...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
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
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
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
...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.
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.
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...
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...
Gettier:
...Smith is therefore completely justified in believing each of these three propositions...
...S is justified in believing Q.
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
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...
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
Gettier literally just says that Smith accepts (g), (h), and (i) as true propositions. In other words, Smith believes they are true...
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.
Smiths belief that:((p v q) follows from (p)) shows it. — creativesoul
...And Gettier characterizes this conditional as a true belief of Smith. That is, p∨q does in fact follow from p...
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
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