Are you using "entail" in some special sense? — Srap Tasmaner

What I'm trying to get at is:

¬p → q ∨ (¬p ∧ ¬q)

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

Whereas unenlightened is saying:

¬p ∧ (p ∨ q) → q

Isn't this what "probably p" already says? Why do this superposition analysis at all? — Srap Tasmaner

Yes, you could use probably and possibly instead. I think the superposition idea is a neat way of doing it, but if we go your way, we have something like this:]S believes {
1. Probably p & possibly ¬p
2. p ⊨ p ∨ q
3. p ∨ q
4. p ∨ q ⊨ ¬p → q
5. ¬p → q
6. p ⊨ ¬p → q) }

7. ¬p (Gettier's stipulation)

Without the unadorned premise "p", one cannot make the move to (p v q), and Gettier's justified true belief that is not knowledge cannot arise.

Yes, you could use probably and possibly instead. I think the superposition idea is a neat way of doing it, but if we go your way, we have something like this:]S believes {
1. Probably p & possibly ¬p
2. p ⊨ p ∨ q
3. p ∨ q
4. p ∨ q ⊨ ¬p → q
5. ¬p → q
6. p ⊨ ¬p → q) }

7. ¬p (Gettier's stipulation)

Without the unadorned premise "p", one cannot make the move to (p v q), and Gettier's justified true belief that is not knowledge cannot arise. — unenlightened

If his belief is just "probably [Jones owns a Ford] and possibly not [Jones owns a Ford]" then his belief is true even if Jones doesn't own a Ford.

Whereas if his belief is false if Jones doesn't own a Ford then the first premise is the unadorned "p".

So what?

So what? — unenlightened

Smith's belief that Jones owns a Ford is true if and only if Jones owns a Ford. So his belief is the unadorned "p".

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.

Smith's belief that Jones owns a Ford is true if and only if Jones owns a Ford. — Michael

Smith's belief that Jones owns a Ford is false, Gettier insists. Jones does not own a Ford.

Smith's belief that Jones owns a Ford is false, Gettier insists. Jones does not own a Ford. — unenlightened

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

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. — Srap Tasmaner

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 (but maintaining the disjunction introduction).

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.

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

I don't think it's really about that.

I believe that if your name is John then your name is John or pigs can fly, but I don't believe that if your name isn't John then pigs can fly. So 3. alone is a problem.

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

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.

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

Yes yes, premise 1 is p, the disjunction applies, and explosion happens such that Smith believes anything and anything because with authorial infallibility, ¬p. I think I have offered a way out that preserves both knowledge and logic. If you don't want to buy it, find your own way out, or a hole in my logic.

"Possibly" is already in the background underwriting "probably". — Srap Tasmaner

Whatever dude. As long as it is clear that you can't derive the disjunction. Annoyingly, Michael's quote ate my vital strike.

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.

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.

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

2. p ⊨ p ∨ q you can keep with my blessing, along with the other rules of inference, because you don't have "p", you only have "probably p" which does not get you to (p v q).

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.

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

Yes yes, premise 1 is p, the disjunction applies, and explosion happens such that Smith believes anything and anything because with authorial infallibility, ¬p. — unenlightened

You seem to be conflating. It isn't:

¬p → Smith believes q

Rather it's (allegedly):

1. Smith believes that ¬p → q

At the moment the truth of any proposition isn't relevant. We're just discussing what Smith believes. Specifically, does he believe p ∨ q?

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.

I'm not sure, but I think it gets you ((probably p) v q) and not (probably( p v q)). Help me out a bit here, but I don't think either is sufficient for Gettier, because probably p includes improbably ¬p, in which case improbably (¬p v q) ? So we have probably (( p v q) v improbably( ¬p v q)), which is a whole 'nother disjunction, with which I have no quarrel.

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 shouldn't or couldn't. I guess you could go with "wouldn't" but that's not especially persuasive. — Srap Tasmaner

I think this is the problem:

1. I believe ¬q
2. ¬q ⊨ ¬p → ¬q
3. I believe ¬p → ¬q

4. I believe p
5. p ⊨ p ∨ q
6. I believe p ∨ q

7. p ∨ q ⊨ ¬p → q
8. I believe ¬p → q

The issue is that the rules of entailment can lead to having contradictory beliefs (3 and 8). unenlightened overcomes the contradiction by denying 6. I overcome the contradiction by denying 8.

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".

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.

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. — Srap Tasmaner

Just an example of how the rules of entailment can lead to contradictory beliefs.

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

The premises are just "I believe ¬q" and "I believe p".

Say "p" is "London is the capital city of England" and "q" is "pigs can fly".

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?

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.