Here is where the bewitchment begins. The key word here is "accept". Accepting (g), (h), and (i) on the basis of (f) requires only realizing and accepting(believing) that the rules of correct inference allow such a thing. That much is indisputable.. — creativesoul

It should be clear that by "accept" Gettier means "accept as true".

They cannot all be true

They could all be true, and would all be true if Jones owns a Ford. Just as both of the below are true:

1. London is the capital city of England and/or pigs can fly
2. London is the capital city of England and/or pigs can't fly

One cannot form and hold three contradictory beliefs simultaneously.

...

One can believe that three contradictory propositions all follow from the same p.

They're not contradictory propositions. Contradictory propositions cannot follow from the same p (unless p itself is a contradiction). It's simple logic.

...Each of these propositions is entailed by (f). Imagine that Smith realizes the entailment of each of these propositions he has constructed by (0, and proceeds to accept (g), (h), and (i) on the basis of (f)...

If by "accept" Gettier means "accept as true", then he certainly means accept as valid. One cannot accept that all three are true, because they all three contradict one another. One can accept that all three are valid.

One cannot accept that all three are true, because they all three contradict one another. One can accept that all three are valid. — creativesoul

They don't contradict each other. I don't know why you keep claiming that they do. Again, both of these are true:

1. London is the capital city of England and/or pigs can fly
2. London is the capital city of England and/or pigs can't fly

No. Both are valid inference from p. You're neglecting the distinction between being valid and being true.

That is precisely the bewitchment.

No. Both are valid inference from p. You're neglecting the distinction between being valid and being true.

That is precisely the bewitchment. — creativesoul

I know the distinction between being valid and true. The sentences are both true, because London is the capital city of England.

Compare with:

1. London is the capital city of France
2. London is the capital city of France or I am in France
3. London is the capital city of France or I am in Germany

2 and 3 are valid inferences from 1, but 1, 2, and 3 are all false.

Precisely. They are not true as a result of being inferred from p. Smith knows that. Thus, he does not accept that all three are true as a result of realizing that all three are entailed by f. He accepts that all three are valid inferences.

He believes that all three are true by virtue of believing p and accepting the rules of valid inference.

Believing (p v q) is true requires only belief that p and belief that p v q follows from p.

It is the case that p v q follows from p, so Smith's belief that p v q is 'true'(valid) is justified by the rules and true - also by the rules. P is false, so Smith's belief that p is justified and false.

He believes that all three are true by virtue of believing p and accepting the rules of valid inference. — creativesoul

Yes, which is what Gettier said. And there is no problem with this.

Furthermore, if he really is justified in believing p, then because those three sentences really do follow from p, he really is justified in believing those three sentences.

See the post above.

You're mixing up truth and validity. To repeat the previous example:

1. London is the capital city of France
2. London is the capital city of France or I am in France
3. London is the capital city of France or I am in Germany

2 and 3 are valid inferences from 1, but 1, 2, and 3 are all false.

Now compare with:

4. London is the capital city of England
5. London is the capital city of England or I am in France
6. London is the capital city of England or I am in Germany

5 and 6 are valid inferences from 4, and 4, 5, and 6 are all true.

If I am justified in believing 4 then I am justified in believing 5 and 6. If I believe 5 and 6 then I know 5 and 6.

And contrary to your earlier claim, 5 and 6 are not contradictory (and neither are 2 and 3).

Show me where I mix up truth and validity.

When you say "so Smith's belief that p v q is 'true'(valid)".

His belief isn't just that p ∨ q is a valid inference from p (as with 2 from 1 above), but also that p ∨ q is true (as with 5 above).

Use Gettier's example.

7. Jones owns a Ford
8. Jones owns a Ford or Brown is in Barclelona
9. Jones owns a Ford or Brown is in Boston

Smith doesn't just believe that 8 and 9 are valid inferences from 7 (as with 2 and 3 from 1 above) but also that 8 and 9 are true (as with 5 and 6 above).

And, again, 8 and 9 are not contradictory.

Furthermore, if he really is justified in believing p, then because those three sentences really do follow from p, he really is justified in believing those three sentences.

No. If he really is justified in believing p, then because those three sentences really do follow from p, then his accepting them as valid is justified and true.

No. If he really is justified in believing p, then because those three sentences really do follow from p, then his accepting them as valid is justified and true. — creativesoul

He's also justified in believing that they are true.

