See my last post.

"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)."

Yes?

That is Smith's deduction.

...for any proposition P, if S is justified in believing P, and P entails Q, and S deduces Q from P and accepts Q as a result of this deduction, then S is justified in believing Q.

(P v Q) is the conclusion to an argument. "Because" operates the same as "therefore." Smith believes (P v Q). Why? Because P and the rule of addition. — Chany

Yes. It's really that simple.

And because you can't assume that Smith knows the law of addition, Gettier specifies that he does; and because you can't assume that he actually makes the inference he is entitled to, Gettier specifies that he does.

A1. I believe that John is a bachelor
A2. If John is a bachelor then John is a man
A3. I believe that John is a man because he is a bachelor
A4. I believe that John is a man
A5. John isn't a bachelor
A6. John is a man

Not like Gettier Case II.

B1. I believe that p is true
B2. If p is true then p ∨ q is true
B3. I believe that p ∨ q is true because p is true
B4. I believe that p ∨ q is true
B5. p isn't true
B6. p ∨ q is true

Again. Not like Gettier Case II.

B1. I believe that p is true
B2. If p is true then p ∨ q is true
B3. I believe that p ∨ q is true because p is true
B4. I believe that p ∨ q is true
B5. p isn't true
B6. p ∨ q is true

Again. Not like Gettier's Case II. — creativesoul

That's exactly Gettier's Case II.

No, it's not...

It is wrong.

...for any proposition P, if S is justified in believing P, and P entails Q, and S deduces Q from P and accepts Q as a result of this deduction, then S is justified in believing Q.

It doesn't follow that thought/belief process...

That's a quote from Gettier explaining why Smith believes (g), (h), and (i). Smith believes (f) and Smith recognizes (g), for example, is entailed by (f) and accepts (g) as true. Therefore, Smith believes (g) on the basis of (f). Therefore, Smith believes (g).

Put another way, Smith believes the P. Smith recognizes (P v Q) follows from P. Michael is correct: Gettier says that Smith believes (P v Q). And what does Gettier say is Smith's reasoning for his belief? Smith believes P and recognizes (P v Q) is entailed by belief in P.

B2 is wrong. — creativesoul

No it's not. It's another way of saying "P entails Q".

Smith believes (f) and Smith recognizes (g), for example, is entailed by (g) and accepts (g) as true. — Chany

You have a typo there. You meant "(g), for example, is entailed by (f)".

Thanks. Fixed.

That's a quote from Gettier explaining why Smith believes (g), (h), and (i). Smith believes (f) and Smith recognizes (g), for example, is entailed by (g) and accepts (g) as true. Therefore, Smith believes (g) on the basis of (f). Therefore, Smith believes (g).

Put another way, Smith believes the P. Smith recognizes (P v Q) follows from P. Michael is correct: Gettier says that Smith believes (P v Q). And what does Gettier say is Smith's reasoning for his belief? Smith believes P and recognizes (P v Q) is entailed by belief in P.

Yes. I know what it is. The full text of Case II preceding his conclusion that Smith is justified in believing Q is below. It warrants very careful attention...

Gettier wrote:

Let us suppose that Smith has strong evidence for the following proposition:

(f) Jones owns a Ford.

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. Let us imagine, now, that Smith has another friend, Brown, of whose whereabouts he is totally ignorant. Smith selects three placenames quite at random and constructs the following three propositions:

(g) Either Jones owns a Ford, or Brown is in Boston.
(h) Either Jones owns a Ford, or Brown is in Barcelona.
(i) Either Jones owns a Ford, or Brown is in Brest-Litovsk.

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). Smith has correctly inferred (g), (h), and (i) from a proposition for which he has strong evidence...

Note the above stopping point. The quote ends at the precise point where Gettier's next line concludes(by necessary implication) that Smith believes Q.

Believing Q is precisely what's at issue here. Q is (p v q). Believing (p v q) is believing that (p v q) is true. Hence, Smith's believing Q is nothing less than Smith's belief that:((p v q) is true).

So, using Case II, Gettier has filled out his earlier formulation. Here it is again...

Gettier wrote:

S is justified in believing P, and P entails Q, and S deduces Q from P and accepts Q as a result of this deduction...

Note here that this quote's stopping point coincides with Case II's, as shown directly above. So, as Gettier says, Smith believes Jones owns a Ford. Smith constructs (g), (h), and (i). All of which are (p v q). So, Smith believes p, and deduces (p v q) from p and accepts (p v q) as a result of this deduction. There is nothing in the above two quotes about Smith's thought/belief process that the first two premisses below cannot effectively exhaust...

p1. ((p) is true)
p2. ((p v q) follows from (p))

Now, it is well worth mentioning here that nowhere in any of this(the above direct quotes from Gettier) is anything at all about Smith's believing Q. That is of irrevocable significance. It is a crucial point to consider here. Smith hasn't yet gotten to the point where he has formed and/or holds belief that:((p v q) is true)...

But oddly enough, Gettier concludes that that is the case, as is shown by his saying...

Gettier:

...Smith is therefore completely justified in believing each of these three propositions...

...and...

...S is justified in believing Q.

