(Since @Michael has made a gambling argument, here's the argument I've put off making on the grounds that it's a lot of trouble for little chance of success. It does complete the record, though. ;-/)

If you believe that p and refuse to believe that p v q, then your beliefs are inconsistent. If you hold inconsistent beliefs then you are vulnerable to a Dutch book, as follows.

You're a bookie and you believe the odds that Jones owns a Ford are 10-to-1, and those are the odds you offer. That is, if Jones does own a Ford, you pay out just $11 on a $10 bet that Jones owns a Ford - Jones owning a Ford is the heavy favorite -and nothing on bets that he doesn't; if Jones does not own a Ford, you pay out nothing on bets that he does, and $110 on bets that he does not.

For some reason, you think it's less likely that Jones owns a Ford or Brown is in Boston than it is that Jones owns a Ford. (No matter where Brown is, the chances are at least equal. You don't agree.) You set the odds that Jones owns a Ford or Brown is in Boston at even money. That is, if either is true, you pay out $20 on a $10 bet that either is true, and nothing on a bet that both are false; if both are false, you pay out nothing on a bet that at least one is true, and $20 on a $10 bet that both are false. (If you think it's irrational to believe that Jones owns a Ford or Brown is in Boston, you might even offer something crazy like 1000-to-1 against. You just have no opinion and offer even money.)

Suppose I strongly believe Jones owns a Ford, and I bet $10 that he does and another $10 that Jones owns a Ford or Brown is in Boston. I figure I'll win both. Here are my actual payouts:
Ford & Boston: $31
Ford & not Boston: $31
No Ford & Boston: $20
No Ford & not Boston: $0
It costs me $20 to play, so my results range from clearing $11 to losing $20.

Now suppose instead I bet $10 that Jones does not own a Ford, and I bet $50 that Jones owns a Ford or Brown is in Boston. Here are my payouts:
Ford & Boston: $100
Ford & not Boston: $100
No Ford & Boston: $210
No Ford & not Boston: $110
The point here is that it only costs me $60 to play. No matter what happens, I clear at least $40. For nothing. With no risk whatsoever. No matter what Jones owns or where Brown is, I am guaranteed to clear at least $40.


Appendix

Assuming a negligible chance that Brown is in Boston and that you're right about the likelihood of Jones owning a Ford, these are the expected payouts:
First player: about $28 for a $20 stake;
Second player, who makes the Dutch book against you: about $100 on a $60 stake.

Gettier's scenario (no Ford, not Boston):
First player loses $20 to you;
Second player takes $50 from you.

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I see nothing especially troubling in this way of speaking. — Cabbage Farmer

The trouble comes this way:
If you have good reason to believe that p, then you have good reason to believe that p v q, and if p v q is true you have a well-founded true belief, but it is possible for p to be false and q true, in which case your reasons for believing that p turn out to be irrelevant.

I predict great success for the hypothesis that it was either coffee or something else. But the issue is getting past tautology and giving some substance to the something else. (We're also in the neighborhood of Nelson Goodman's discussion of ceteris paribus in FF&F.)

Here's another tack. Smith's original approach to Jones's car ownership is in part an abduction: if Jones owns a Ford, he'll drive a Ford. There's additional support for the abductive thesis in Jones's history of Ford ownership. An alternative abductive thesis would have been that Jones is renting a Ford, but there's no additional support for that.

The two abductive theses are connected by having a common result, that Jones drives a Ford. Which brings us to an issue we haven't specifically discussed, which is disjunction elimination. That works like this:
1. A→C
2. B→C
3. A v B
4. C
Given that Jones drives a Ford, we could form the abductive hypothesis that Jones owns a Ford or Jones rents a Ford. That's clearly an improvement, as it would in fact be true. But it's not a foolproof method. Maybe Jones borrowed a Ford. How can you be sure you've thought of every possible explanation of Jones driving a Ford? (Or coffee drinkers getting cancer.)

On the other end, I think what bothers people about (g), (h), and (i), the arbitrariness of those distinctions, is that it's not obvious how you could eliminate them. What would be a consequence either of Jones owning a Ford or Brown being in Barcelona?

This is actually the same problem as above. It's the sort of thing that the TV show House relied on. "By any chance, have you been to Barcelona recently?" Wildly unconnected underlying issues can produce similar symptoms.

