Well let me ask you a question in return. If you have a reasonable belief p, and a reasonable unconnected scepticism q (say p - that aspirin is an effective painkiller, and q - that Bluebeard's treasure is buried on Easter Island), what is to be gained by forming the disjunction, (p v q) ? How does S advance his knowledge, or understanding or in any way profit from forming his disjunction? Does it enable a test of p, or the building of a deeper theory or something? — unenlightened

Yes! It appears to be totally unmotivated, doesn't it? At the very least, it violates Grice's "Be relevant" maxim. It even seems to edge toward the logically tenuous mental gymnastics we associate with conspiracy theories. If this is science, it's pretty bad science, right?

I see the main issue as coincidental confirmation, so I'm ignoring Gettier's agenda most of the time. I imagine Smith learning that "Jones owns a Ford or Brown is in Barcelona" is true, but not learning what makes it true, and thus treating it as confirmation of his hypothesis that Jones owns a Ford. I have also imagined Smith not forming the disjunction at all, but simply making a test that he thinks is of Jones owning a Ford but is actually a test of "Jones owns a Ford or Brown is in Barcelona".

Suppose that's what happens, but then through other channels Smith discovers Jones does not own a Ford.* He might forget all about it, but if he is a good scientist, that unexplained positive will bug him. To get to the bottom of that, he'll have to be able to form this goofy disjunction.

Think about the discovery of cosmic microwave background radiation, how it went and how it could have gone. Suppose, contrary to fact, there is no CMB, and Penzias and Wilson were looking for it. They get this noise, check their equipment out, and think they've found it, but the source of the noise was actually pigeons nesting in their dish. What actually happened is the opposite: they weren't looking for it, checked their equipment, chased off the pigeons, and it was still there. They determine its characteristics as best they can, but have no idea what it is until someone tells them about the prediction that the Big Bang would cause such a thing.

Brown being in Barcelona is the pigeons in the first scenario and the CMB in the second. It's the unknown unknown. And when there are unknown unknowns, you can mistake noise for signal and signal for noise. To suss out what's going on you may eventually have to form odd disjunctions involving pigeons and the Big Bang.

That's my big picture version of what's going on. I think Gettier's examples are outlandish in order to make any claim of knowledge implausible, but he could have made them simpler. For example:

I leave my keys on the table, you mistakenly grab them on your way out, realize you have my keys rather than yours and put them back. I have no idea my keys ever moved. Do I know where my keys are? It's luck: I "know" but only because you put them where I put them, not because they stayed there. You chose the salient location for my keys, but you might not have, and I have no idea I'm relying on our shared rationality ...

My hunch is that Smith's disjunctions are unacceptable in part as a matter of linguistics, and insofar as that supports our communal rationality, he is violating a norm of some kind. They can also be criticized as you have done here, as being unmotivated, even pointless. In fact, I mentioned this about a week ago: if a disjunction is part of an argument from cases, what is the result both of these produce that could eliminate the disjunction? (Compare the CMB story, where pigeons and Big Bangs both result in noise.)

I have no idea-- finally answering your specific question-- and I think the lack of apparent rationale is why we are inclined to reject them. My thought when I brought this up before is that it explains our feeling that these disjunctions are arbitrary. And I suspect we could even measure that: how unlikely would an event be that could be caused (or enabled?) either by Jones owning a Ford or Brown being in Barcelona? I think most of us would guess pretty dang unlikely. (But keep in mind the Connections TV show!)

I think this is the neighborhood where most discussion of Gettier sets up shop. Is there something about the conditions (beyond Smith's control) that makes this not knowledge? Is it something in Smith's behavior, some norm of rationality he has violated?



* Coincidentally, I have driven three different Fords for several years each without owning any of them.

So what about the disjunctive syllogism?

If I assign to A a probability of r, and to B a probability of s, what probability should I assign to ~A & B? (That is, to ~A & (A v B).) That would be (1 - r)s. Since the probability of A v B is r + s - rs, it's also pr(A) + pr(~A & B). If pr(A v B) = 1, then if pr(A) goes to 0, pr(B) = 1. So there's no weirdness treating the usual disjunctive syllogism as a special case of standard probability.

I don't have Smith assigning a probability of 1 to Jones owning a Ford, and I don't have him assigning a probability of 0 to Brown being in, say, Barcelona. Those are assumptions of mine that I think are defensible from the text-- and from life-- but there's certainly room to argue otherwise.

