Proof Isn’t All That
The first section can be skipped for those who know what necessary versus conditional truth is.
I recall an anecdote about John von Neumann which had a fellow asking von Neumann for the proof of some mathematical proposition. Von Neumann asked the fellow which of other several theorems the fellow might already know, and he mentioned two, whereupon von Neumann proved the proposition twice, along the two different paths. Implicit in the story is that he could have proven it upon the other paths as well.
We don’t know what this proposition was, so call it X. Since X is necessarily true, we can have knowledge of it, where knowledge, as some philosopher define it, is “justified true belief.” They’d say the justification comes from the proof and the belief comes from us as an act of our intellect.
But does truth come from the proof? Von Neumann showed there were many different ways of knowing a proposition was true, but the multiplicity did not add to the truth of X. X was just true, and always was, regardless whether anybody knew it or believed it. So there is a difference between the truth of some thing and our knowing it; or rather, there seems to be a difference in the justification of our belief the thing is true and its truth.
Let’s clarify. Take our old standby argument with premises E = “All Martians wear hats and George is a Martian” relative to the proposition Y = “George wears a hat.” Y given E is true; that is to say, we know that Y given E is true, that it follows. We may therefore believe Y given E, as a sort of joint proposition, say, Y-given-E. But Y by itself, sans E, is not a necessary truth. Neither is E by itself a necessary truth. But Y-given-E is. Y therefore is a conditional truth, given or accepting or believing or having faith that E.
A necessary truth is one which is true no matter what. Take non-contradiction. It cannot be true that Z = “X is true and so simultaneously is not-X true”. In other (and confusing words), not-Z is true. There isn’t any way to think that Z (except, as many do, by changing it so that Z is no longer Z, and then forgetting they made changes). Why is Z false? Who knows? God made it that way. Why is it true that W = “For every natural number r, r = r”? I have no idea. God made it that way. What is our justification for believing W? Faith? Or is it that we’re too light in gray matter to discover a proof—or, worse, a counter proof?
Actually, we do have reasons for believing not-Z and W. That we cannot think of how Z is true is a dandy reason for thinking it false, and all experience is that for every natural number r, r does indeed equal r. Induction supplies the rest. From our senses to the truth!
All this is just a sketch, which we needed for the real meat which follows.
A man hears his wife say she bought him a lottery ticket and he thinks to himself, R = “I now have a chance to win”. Unbeknownst to him, his wife was teasing. We know this, his wife knows this, but the man does not. The man accepts his wife’s word, conditional on which he believes R. R given the premise “Wife bought ticket” is thus a conditional truth. A believable truth, too, given he accepts (unconditionally) his wife’s word. R is not necessarily true, however, as is obvious.
Now Edmund Gettier famously claimed there were situations in which a person has a justified true belief, yet that belief did not meet the test of knowledge. Our lottery situation isn’t quite what he had in mind, because everybody would agree that R is a conditional but not necessary truth. To make this a “Gettier problem”, let’s add the premise “The man’s mother bought him a ticket for the same drawing but told nobody”. It is clear that R is now true, say Gettier followers, and the man is should believe it, but his claim doesn’t rise to the level of knowledge because his accepting R is based on his believing something which is false in fact (his wife’s joke).
But R is still a conditional truth to us and to the mother, who know of her actions. R, being contingent, can never be a necessary truth.
Gettier “problems”, I think, are based on forgetfulness. We forget who knows what and we forget what question is being asked of the evidence. To the man, R is conditionally true based on one set of premises, and to us it is conditionally false based on one set of evidence (just the wife’s statement) true based on another set (adding the mother’s). R is never true is the necessary sense. Plus, there are any number of premises which can exist, and which can be believed, that make it conditionally true. Even conditioned on the premise, D= “I, the man in this example, bought my own ticket” R is still only conditionally and not necessarily true.
In short, Gettier “problems” aren’t. This, incidentally, is one of the few cases where symbolic logic helps; I mean, being able to write the story down in symbols makes it much easier to see what goes where and who knows what, so that it is less easy to slip up.
I’m taking this example from Wikipedia, which (yes) does a good job explaining the set up.
The [justified true belief] account of knowledge is the claim that knowledge can be conceptually analyzed as justified true belief — which is to say that the meaning of sentences such as “Smith knows that it rained today” can be given with the following set of necessary and sufficient conditions:
A subject S knows that a proposition P is true if and only if:
- P is true, and
- S believes that P is true, and
- S is justified in believing that P is true
Recall von Neumann’s example and that X being true and anybody knowing X and the proof or belief of X are not the same thing. And also note that this definition mistakenly forgets to emphasize whether P is a conditional or necessary truth.
Here is a Gettier problem (also Wikipedia):
Smith has applied for a job, but, it is claimed, has a justified belief that “Jones will get the job”. He also has a justified belief that “Jones has 10 coins in his pocket”. Smith therefore (justifiably) concludes (by the rule of the transitivity of identity) that “the man who will get the job has 10 coins in his pocket”.
In fact, Jones does not get the job. Instead, Smith does. However, as it happens, Smith (unknowingly and by sheer chance) also had 10 coins in his pocket. So his belief that “the man who will get the job has 10 coins in his pocket” was justified and true. But it does not appear to be knowledge.
What has gone wrong?