Sorry, I mean I don't understand what this says:

That the equations that are used now to measure some results are the same results that were observed 5 billion years ago. — Rich

That the equations that are used now to measure some results are the same results that were observed 5 billion years ago. — Rich

What?

Now all you have to show is that this would have been the same observation 50 million years ago. — Rich

Why is that what he has to show?

There's coffee.

There had been, a long time ago, a study linking coffee consumption to increased risk of cancer. But coffee drinkers are more likely to be smokers. Controlling for smoking, coffee's risk was downgraded. Then it went back up. The latest I think is that there's a risk associated with very hot drinks, not coffee per se.

Tests produce results, but they don't tell you why they produce the result they do. That's why justification can point away from the truth instead of toward it.

Science doesn't formulate unconnected disjunctions and then try and establish which arm is true. — unenlightened

That misses the point.

it could always turn out to be the truth of q reinforcing your belief that p. — Srap Tasmaner

Gettier just constructs an artificial example to show how this works. It happens when you think you're testing p but you're actually testing p v q v r.

S doesn't believe (p v q) unconditionally as Gettier and others here claim, but the conditional, "If really p, then (p v q)". — unenlightened

My version of the story goes like this: Smith puts the odds of Jones owning a Ford at 9-1, and the odds of Brown being in Barcelona at 1-99, so the odds of at least one being the case are a little better than 9-1. Then what? He makes a prediction, and he acts on that prediction. If you never actually place your bet, you don't get paid. If you never actually test a hypothesis, you don't learn anything.

Gettier tells us that Smith accepts (g), (h), and (i). Suppose Smith could ask someone who knows. If he started with (h), he would get the answer he expected, and continue to believe (h). But if he also asked about (g), and then about (i) too, he would not get the answers he expected.

Once he gets "false" for (g), he knows Smith doesn't own a Ford, right? The result for (i) confirms this, and now Smith will be strongly inclined to believe that Brown is in Barcelona.

But it's not that simple. There could be another factor here: can Smith tell the difference between Brown being in Barcelona, on the one hand, and his informant telling him the truth about (h) but lying about (g) and (i)? Can Smith tell the difference between testing "p v q" and testing "(p v q) & (z v x)"? There's nothing for it but to keep forming hypotheses and keep testing.

And so must we. We have to actually plump for "p v q" and get on with it, and be prepared to revise our beliefs as we go. That's why Gettier is useful: it could always turn out to be the truth of q reinforcing your belief that p. Science is hard.

Now, how do you test "If really p, then p v q"?

In my last response, I ignored -- God knows why -- that in my model, we've got a probability for q, so some of that was crosstalk.

I am actually interested in what you were getting at with "Probably p, but if not then definitely q", the crossover from probable or partial belief to belief, assertion, placing a bet, answering a question on a test, or any other way of acting on a belief that commits you to accepting the consequences of so acting, positive or negative.

But I don't think it's relevant to Gettier's argument. We clearly can and do and probably should and must commit in this way. Smith does, and does so with some justification. That's all Gettier needs.

The point is that if I'm asked what would follow if ¬p then I would withdraw the disjunction rather assert q. — Michael

This is worth quoting again.

If someone tells Smith that as a matter of fact Jones does not own a Ford, what happens to my jars?

We dump all the reds out of "Jones" and leave just a blue; "Barcelona" still has 99 blues and the one red. Chances, drawing one from each jar, of getting at least one red? 0 + 0.01 - (0)(0.01), so just 0.01.

Yeah, that's a belief you're unlikely to hold.

No, evidence is apprehended as being correlated with the belief which it is evidence for. So you have two things wrong here. First, the thing which the evidence is evidence of, is a belief it is not a fact. It cannot be called a fact, because the purpose of evidence is to convince someone of something which may or may not be true. Second, in order for it to be called evidence, it need not be intimately related to the belief, it needs only to be perceived as such. This is what makes it evidence of the thing, the fact that it is perceived as being related to the thing, whether or not it actually is, is irrelevant. — Metaphysician Undercover

I think I take the ground being wet as evidence that it rained recently because rain makes the ground wet.

How do you describe this scenario?

Right, I mean exactly that: if you believe that p, there's nothing contradictory about believing that ¬p→q and believing ¬p→¬q. It just means you get material implication.