So what should Smith's view be of the possibility that Jones does not own a Ford but Brown is indeed in Barcelona? Given probabilities of 0.90 for the Ford and 0.01 for Barcelona, he should assign a probability of 0.001 to Barcelona but no Ford. As it should be, since pr(Ford & Barcelona) = 0.901. So that's at least consistent.

But it has to be admitted that what I'm doing here is not-- what should we call it?-- "simply" inferring one belief from another. I allow Smith to form the prediction that Ford or Barcelona based on his hypothesis that Jones owns a Ford, but then in order to assign probabilities to it and to the disjunctive syllogism (to its premises actually, since he already has a prior for Barcelona), he does the math.

Thus I never see Smith being in the position of saying, "Probably A, but if not then definitely B."

Where do you derive this principle from? It isn't a law of logic. If I might use an analogy, the higher you want to build, the more secure you need to make your foundations. — unenlightened

Yeah, I have no justification for that (my thing about the credence you give a conclusion). I think it's a reasonable rule of thumb, something like Hume's saying that "the wise man proportions his belief to the evidence." For instance, in the case at hand of addition, the likelihood of A v B is higher than the likelihood of A, but that's because I'm smuggling in a prior for B.

As a matter of fact-- and this gets to your second point-- if A entails B, then the likelihood of B is at least as high as that of A. (If all F are G, there are at least as many G as F.)

So while there's intuitive support for the general idea of firm foundations and less and less certainty the farther your chain of inference carries you from those foundations, you have to be careful. If your theory as a whole is thought of as just a big conjunction of all of your current beliefs, and if some of those are less than certain, then all of them being true is less likely than some of them taken alone, because when you multiply the independent ones, their product is necessarily smaller. Sure. But our theories are more complicated than big conjunctions. There's a lot of dependence, entailments, conditional probabilities and disjunctions in there.

So I don't see Smith as overstepping the bounds of reason and landing in a puddle of nonsense. I see him as a victim of chance. Something extraordinarily unlikely happens, and it will challenge his otherwise orderly process of belief formation.

I'll just note that science, probability, and induction/abduction are what S does to arrive at his belief p. No quarrel with him there. — unenlightened

Yes absolutely.

Broad Agreement feels good.

No, it's always sound, because it already has your 'if' incorporated. I put it in capitals so you would notice. It is contentless because it does not claim that p is true. S wrongly makes the claim, that p is true, and then uses this formula to arrive illegitimately at (p v q). — unenlightened

Here's an argument:
Everyone in this room is happy.
Steve is in this room.
∴ Steve is happy.

That's a valid argument, whether or not either of the premises are true. If both of the premises are true, then it is also a sound argument.

I have taken you to be saying that Sharon is only entitled to make such an inference as shown above if the premises are true. It is conceivable that the premises are true but Sharon does not know this, in which case she is entitled to make an inference that she does not know she is entitled to make. And so it may be. If Sharon does know that the premises are true, then she also knows she is entitled to make the inference.

I have argued that making such valid inferences is one of the ways Sharon will try to determine whether the premises are true, by further exposing them to confirmation or disconfirmation. In this example, she would determine that Steve is in this room and then try to determine whether Steve is happy. If he is, the universal premise is partially confirmed; if he is not, then the universal premise is partially disconfirmed, but may still survive a reformulation like "Everyone in this room but Steve is happy," or "Almost everyone in this room is happy."

On my approach, Sharon knows the argument is valid, so she can come to know whether "Everyone in this room is happy" is true by assuming it true, hypothetically, making an inference and thus a prediction about each person in the room, and then testing those predictions.

She could also make no such hypothesis, and no such predications, but just ask everyone, tally them up and find that everyone or everyone but Steve is happy, whatever. There's really not much difference in this case.

The difference is that my approach obviously scales up and allows the use of statistics and probability, besides recognizing that raw fact-gathering is not the only thing we care about. We also need to make predictions, so we need to be good at it.

The curious thing is, by the time Sharon knows she is entitled to make the inference, she no longer needs to.

Suppose now Sharon shares her results with a colleague, Carol. Carol notices Steve's name on the list of people in this room, and, knowing Sharon's results, infers that Steve is happy. But do we just say that Carol knows that everyone in this room is happy the same as Sharon does? Isn't it rather the case that Carol is taking Sharon's results as given, for whatever reasons good or bad? That she is assuming Sharon's result is correct? And then she could test it, by, for instance, asking Steve if he is happy, thus confirming or disconfirming Sharon's result.

