No.

Believing Q, when Q is a disjunction deduced from believing P...

p1. ((p) is true)
p2. ((p v q) follows from (p))
p3. ((p v q) is true if...(insert belief statement(s) regarding what makes this particular disjunction true))
C1. ((p v q) is true because... (insert belief statement(s) corresponding to the prior 'if'))

His believing Q is exhausted by C1.

Salva veritate

See p3 and C1???

Gettier and you have not - cannot - properly take account of that and yet you agree that it's necessary.

His believing Q is exhausted by C1. — creativesoul

No it isn't. There's also:

C2. p v q is true

My belief that Donald Trump is the President isn't exhausted by:

C1. Donald Trump is the President because he won the popular vote

There's also:

C2. Donald Trump is the President

I've already refuted that attempt.

I've already refuted that attempt. — creativesoul

No you haven't. You've denied it. But it's a fact of logic that C1. p ∨ q ∵ p entails C2. p ∨ q.

Knowing what a disjunction means requires knowing what makes it true.

Show where Gettier or your report of Smith's thought/belief process accounts for this.

p3 and C1 do so very nicely.

p1. ((p) is true)
p2. ((p v q) follows from (p))
p3. ((p v q) is true if...(insert belief statement(s) regarding what makes this particular disjunction true))
C1. ((p v q) is true because... (insert belief statement(s) corresponding to the prior 'if'))

His believing Q is exhausted by C1.

Salva veritate

See p3 and C1???

Gettier and you have not - cannot - properly take account of that and yet you agree that it's necessary.

You're just repeating the same nonsense ad nauseam. I can see I'm wasting my time.

Take it from the top...

Gettier states:

I shall begin by noting two points. First, in that sense of "justified" in which S's being justified in believing P is a necessary condition of S's knowing that P, it is possible for a person to be justified in believing a proposition that is in fact false.

I would concur.

Secondly, 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.

This is not always true. To be as precise as ordinary language allows:S must first arrive at a belief before we can say that S is justified in forming/holding that belief. In Gettier Case II, the above formulation is utterly inadequate for S's arriving at belief that:((p v q) is true).

Keeping these two points in mind I shall now present two cases in which the conditions stated in (a) are true for some proposition, though it is at the same time false that the person in question knows that proposition.

This I outright deny.

Gettier's aims at a case that Smith forms/holds a Justified True Belief that:((p v q) is true) by virtue of going through the thought/belief process set out in the above formulation beginning with "Secondly..." Belief that:((p v q) is true) is the only value appropriate for Q in that formulation, for Q is (p v q) and believing Q is nothing less than belief that (p v q) is true. Hence, believing Q is belief that:((p v q) is true).

I will show that Gettier's formulation is inadequate regarding it's ability to take proper account of the thought/belief process required for S's belief that:((p v q) is true). S cannot arrive at that without another step that Gettier leaves out. To be clear, if the astute reader looks carefully at that formulation, s/he will note that only one deduction is purportedly necessary in order to satisfy the formulation. Namely, S's deducing Q from P.

I'm strongly asserting that it takes more than one deduction for S to arrive at belief that:((p v q) is true), and since that is the case, it only follows that Gettier's criterion is inadequate. That will be clearly shown.

To be clear, 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 not necessarily justified in believing Q, for - in this case in particular - believing Q is nothing less than belief that:((p v q) is true) and S cannot arrive at that following Gettier's formulation. Belief that:((p v q) is true) requires yet another deduction that is left sorely unaccounted for in Gettier's formulation. It's been said heretofore, but it now bears repeating...

S must first arrive at a belief before we can say that S is justified in forming/holding that belief. In Gettier Case II, the above formulation is utterly inadequate for S's arriving at belief that:((p v q) is true). The following argument represents the process of thought/belief that is necessary prior to even being able to arrive at believing Q and is an exhaustive account thereof. The term "because" in C1 is the necessary but missing deduction in Gettier's formula.

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)

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. As Gettier says, Smith believes Jones owns a Ford. Smith constructs (g), (h), and (i); all of which are (p v q). Smith believes p, and deduces (p v q) from p and accepts (p v q) as a result of this deduction. There is nothing 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 has yet to have gotten to the point where he has formed and/or holds belief that:((p v q) is true). Gettier thinks otherwise, 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...

