I was rereading Bill Lycan’s paper “On the Gettier Problem Problem”:http://www.unc.edu/~ujanel/Gettier.htm and I noticed a couple of odd things about his view that knowledge justified true belief with ‘no false assumptions’. I think Lycan does a pretty good job in arguing that the cases that were traditionally thought to pose a problem for this view do not really pose such a problem. But still there are three kinds of cases where I think something needs to be said.

First, it seems to me much too strong to rule out all cases of inference from false premise. It doesn’t seem plausible to say that an engineer doesn’t know his bridge will stand up because he presupposed a little bit of Newtonian theory in his calculations. This is a fairly well known case and I think people generally think there’s a fairly easy fix. (Take the assumption not to be the perfect accuracy of Newtonian theory, but the approximate accuracy, and now the assumptions the engineer makes are all true.) But I suspect the problem here is much more general than this one case, and the solution won’t in fact generalise.

For a striking case, consider “Kevin Davey’s”:http://www.ku.edu/~philos/faculty/kdaveycv.html discussion in the “British Journal for the Philosophy of Science”:http://bjps.oxfordjournals.org/content/vol54/issue3/index.dtl of path integrals, as introduced by Feynman. We know there is some false assumption going on here, because the mathematical theory is contradictory. It seems to turn out that provided we are careful with how we apply the theory, provided we are ‘inferentially restrictive’ in Davey’s terminology, we avoid contradictory results. I’d even say that we gain knowledge this way. The difficulty is that no one knows what a consistent (let alone true) mathematical theory that does the work path integrals currently do might look like, certainly no one knows how to replace the theory with a true version, and it doesn’t look (at least to me) like this is a case where the theory we use is in any sense an approximation to the theory that is true. (Could a contradictory theory approximate the true theory?) Either we have to say that the physicists here are not getting knowledge about the world, or we have to put quite sharp limits on what we say about inference from a false premise.

For a second worry, consider a case where I’m in a state that we’d usually call a paradigmatic instance of knowing that p, but I also have some false belief that q. Presumably the fact that in learning that p I acquire the false belief that p & q doesn’t mean I fail to know p. But it isn’t clear why this doesn’t follow on Lycan’s theory.

Finally, there seem to be really difficult cases of existential Nogot which require quite a bit of fancy footwork. Here are two.

_Testimonial Nogot_
A acquires a justified false belief that Smith owns a Ford. In fact Smith’s workmate Jones owns a Ford, but A’s only reason for believing that someone in the office owns a Ford is that he believes Smith owns a Ford. B asks A, what brands of car are owned by people in the office, and A says “Holden and Ford.” B comes to truly believe that someone in the office owns a Ford.

I guess the false assumption there is that A knows that someone in the office owns a Ford.

_Faulty Machine_
A machine at the front door of the office measures whether anyone going in is over 2 metres tall, and a light goes on if that is the case. (The light goes off at the end of each day.) Smith, who is just under 2 metres tall, walks in and the machine malfunctions, measuring him as over 2 metres tall, and lights up. Jones, who is over 2 metres tall, walks in through the back door. A sees the light and infers that someone in the office is over 2 metres tall.

Here I think the false assumption is that the person who caused the light to turn on is over 2 metres tall. But note that this implies a lack of knowledge in the following case.

_Faulty Machine part II_
A machine at the front door of the office measures whether anyone going in is over 2 metres tall, and a light goes on if that is the case. (The light goes off at the end of each day.) Smith, who is just under 2 metres tall, walks in and the machine malfunctions, measuring him as over 2 metres tall, and lights up. Just after that Jones, who is over 2 metres tall, walks in through the front door, and he would have caused the light to go on if Smith had not. A sees the light and infers that someone in the office is over 2 metres tall.

It seems a little wrong to me to deny knowledge in that case, but I’m not sure how a theory like Lycan’s can distinguish the two.

The reason for all this is that the tweaks I’m using to turn my theory of justified belief into a theory of knowledge bear a strong resemblence to Lycan’s theory of knowledge, so any bugs in his theory might be replicated in mine. So it’s find ’em and exterminate time around here!