This hypothesis-prediction-test-revision cycle seems eminently rational to me and depends on making valid hypothetical inferences of unknown soundness.

What here do you disagree with?

[Disjunctive syllogism stuff in a future post.]

1. Substitution into belief statements can fail salva veritate
2. Substitution into Justification statements can fail salva veritate
3. S has in inadequate justification for (p v q). — Banno

I don't think we're talking about substitution here exactly. No one thinks p v q is equivalent to p; it's inferred from p. And explicitly we're not inferring what Smith believes. Smith makes these inferences and we're just told that he does.

This gives a weighted disjunction, (p(99%) v ~p(1%)). — unenlightened

Not that it matters, but I think you want that to be a conjunction.

IF p, then (p v q). That's valid, sound, true and contentless — unenlightened

It's only sound if p is true.

A person is to acknowledge the fallibility of his beliefs and refrain from making arbitrary unconnected pointless disjunctions of them as if they were necessarily true, because they ain't. — unenlightened

The conclusion of an inference merits no more or less credence than what you grant your premises. If you're uncertain about your premises, then you should be just that uncertain about your conclusions.

Why in your view is that not a sufficient acknowledgement of fallibility?

I agree, of course, that there is something odd about Smith's inference. Maybe there should be another rule brought to bear here. I just don't know what that rule would be.

No, only if it is an actual truth. Only if it is known, because then it is true. If it is only believed then it may not be true — unenlightened

Aren't you just conflating validity with soundness? I just don't understand the idea that inference is only possible from actual truths.

But in practice, beliefs are normally not known to be knowledge unless they are necessary. — unenlightened

What's a person to do then? Suppose I think I know that A. Should I infer B from it? Or only if I know that I know? Maybe I only think I know that I know ...

This gives a weighted disjunction, (p(99%) v ~p(1%)). And that does not lead to (p v q). It's so simple it seems to be invisible to everyone, but as soon as it is possible that ~p, the damaging disjunction (p v q) cannot be made at all. — unenlightened

So in your view we are only entitled to infer p v q from p if p is a necessary truth.

So one big flaw in Gettier is that he takes the justification to be one or two supporting propositions, and not holistic. — Banno

I don't think he does. The way Gettier sketches in the "strong evidence" Smith has is clearly just a gesture toward whatever we would generally count as strong evidence:

Smith's evidence might be that Jones has at all times in the past within Smith's memory owned a car, and always a Ford, and that Jones has just offered Smith a ride while driving a Ford.

So do you have some thoughts on justification? What is this holism of which you speak?

From "defeasibly p", (p v q) does not follow. In logic a thing follows or it doesn't; there is no 'defeasibly follows'. — unenlightened

We agree that whether B follows from A is purely a matter of logic, nothing to do with belief. I don't know why you think I'm arguing for extra logical constants. I've never ever said anything remotely like that.

If you infer B from your belief that A, and if your belief that A is defeasible, then so too should your belief that B be. That's normal inference, not some special kind just for defeasible beliefs.

Suppose you think A might be the case, but it's difficult to test for directly, but you know that A→B, and it's comparatively easy to test for B. If B comes back false, you apply modus tollens and conclude ¬A.

One way to look at this is like so:
1. A→B
2. ¬B
∴ 3. ¬A
That is, in the Great Book of Established Truth, A was never entered at all.

But in practice what happens is more like this:
1. A→B
--- 1.1. Assume A
--- 1.2. B
--- Test B
--- 2. ¬B
--- 2.1. A→⊥
3. ¬A
To get to the point of having something to test, you take A hypothetically, and infer B from it.

I think most scientists would be inclined to say there is no Great Book of Established Truth. Instead there is the Great Book of Not Disconfirmed Hypotheses. If B comes back true (as best we can tell), that's another tally mark for A, and that's the best we can do. That's why we have the book, to keep track of the A's that are doing pretty well in the confirmation department. As Hume said, all arguments about matters of fact are probable, not demonstrative. You can have a Great Book of Established Truth if you want, but you might as well leave it in the Math Dept. because they're the only ones who'll ever put anything in it.

If later there's even better evidence against B, they may both have to be moved to whatever the current volume is of the Great Book of Discredited Hypotheses.


** ADDED: We could of course have a "defeasibly follows from" in the sense that we could hold A→B to be only probable. It's a premise too, no real difference from holding A to be only probable. That's not our situation here, because though we can write P→P ∨ Q if we want, that's not really a premise. We're not entertaining the option that it could be false. It's an inference rule rather than a premise, and should really be written P⊢P ∨ Q, or $\small \frac{P}{P \lor Q}$.