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)


Thus, we can clearly see that Gettier's formulation is inadequate to account for the belief that he needs for Smith to hold in order to make his case. Getting to belief that:((p v q) is true) requires both p3. and C1. Further we can also see that Smith's belief is not true, for he does not ever get to belief that:((p v q) is true). Gettier wants us to believe that Smith holds the belief that:((p v q) is true). This post has shown all sorts of problems with Gettier's formulation, and the aforementioned want of Gettier is just yet another.

Belief that:((p v q) is true) is not equivalent to belief that:((p v q) is true because (p)). The former is existentially contingent upon the latter and has a different set of truth conditions. The latter consists in part of the deduction missing in Gettier's account. The missing necessary deduction clearly shows that Smith's belief is false, Gettier's formulation is inadequate, and the 'problem' regarding Case II is non-existent.

Salva veritate

Smith believes Jones owns a Ford. Smith believes that 'Jones owns a Ford' is true. Smith believes that 'Either Jones owns a Ford or Brown is in Barcelona' follows from 'Jones owns a Ford'. Smith believes that 'Either Jones owns a Ford or Brown is in Barcelona' is true if either 'Jones owns a Ford' or 'Brown is in Barcelona' is true. Smith believes that 'Either Jones owns a Ford or Brown is in Barcelona' is true because Jones owns a Ford.

That is Smith's believing Q as the result of another deduction.

QED

Knowing what a disjunction means requires knowing what makes it true. You agree to this. So...

Show where Gettier or your report of Smith's thought/belief process accounts for this.

A rational person doesn't believe statements if they do not know what the statement means. Smith is rational. Believing Q requires Smith's knowing what makes it true.

Where is any of that accounted for in Gettier's paper?

p3 and C1 do so quite nicely.

I agree Smith knows what the disjunction means.

Knowing what a disjunction means requires knowing what makes it true.
— creativesoul

Yes, and he believes it to be true. And it's true. So he has a true belief. — Michael

Michael knows what this means:

(c) If unenlightened does the rain dance, it will rain tomorrow.

He also believes it and believes it is justified (because it rains every day) and true.

The guy's clearly not worth talking to. :D

(p1) Whatever unenlightened does, it will rain tomorrow.

(p1) is true, and entails (c). (c) is patent nonsense. Michael has to claim that (c) is not patent nonsense, and therefore has to claim that it does not mean what we all understand it to mean. Only by denying the common meaning can he maintain that logic preserves truth.

But if logic cannot follow the workings of language, and help us untangle sense from nonsense, we might as well forget logic. Language and meaning is prior to logic; logic must enhance our understanding, not ride roughshod over it.

Looking at the problem again, notice that (p1) means that what unenlightened does is unconnected to the rain, whereas (c) makes just such a connection. It is the making of this connection that is illegitimate, and makes a nonsense from sense. So I propose a new rule:

Thou shalt not connect the unconnected. (p v q) for example is empty rhetoric - empty of meaning or false, unless there is a connection between p and q, because it declares that there is such a connection.

Thus we allow, 'Socrates is a man, and all men are mortal', and 'the glass contains water and/or vodka', but not, 'the glass contains water, and/or all men are mortal'.

Looking at the problem again, notice that (p1) means that what unenlightened does is unconnected to the rain, whereas (c) makes just such a connection. — unenlightened

If c makes a connection then c isn't justified by the fact that it rains every day. You're equivocating.

Either "if ... then" implies a causal connection, in which case the claim isn't justified, or it's justified, in which case it doesn't imply a causal connection.

So you need to disambiguate into one of these two:

1. If unenlightened does a rain dance then it will cause it to rain tomorrow
2. If unenlightened does a rain dance then perhaps incidentally it will rain tomorrow

The latter is what I believe to be true (assuming that it rains everyday, and that this justifies believing that it will rain tomorrow).

But again, I don't understand the relevance of this. You're talking about a material conditional, whereas the issue at hand is a disjunction.

"London is the capital city of England or pigs can fly" is true if London is the capital city of England or if pigs can fly, and so if I believe that London is the capital city of England then I will believe that "London is the capital city of England or pigs can fly" is true.

"London is the capital city of England or pigs can fly" is true if London is the capital city of England or if pigs can fly, and so if I believe that London is the capital city of England then I will believe that "London is the capital city of England or pigs can fly" is true. — Michael

I believe that there is no connection between the name of the capital of England and the aerial abilities of pigs. So I believe you are making an unjustified disjunction devoid of meaning. All you really believe, and all you can honestly assert is that London is the capital of England. As it happens, I can assure you that pigs can and do fly on a regular basis, but they invariably fly as baggage, so you are unlikely to have noticed them unless you are involved with baggage handling.

