to "There is a sheep in the field". In that case you at least have a false lemma. Now I suppose it's an open question whether it's possible to operate (or for outsiders to ascribe) a system of beliefs of the form "The WFOTU is arranged ...ishly" where the "..." has no logical structure; and if not, what logical structures are needed. Perhaps the hypothesis that "I believe neither in sheep nor in particles, only the WFOTU" cannot be maintained.
I didn't mean the first paragraph to challenge the idea that there is a false lemma, but to challenge the idea that there is a degreed failure or shift of reference. For an agent with the view in question only ever refers to the WFOTU, and that reference doesn't fail or shift.
That said, I have for a long time been thinking that if you are a good Bayesian, standard Gettier cases don't work against you. In the standard sheep/cutout case, your observations give you good reason to believe something like: (SC) "There is just a sheep or just a cutout or a cutout in front of a sheep or a sheep in front of a cutout". And then your priors tell you correctly that P(just a cutout | SC) is low, and hence P(just a sheep or a cutout in front of a sheep or a sheep in front of a cutout) is high. You then correctly conclude that P(sheep) is high. Nothing went wrong with your reasoning.
Granted, you would also have a high credence in "just a sheep", but you don't simply derive "sheep" from "just a sheep", but from "just a sheep or a cutout in front of a sheep or a sheep in front of a cutout". So I think you're not Gettiered, and maybe you even know that there is a sheep.
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