Have not read.

Hoffman talks about Maxwell's demon and Landauer. But my memory is that a lot of the discussion of RNA, DNA, ribosomes, etc. is just focused on how they work. It seems like there should be another chapter or two to put some pieces of the puzzle together, but I guess that's because the idea here is to show what life is, rather than how it arose. It's not really intended to be a brief for abiogenesis. I'll probably check out Nick Lane's book, but I'm already reading four or five books, so it'll have to wait.

Enjoy Life's Ratchet.

I also read Life's Ratchet on apo's recommendation and it's excellent. There is relatively little about information** in it, and nothing I recall about semiotics, but the explanation of how the thermodynamics works is fascinating. Gets a little deep in the weeds on transport molecules, since I think that's what Hoffman spends a lot of his time doing, but worth powering through.

Once you can picture a kinesin binding an ATP to clamp one foot to a track, allostery causing the back foot to hydrolyze its ATP, and then relying on thermal motion to get the back foot to its new position -- wow. The combination of consuming free energy and "capturing"-- putting to use rather than resisting-- energy from what Hoffman calls the "molecular storm" of randomly moving water molecules, it's extraordinary.

** ADDED: But some nice stuff about regulation, which is, as @MikeL has been noting, what we're often looking to do with information.

He believes p with good reason, but he is not thereby entitled to argue formally from p, but only from (p v I am mistaken about p). — unenlightened

@Cabbage Farmer describes Smith somewhere as having a defeasible warrant to assert that p, and that's all he needs.

Think about how reductio works: assume that p, and then show that p leads to a contradiction or a known falsehood. And then plain old modus tollens.

It's the same thing here: Smith has strong evidence that Jones owns a Ford, but his belief is defeasible. If Jones walked in the office one day and said, "I don't know why I bought Fords all these years, this Chevy I just bought is fantastic!" Then poof, there goes Smith's belief that Jones owns a Ford. Why? Because Smith can immediately see that his theory would predict Jones not saying this. (The switch from material to subjunctive conditionals has a little wiggle room.)

There is all the reason in the world to reason from premises not known to be true, precisely so you can test them, expose them to possible defeaters. You can reason from premises you believe false as if they were true, precisely in order to show that they are false. (Again there may be a switch to counterfactual conditionals.)

In short, I think you're always entitled to inference. Inference is innocent. The credence you give your premises is usually the issue. If you hold certain what you shouldn't, then you block the modus tollens that would revise your belief. If you never infer, you block the modus tollens that would revise your belief.

I really have no idea what you're up to.

Is that so? Who, before Gettier, took it seriously? — Banno

Gettier cites the Theaetetus, Chisholm, and Ayer, for starters. Obviously the stars of Chisholm and Ayer have dimmed somewhat since then. Besides explicit support, I think the thrust of foundationalism, of empiricism in general, is toward such a position: a true belief grounded in experience, in the testimony of your senses, is the foundation of knowledge, that sort of thing.

Also, I think Gettier spurred many to consider defending the theory, because it feels like his argument is a parlor trick in some way.

If we understand the arrow above to be justifies or some cognate...

Then if the consequent is true, it is justified by any antecedent, true or false.

So anything justifies a truth.

Just sayin'. — Banno

The problem here is that you're switching in the middle from talking about justification to talking about truth. We expect both of these to hold, given that p→q:

(1) If p is true, then q is true;
(2) If I am justified in asserting (or believing) that p, then I am justified in asserting (or believing) that q.

But we cannot expect to freely mix and match:

(1*) If p is true, then I am justified in asserting q;
(2*) If I am justified in asserting that p, then q is true.

(1*) fails because I may have no idea that p; (2*) fails because I may be justified in asserting p though p and q both be false.

Inference does not confer either truth or justification; it only preserves whatever truth or justification is to be found in the premises.

What irks me about Gettier is that he appears to be assaulting a straw man. Who is it that believes knowledge is exactly justified true belief? — Banno

Well, it was over fifty years ago. A simpler time.

Plus, it's a theory with some pedigree.

Plus, "justification" is a pretty flexible word. Informally, I think of JTB just as getting the right answer for the right reasons. That sounds plausible doesn't it?

IS that the only choice, certainty nothing? — Banno

I don't know. But you can arbitrarily strengthen the justification and still be vulnerable to Gettier. At least it seems that way. I think the impulse to say that if a belief were really justified, you know, really, properly justified, the way God intended and no cheating, then it would have to be true -- I think this impulse is mistaken.

