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.

Belief, nor justification, nor inference confer truth. — unenlightened

Yes. Should have made it clearer that inference preserving truth is something like a precedent for what we expect inference to do with justification.

On belief and assertion, I defer to Moore's paradox: asserting that P appears to carry with it a non-cancelable implicature that I believe that P. You can modify your degree of belief with "I'm not sure but I think" and the like, but you can't set it to zero.

Naturally if your belief in the premises of an inference is less than 100%, your belief in the conclusion should be less than 100%. Being the conclusion of an inference doesn't add or subtract certainty. -- We're talking here about perfect entailment. If you only have "If P, then it's likely that Q", that's a whole 'nother deal.

When I first saw Gettier, I had a similar reaction as creative, and as you're hinting at here -- that the "if p" tags along. As it turns out, this is the aspect of modus ponens that Tarski highlights by calling it the "rule of detachment" -- you get to detach the conclusion from the argument for it.

And I think that's right, with the proviso that your degree of belief in the conclusion, or the degree to which belief is justified, will track your degree of belief in your premises, or the degree to which those beliefs are justified.

I think there's a misunderstanding here.

I wasn't claiming that, to take your example, people can simply be pegged to a spot on some approval scale. I was saying that part of classifying their qualitative judgments as "approve" or "disapprove", say, is that we can count them -- four approve, six disapprove. Predicates need quantifiers. Seven say this tastes good, nineteen say it doesn't.

4. Logical implication is a justification. — unenlightened

I'm not crazy about this one. (But agree with everything else.) I'd rather say something more like what Gettier says: whatever justification the premises have, the conclusion inherits. As I said earlier, inference is not expected to confer truth, but to preserve it.

There isn't a one-to-one map between an organism's features and its genome. — Srap Tasmaner

Should also have explicitly said there's not a single genome to find its way around the valley. You get whatever you get when some part of the population's gene pool makes it over there.

I'm far from being an expert — Srap Tasmaner

Heh. I've garbled the science by the way I'm using the word "gene" but I think what I was trying to say is okay. Trying to unrust.

. This would require that every mutation was either positive or neutral. There could be no negative mutations, as this would wipe out the organism — MikeL

I'm not sure this is the right way to look at it, although I'm far from being an expert. There isn't a one-to-one map between an organism's features and its genome. How and whether a gene is expressed depends on lots of factors, including other genes. Mutation is the factor we're most interested in, and carries the greatest share of the randomness burden, but in addition to environment (including the developmental environment) there's the shuffling of genes in sexual reproduction.

Every variant in the population is either more suited or less suited to its environment or has not net change — MikeL

I think again that's assuming all changes in the genome show up immediately in way that can be readily judged based on the current environment. Suppose part of a population carries an unexpressed gene that's useful when the environment changes, and part of that sub-population carries a variant of another gene that allows that useful gene to be expressed, then they'll end up winning.

Anyway, I agree with almost everything in your last two posts. Did you think I wouldn't for some reason?

But doesn't classifying or categorizing things presuppose the possibility of counting them? The qualitative and quantitative are different, yes, but you can't have one without the other.

, there are not billions and billions of different lifeforms on earth, there is only one organism covering the adaptive landscape like a mat. — MikeL

That's really not a bad starting point in my opinion. The first three billion years of life on Earth is single-cell organisms. The last billion is multicellular, and the cells of any one multicellular organism are not that different from the cells of another. DNA, ribosomes, microtubules, membranes, all the machinery developed over three billion years is common to everyone.

@creativesoul has nothing to say about justification, as he will tell you himself. His issue is something about the psychology of logic, I think.

Right, that's part of Gettier's setup. The only belief he attributes the Smith is the belief that p v q; he seems purposefully to avoid attributing p, which will turn out to be false anyway. Gettier only relies on p for justification, not for truth.

This is what I see as the principal deficiency in describing evolution in terms of survival. There is no being, or thing which survives, they all die. There is no survival. Evolutionary theory attempts to get around this problem by assuming the real existence of an abstract thing, a variety, or species, which survives. — Metaphysician Undercover

Well, insofar as the phrase "survival of the fittest" has any use, it's just this: you don't get to reproduce if you don't survive. Evolution is about populations; not individual organisms. The concept of species only comes into the theory as the question of whether two populations can interbreed.

Do you have a different understanding of evolution by natural selection?

You, more or less, agree with me. — TheMadFool

Except for the part where I didn't.

Got it. Thanks. (I was confused because I hadn't seen the earlier reply.)

So how does creative evolution handle a case like this?

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