Can Inferences Sometimes Lead to Knowledge Even if Their Premises Are False?
It seems a little strange to think so, and the philosophers I've asked about this in the last few days tend to say no. But here are three possible examples:
Inferential Ascent:
Consider the following rule: If P is true, then conclude that I believe that P is true. Of course, it's not generally true that for all P I believe that P. (Sadly, I'm not omniscient. Or happily?) However, if I apply this rule in my thinking, I will almost always be right, since employing the rule will require judging, in fact, that P is true. And if I judge that P is true, normally it is also true that I believe that P is true. So if by employing the rule I generate the belief or judgment that I believe that P is true, that belief or judgment is generally correct. The rule is, in a way, self-fulfilling. (Gareth Evans, Fred Dretske, Richard Moran, and Alex Byrne have all advocated rules something like this.)
And of course the conclusion "I believe that P is true" (the conclusion I now believe, having applied the rule) will itself generally be true even if P is false. I'm inclined to think it's usually knowledge.
One question is: Is this really inference? Well, it looks a bit like inference. It seems to play a psychological role like that of inference. What else would it be?
Instrumentalism in Science:
It's a common view in science and in philosophy of science that some scientific theories may not be strictly speaking true (or even approximately true) and yet can be used as "calculating devices" or the like to arrive at truths. For example, on Bas Van Fraassen's view, we shouldn't believe that unobservably small entities like atoms exist, and yet we can use the equations and models of atomic physics to predict events that happen among the things we can observe (such as tracks in a cloud chamber or clicks in a Geiger counter). Let's further suppose that atoms do not in fact exist. Would this be a case of scientific inference in which false premises (about atoms) generate conclusions (about Geiger counters) that count as knowledge?
Perhaps the relevant premise is not "atoms behave [suchly]" but "a model in which atoms are posited as fictions that behave [suchly] generates true claims about observables". But this seems to me needlessly complex and perhaps not accurate to psychological reality for all scientists who'd I'd be inclined to say derive knowledge about observables using atomic models even if some of the crucial statements in those models are false.
Tautologous Conclusions:
In standard two-valued logic, I can derive "P or Q" from P. What if Q, in some particular case, is just "not P"? Perhaps, then, I can derive (and know) "P or not P" from P, even if P is false?
What's the problem here? Why do philosophers seem to be reluctant to say we can sometimes gain knowledge through inference from false premises?