In fact, even if he isn't justified in believing p, his accepting the inference as valid is justified and true, because validity doesn't require true premises.

But again, Gettier isn't saying "accepting as valid inferences", he's saying "accepting as true".

Yes. But be careful here...

His believing that they are true is nothing more and nothing less than his believing that the rules of disjunction allow him to randomly add any other statement to his belief that p and then call it "true" as a result of his believing that p.

Yes. But be careful here...

His believing that they are true is nothing more and nothing less than his believing that the rules of disjunction allow him to randomly add any other statement to his belief that p and then call it "true" as a result of his believing that p. — creativesoul

Or it's that he believes that the disjunction describes some fact about the world, like the disjunction "either there's something in your pocket or you're just happy to see me".

So it's not clear to me how your argument addresses Gettier at all. There's a true disjunction that is justifiably believed to be true. Under the JTB definition of knowledge, Smith knows that either Jones owns a Ford or Brown is in Barcelona. Nothing you've said refutes any of this.

The argument is really straightforward, as I explained here, and there's a difference between accepting that something is a valid inference and accepting that something is true, as I explained here.

But, of course, the main thing I've been trying to show you today is that the three propositions are not contradictory.

Jones owns a Ford or Brown is in Barcelona
Jones owns a Ford or Brown is in Boston

These are both valid inferences from a belief that p. Because p is false, they cannot both be true. They cannot both be true, because p is false and they state contradictory locations regarding the whereabouts of Brown. Smith believes that p, and has no idea of Brown's location. So, to make a statement about Brown's location is to state something that Smith, himself, does not believe.

Smith can believe that the rules of disjunction allow him to randomly add any other statement to his belief that p and then call it "true" as a result of his believing that p.

Smith's belief is justified by the rules and true by the rules.

None of that addresses Gettier.

That totally addresses Gettier. I'm showing that Gettier's claim that Smith's belief that g, h , and i are true aren't at all about the content of g, h, and i, but are rather Smith's belief about the rules of correct inference.

That totally addresses Gettier. I'm showing that Gettier has claimed that Smith belief that g, h , and i are true aren't at all about the content of g, h, and i, but are rather about Smith's belief about the rules of correct inference. — creativesoul

Of course it's about the content of g, h, and i.

I believe that "London is the capital city of England and/or pigs can fly" is true because of the content of that statement. It describes what I believe to be the fact that London is the capital city of England and/or pigs can fly.

Smith believes that "Jones owns a Ford and/or Brown is in Barcelona" is true because of the content of that statement. It describes what he believes to be the fact that Jones owns a Ford and/or Brown is in Barcelona.

Jane believes that "Jim either has something in his pocket or is happy to see Sarah" is true because of the content of that statement. It describes what she believes to be the fact that Jim either has something in his pocket or is happy to see Sarah.

Your objections just don't make any sense, and seem to rest on your own peculiar logic, given that you think that the three statements that can be inferred from p are contradictory.

One cannot believe that Brown is in three different locations. Thus, the only belief of Smith's that is operative in his deriving disjunction from his belief that p involves his belief about the rules of correct inference. Gettier even admits this much.

Try again.

One cannot believe that Brown is in three different locations. — creativesoul

Smith doesn't believe that Brown is in three different locations. Your argument is based on a false premise.

Try again.

For Pete's sake Michael...

Read the whole post and think about it.

I have read it. I addressed your confusion here.

There's a difference between just believing that p ∨ q is a valid inference from p and also believing that p ∨ q is true.

Smith believes that p is true.
Smith believes that p ∨ q is a valid inference from p.
Therefore, Smith believes that p ∨ q is true.

The full argument is here. Nothing you've said has refuted it.

The only belief of Smith's that is operative in his deriving disjunction from his belief that p is his belief about the rules of correct inference.

Do you disagree?

I don't even know what you're asking. He believes that p ∨ q is true because he believes that p is true. What more is there to say? If his belief that p is true is justified then his belief that p ∨ q is true is justified. If q is true then p ∨ q is true. He has a justified true belief. He knows that p ∨ q is true. The logic is very simple. Nothing you're saying refutes it.

He believes that p v q is true because he believes that p, and he believes that the rules of correct inference allow him to derive p v q based upon p.

The logic is inadequate in it's explanatory power.