We can leave out my habitual translation:
9. (¬p→¬q) & (¬p→q)
10. (F→T) & (F→F)
11. T

As I said, it's all of those false premises in 10 that are annoying.

3. p v ¬q
8. p v q
9. (p v ¬q) & (p v q)
10. (T v T) & (T v F)
11. T

3 and 8 are not contradictory. You have p as a premise, so you can get anything you like from ¬p. Also, you could save some time by just deriving both p v q and p v ¬q from p. q doesn't matter since you've already got p. That is, you don't need ¬q as a premise here at all.

Once again, I think we're really taking about Material Implication: Miracle or Menace?

Okay. I'll look again, but doesn't that mean your premises must have been inconsistent?

Where does 1 come from? Smith does not believe Brown is in Barcelona, but he doesn't believe Brown is not in Barcelona.

If he did, the whole exercise makes no sense.

I think if you believe probably A, then you (should) believe probably what's entailed by A, and that's how Gettier treats justification.

Look at the jars again. "Jones" is likely to give red, so "Jones" or "Barcelona" is likely to give red. It's just not true that if you get a blue from "Jones" you'll definitely get a red from "Barcelona". There's no reason to think that and I don't think Smith does as a matter of fact. **

Similarly probably (p v q) includes improbably ¬(p v q), not improbably (¬p v q). The "or" is inside what is believed probably, not outside, as above.

Taking a step back: if I have reason to believe you own a Ford, then I have reason to believe you own a vehicle, because owning a Ford entails owning a vehicle. That's Gettier's claim, that entailment preserves justification just as it preserves truth, however much or little there is.

** ADDED: Again, it's a bizarre bit of luck that Smith draws the one lonely red from "Barcelona".

We're just discussing what Smith believes. Specifically, does he believe p ∨ q? — Michael

And I still find this peculiar. Gettier tells us in so many words that he accepts (g), (h), and (i). The argument has to be that he shouldn't or couldn't. I guess you could go with "wouldn't" but that's not especially persuasive.

But it gets you "probably p v q" doesn't it? That's all Gettier needs p for.

But my point is that premise 1 is "p", not "probably p and possibly not p". — Michael

I'd still say this is unclear in Gettier's text, and what's more it's an interesting case, because we often do want to reason from premises we only hold probable.

As long as it is clear that you can't derive the disjunction. — unenlightened

And I'm not willing yet either to give up using or forbid others from using standard rules of inference.

We don't like the result, agreed. So we need some other rule to override here. The natural choice, to almost everyone, is to say that the belief of Smith's that turns out to be true is not in fact justified.

Perhaps relevance logic is more appropriate here — Michael

Don't know anything about relevance logic, but my intuition throughout has been that the justification for believing p turns out to be irrelevant to the truth of p v q.

But this is not an issue with logic per see, but something else. That something else could be Grice's maxims, for instance. — Srap Tasmaner

Hmm. Not right.

We do want the other principle at work to relate directly to our standards of rationality.

But that's just the usual issue with material implication.

Yes, you could use probably and possibly instead. — unenlightened

If you have "probably", you don't need "possibly" to stand in for "improbably": "probably" already covers both. "Possibly" is already in the background underwriting "probably".

I think the issue is that whereas this is valid:

1. p
2. p ∨ q
3. ¬p → q

This probably isn't:

1. B(p)
2. B(p ∨ q)
3. B(¬p → q)

Perhaps relevance logic is more appropriate here, denying the disjunctive syllogism. — Michael

Well this second 3 is still a conditional. We don't yet have something like
4. B(¬p)
That would force us to conclude that q. Once we get 4, our beliefs are inconsistent and something must be done.

The point is that if I'm asked what would follow if ¬p then I would withdraw the disjunction rather assert q. — Michael

Now this I agree with completely!

But this is not an issue with logic per see, but something else. That something else could be Grice's maxims, for instance.

So this is similar to the path of constraining justification: there are other rules besides logic in play.

The whole point of Gettier's Case II is that it's a bizarre coincidence that Smith's belief is true, and true for reasons that have nothing to do with his reasons for holding that belief.

It just doesn't matter if his holding that belief is also bizarre. There are two elements to a coincidence. Of course his holding that belief is bizarre! It's an abuse of logic. But if, for whatever peculiar and idiosyncratic reasons, he is inclined to form such a belief, it will be true and justified but not knowledge. You either accept that, and scrap JTB, or you block the supposed justification.