The time I've spent in this thread (time I will never get back) has led me me to think that the point of Gettier is this: justification can point away from truth instead of toward it. I find that pretty interesting.

A belief isn't justified if the justification is false.

But you are right; instead of being false, in this case the justification is insufficient. — Banno

All Gettier says is that the Smith has "strong evidence" for the proposition that Jones owns a Ford.

Do you recommend holding out for justification that absolutely guarantees truth?

As far as I am aware statistics was created by humans. — Jeremiah

Sure. Do you draw some conclusion from this? For instance, do you have an answer to the question you posed:

Does the string have length because that is an objective property of the string, or does it have length because we created the ruler? — Jeremiah

If you just mean the usual way, then we choose from among things that already exist, but things we make don't exist until we make them. If this what you mean?

@apokrisis
Have you looked at this challenge to Landauer? Odd coming right on the heels of this.

What grounds this move to split up smith's belief? On what basis do you posit Smith holding two beliefs? — creativesoul

Just "because"?

Quick note on the linguistics here:

(1) Mary gave me £10 because I won; and
(2) Mary didn't give me £10 because I won

usually both presuppose that Mary gave me £10.

The point of the statement is to highlight the reason for the action. The implication of the negative statement is that there was some other reason for the action taken. (Compare: "Mary didn't give me £10 because I didn't win.")

You get this a lot in Hollywood screenplays: "I didn't put you in the game because you're my son. I put you in the game because you earned it."

Similarly, someone could tell Smith, "Mary didn't give you £10 because Jones owns a Ford; she gave you £10 because Brown is in Barcelona."

Weasel words these, if you don't mind my saying. If I'm right about something, probability no longer applies. — unenlightened

?

I mean something as simple as this: I think there are 4 beers in the fridge because I think there are 3 Guinness and 1 Bud Light. Sadly, there's 1 Guinness and 3 Bud Light. I'm right about how many, but I've got the proportion wrong.

If you think about my stupid jars, it's obvious how this works.

If I'm weaseling about anything, it's that there are objective probabilities to get right or wrong. Sometimes there are, as with the jars. And sometimes you can be wrong about how probable an event is. I'm not offering a position in which there is no objective truth. Gettier gets dramatically reworked if you do that.

Where I'm at with this at the moment is that Smith does not arrive at his belief 'p' by formal logic, but by informal induction, — unenlightened

Together with abduction, yes. This is how we reason about matters of fact, sure.

and therefore he is not entitled (by logic) to treat his belief as a certainty, — unenlightened

Which he needn't; he only needs his belief to be justified. As Gettier puts this, he has "strong evidence" for (f).

which is required to form the disjunction with a random 'q'. — unenlightened

No, it's clearly not, as my jar model shows. You can form a disjunction of beliefs held only probable, not certain.

If Smith had the humility to assert in the first place, not 'p', but '(p v (I falsely believe p))' — unenlightened

But that just is to assert that p. Of course to say something is the case is to recognize that it might not be - it's the main reason we bother to make assertions. They are informative precisely because the facts they communicate are usually contingently so. And to recognize that is to recognize that you could be wrong. No one takes everything he says to be a necessary truth. But by asserting you commit yourself to the consequences of being right or being wrong. You place your bet, you answer the test questions, you test your hypothesis. What alternative is there?

I don't think humility saves you from Gettier. Suppose Smith never assents to p v q unadorned, but only as probable. We could still have a situation where Smith is right about the probability of p v q (and this is all we're talking about, not its truth) but his subjective probabilities are swapped: it's actually the "Jones" jar that is nearly all blue and the "Barcelona" jar that's nearly all red. Same problem, even without any claim of certainty or any belief held unconditionally true.

Is there some reason to insist that we cannot have justified true beliefs of this sort? — Cabbage Farmer

No. But for many philosophers the intuition here is that the justified true beliefs in Gettier cases are not knowledge, so it's a problem for such accounts of what knowledge is.

My reasons for believing that p are obviously relevant to my believing that p v q, but it will turn out they have nothing to do with what makes p v q true. It's a bit of luck that I believe p v q for one reason but it turns out to be true for another. (Abusing the word "reason", I know.)

This thread is almost entirely about the B in JTB, for reasons that pass understanding.

(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.

It's the "Reply" button.

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Replying (without quotation) or quoting will both notify the poster of your response.

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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.