He lost sight of exactly what believing Q requires. It requires precisely what follows...

p3. ((p v q) is true if either (p) or (q) is true)
C1. ((p v q)) is true because (p))(from p1,p3)

Now, it is well worth mentioning here that nowhere in any of this(the above direct quotes from Gettier) is anything at all about Smith's believing Q — creativesoul

"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)."

Gettier literally just says that Smith accepts (g), (h), and (i) as true propositions. In other words, Smith believes they are true; Smith believes (g), (h), and (i). These propositions are Q in this quote:

S is justified in believing P, and P entails Q, and S deduces Q from P and accepts Q as a result of this deduction...

S is Smith. P is (f). Q is (g)/(h)/(i) (they all work). Gettier wants to emphasize that Smith's beliefs are all justified. Smith is justified in his beliefs and makes valid inferences from these beliefs towards different beliefs.

Smith believes C1. From C1, we know that Smith also believes C2: (p v q)- its the conclusion in Smith's internal argument, an argument indicated by "because" in C1. Smith's C1 is false because (p) is not true. However, Smith still believes C2- (p v q)- and is justified in that belief. And, it turns out, that (p v q) is true, as (q) is true. Smith has justified true belief, but he does not have knowledge.

The above quotation is about justification, not about the truth value of the statements. Smith doesn't say anything about Smith's beliefs as he is explaining Smith's justification because he already showed Smith's beliefs by saying Smith accepts (believes) the propositions (f) and (f)'s entailments: (g), (h), and (i).

So, as Gettier says, Smith believes Jones owns a Ford. Smith constructs (g), (h), and (i). All of which are (p v q). So, Smith believes p, and deduces (p v q) from p and accepts (p v q) as a result of this deduction. There is nothing in the above two quotes that the first two premisses below cannot effectively exhaust...

p1. ((p) is true)
p2. ((p v q) follows from (p)) — creativesoul

Except you don't show the actual deduction of p∨q. In truth, it's barely a deduction at all. It's just or introduction.

Gettier literally just says that Smith accepts (g), (h), and (i) as true propositions. In other words, Smith believes they are true...

Ah. There it is once again.

The conflation of being true and being called "true" as the result of being the conclusion of a valid inference. Validity is insufficient for truth.

If Smith is rational, he wouldn't do that.

You wrote:

Except you don't show the actual deduction of p∨q. In truth, it's barely a deduction at all. It's just or introduction.

Why ought I need to show it?

Smiths belief that:((p v q) follows from (p)) shows it.

Smiths belief that:((p v q) follows from (p)) shows it. — creativesoul

No it doesn't. That's a conditional. It says only that if p, then p∨q. We have p, therefore we have p∨q.

And Gettier characterizes this conditional as a true belief of Smith. That is, p∨q does in fact follow from p -- it's not "merely", so to speak, a belief of Smith. It's one of Gettier's conditions that the entailment be correct.

1. If P, then Q.
2. P.
Therefore,
3. Q

If I say, "I accept Q because of the argument is valid and sound," it is equivalent to saying "I believe Q is true." Q follows from the argument above.

...And Gettier characterizes this conditional as a true belief of Smith. That is, p∨q does in fact follow from p...

That's what I said...

Smith holds the belief that:((p v q) follows from (p)).

That is a true belief, not that it matters.

Fill out Smith's belief using modus ponens...

Do you understand that the argument I'm providing is an account of Smith's thought/belief process, as Gettier sets it out?

But your missing the crucial detail: the "because" part is just an explanation of how and why Smith arrived at his belief "p v q." It's not the belief itself.

For example, I believe the government should legalize pot. Why? Because of various arguments I've seen for it. My belief is legalizing pot is the conclusion of arguments. But I still believe the proposition "the government should legalize pot."

Again, please write your proposition out in formal logic or explain how the word "because" functions in logic.

I'm not even understanding what you're trying to say...

I've put forth Smith's thought/belief process...

p1. ((p) is true)
p2. ((p v q) follows from (p))
p3. ((p v q) is true if either (p) or (q) is true)
C1. ((p v q)) is true because (p))(from p1,p3)

That is Smith's belief. Gettier wants to say that Smith believes that:((p v q) is true), and he wants us to think/believe that Smith can arrive at that from, or as a result of arriving at p2. He neglects to consider that p3 and C1 are necessary. And, as I've argued ad nauseum...

Salva veritate

Belief that:((p v q) is true) is not equivalent to belief that:((p v q) is true because (p)).

That notation is as formal as it can be, as far as I know. I've packed as much as possible within the belief statement. Note that C1 leaves out "is true". That is by design, not accident. That's how this particular kind of thought/belief process works.

The "because" in C1 is the deduction that is missing from the Gettier formulation. It is a necessary step in believing this particular Q.

First, it's not. If it was, it would be only symbols with no words.

In C1, what does "because" translate to in formal logic? Does it signify a conjuction, like "and" does? Does it signify a conclusion, like "therefore" does?

You're assuming that logical notation can properly account for belief.

If it can... great. That's the best I can do, and it is perfectly intelligible.

p1. ((p) is true)
p2. ((p v q) follows from (p))
p3. ((p v q) is true if either (p) or (q) is true)
C1. ((p v q)) is true because (p))(from p1,p3) — creativesoul

Just to be clear, you are claiming that Smith does not actually believe that p∨q is true, right?

I'm saying that:

Smith's belief that Q is nothing less than belief that:((p v q) is true because (p))