CASE 1: (due to Jay Allman)
Prof. Nitpick teaches an undergraduate class in propositional/predicate logic. On Monday, he gives a homework assignment which requires his students to evaluate a completed proof, marking whether each step of the proof follows from a previous step (or steps) via the rule cited in that step's justification.
That night, Shirley Knott, who is a competent student in the class, carefully evaluates the proof, and she decides that one of the steps misapplies the rule of universal generalization. So she marks the proof as being invalid. She is correct that the proof makes this mistake, and she is justified in believing the proof is invalid because she has carefully checked each step against her (carefully taken) notes about how the rules are supposed to be applied.
On Tuesday, Shirley compares her homework with that of her boyfriend, Randy McGlib. Randy is a careless student and not very good at logic, but he has a radiant (and completely unjustified) confidence in his own logical abilities, and he has far more confidence in his abilities than Shirley has in her own. Randy thinks the proof makes no errors, and he soon convinces Shirley that she was wrong. So she gives up her belief that the proof is invalid and decides that it is valid, going so far as to alter her own notes on the rule of universal generalization to bring them into line with her new (and faulty) understanding of the invalid proof.
On Monday night, did Shirley know that the proof was invalid?
Mr. Nickfit, a smoker, has been considering the question of whether or not smoking causes cancer. He was told by his doctor that it does. He has also read (and understood) some authoritative meta-analyses of the evidence in professional medical journals which support the claim. On balance, therefore, Mr. Nickfit has some fairly good evidence (probably far better than you or I) that smoking causes cancer. In addition, Mr. Nickfit understands that this evidence supports the claim that smoking causes cancer and comes to believe as a result that smoking causes cancer. Suppose further that smoking does cause cancer. So, Mr. Nickfit has a justified true belief that smoking causes cancer.
Suppose, however, that Mr. Nickfit is currently so disposed that fairly weak counterevidence to the contrary will result in his changing his belief (i.e., believing that smoking doesn’t cause cancer). Think of the case as one in which Nickfit correctly uses his evidence to justify his belief while nevertheless being confused in various ways about how justification works. Specifically, he understands that his evidence gives him a reason to believe that smoking causes cancer; what he fails to understand is that this evidence ought to continue to compel him to believe that smoking causes cancer even in the face of relatively weak counterevidence. In some sense, Mr. Nickfit fails to make use of a rational diachronic update procedure on his evidence.
Does Mr. Nickfit know that smoking causes cancer?