ADDED: Or I guess you could say that JTB "almost always" works, or "usually" works, or works for "normal" cases -- @Fafner is making a related argument elsewhere.

1a. " Believably p, but conceivably ¬p." — unenlightened

Isn't this what "probably p" already says? Why do this superposition analysis at all? Do belief and conception vary freely, or is there some relation there? If you just stick with probability, it's direct: as p seems less probable, ¬p seems more probable. Isn't that more sensible?

2b. Believably (p v q) but conceivably (¬p v q) — unenlightened

That says "Believably ¬p→q but conceivably p→q", but again manages to lose the connection between them. For any q, either p implies it or ¬p does, and if one doesn't, then the other definitely does.

Are we trying to reinvent "or" here?

p ∧ (p ∨ q) doesn't entail ¬p → q, — Michael

?

Are you using "entail" in some special sense?

Dang. I knew it was a mistake to speak for someone else.

I (and I think @T Clark) don't see how you and Rich can believe in a life force, and y'all don't see how we can't.

If there's some common ground, there's a basis for discussion. What's our common ground?

Agreed. People find material implication and inclusive disjunction counterintuitive, and then mistake their objections to them for objections to arguments that use them.

Because A ∨ B ↔(¬A→B), I guess.

That means ∨-introduction comes to P→(¬P→Q) for any Q, which, duh.

"Probably A, but if not A then definitely B" — unenlightened

That puts the probability of A ∨ B at 1. I put it at 0.901. Why would you put it at 1?

Suppose you're also pretty confident that Brown is in Barcelona, and we put 90 reds and 10 blues in "Barcelona" as well. Then the probability of getting at least one red is 0.90 + 0.90 - (0.90)(0.90), which is 0.99. Still not 1.

For comparison, if "Jones" has only red marbles in it, guess what the probability is that, drawing a marble from each jar, at least one of them will be red.

Gettier nowhere says that Smith believes Jones owns a Ford, only that he has good evidence for this belief. Let's say he thinks it highly probable.

We can represent Smith's belief thus: label a jar "Jones", and put 90 red marbles and 10 blue marbles in it. Red will represent "true" and blue "false".

Smith has no reason to think Brown is in Barcelona, so let's label another jar "Barcelona", and in this one we'll put 1 red and 99 blues. It's a long shot, but possible.

Smith should expect that if he draws a marble from "Jones" that the chances of it's being red are 9 in 10. If he draws a marble from "Barcelona", the chances of it's being red are 1 in 100.

What are the chances that, if he draws a marble from each, at least one of them will be red? I can tell you: it's 0.90 + 0.01 - (0.90)(0.01), which is 0.901.

No rational person would think it's reasonable to believe A but unreasonable to believe A ∨ B.

Gradations of your personal and subjective "likes" is not quantitative, as it is not an intersubjectively verifiable numerical measurement with meaningful units. — Jeremiah

Well what we'd look for if we did want to head down this road is behavior.

For example, there's Ramsey's famous suggestion about how to measure degree of belief. Suppose you're walking from one town to the next and come to a fork. You're not certain which is the correct way, but you think it's to the right. Now suppose you see a farmer out in a field. How far would you be willing to walk to ask him if you're going the right way? The more confident you are you're going the right way, the shorter that distance, and vice versa.

If you say like A&W better than Hires, we'd expect you to buy A&W more often, be willing to pay a little more for it, drive a little further to a store that carries it if you have to. How much further? How much more are you willing to pay?

Say you're a movie critic, and at the end of the year you publish a top-ten list. It's natural to attach numbers to the list precisely because to you the list is already well-ordered under the relation "better than".

There's a sort of implicit "unit of preference" here, but that's less important than being able to order the set.

You are not even looking at the variable of interest any more. — Jeremiah

That's a fair point. In responding I conflated two different acts of categorizing. I ended up talking about counting acts of categorizing, which wasn't helpful.

Back to the question at hand, what do you make of the fact that people do arrange their qualitative judgments comparatively? For instance, with your movie example: people say things like, "It wasn't as bad as the third Batman movie, but it was pretty bad." "Hires root beer is okay, but I'd rather have A&W", etc.