The relevance of this is that it solves the problems raised by Gettier, and prevents people from claiming as 'logical truth' certain things that are patent nonsense.

I believe that there is no connection between the name of the capital of England and the aerial abilities of pigs. So I believe you are making an unjustified disjunction devoid of meaning. All you really believe, and all you can honestly assert is that London is the capital of England. As it happens, I can assure you that pigs can and do fly on a regular basis, but they invariably fly as baggage, so you are unlikely to have noticed them unless you are involved with baggage handling.

The relevance of this is that it solves the problems raised by Gettier, and prevents people from claiming as 'logical truth' certain things that are patent nonsense. — unenlightened

Except "London is the capital city of England or pigs can fly" isn't nonsense. It's a meaningful English statement which is true if London is the capital city of England or if pigs can fly.

There doesn't need to be a connection between the operands for a disjunction to make sense.

Surely you understand what I mean when I say that one or both of "London is the capital city of England" and "pigs can fly" is true? Because that's all the disjunction is saying.

Surely you understand what I mean when I say that one or both of "London is the capital city of England" and "pigs can fly" is true? Because that's all the disjunction is saying. — Michael

Indeed, I understand and accept your conjunction as phrased in the first sentence, because there is a connection made between the statements mentioned as distinct from used concerning their truth or falsity. But this connection is not made when they are conjoined in use in the disjunction. A claim that mentions statements is not identical to a claim that uses them, and your use of quotation marks indicates that you understand that.

""London is the capital of England" is true" means the same as "London is the capital of England" - or so it can be argued, anyway. But it cannot be argued that ""Pigs can fly" is false" means the same as "Pigs can fly". And since the claim is that one statement can be false, the identity you propose cannot be maintained.

Indeed, I understand and accept your conjunction as phrased in the first sentence, because there is a connection made between the statements mentioned as distinct from used concerning their truth or falsity. But this connection is not made when they are conjoined in use in the disjunction. A claim that mentions statements is not identical to a claim that uses them, and your use of quotation marks indicates that you understand that. — unenlightened

So you're saying that the following two propositions are different?

1. "London is the capital city of England or pigs can fly" is true
2. One of both of "London is the capital city of England" and "pigs can fly" is true

Because it seems to me, to channel my inner creativesoul, that this is a case of salva veritate.

Regardless, you can always apply Gettier's reasoning to the second.

Regardless, you can always apply Gettier's reasoning to the second. — Michael

I don't think you can. Smith's belief that "at least one of two statements, 1 and 2, is true" is not the same as the belief that "statement1 and/or statement 2", for reasons that you have dismissed without criticism. Now Smith, by hypothesis, does not know or understand unenlightened's law, so we must forgive him if he conflates them. Nevertheless, if we asked him to justify his belief, he would say something like, " well I've no idea about 2 but I'm sure of 1 because... " you know the story.

And that would satisfy you, but not me, creative, or Gettier. Gettier says Smith has a justified true belief that is not knowledge, creative says that he does not believe what he says he believes, And I say it's all you logicians fault for neglecting the meaning of language and only looking at the form.

I don't think you can. Smith's belief that "at least one of two statements, 1 and 2, is true" is not the same as the belief that "statement1 and/or statement 2", for reasons that you have dismissed without criticism. — unenlightened

I don't understand your reasons. To say that "London is the capital city of England" is true is to say that London is the capital city of England and to say that "pigs can fly" is true is to say that pigs can fly. And so to say that one or both of "London is the capital city of England" and "pigs can fly" is true is to say that London is the capital city of England and/or pigs can fly.

1 and 2 say the same thing. Compare with:

3. "London is the capital city of England and pigs can't fly" is true
4. Both "London is the capital city of England" and "pigs can't fly" are true

T(p) ∧ T(q) is equivalent to T(p ∧ q).

Now do it with false.

Why? The propositions I'm comparing are ones that predicate truth, not falsehood. I'm not saying that these two are equivalent:

5. "London is the capital city of England or pigs can fly" is false
6. One or both of "London is the capital city of England" and "pigs can fly" is false

That 5 and 6 are not equivalent is not that 1 and 2 are not equivalent. The semantics of "true" and "false" make a difference.

But again, you can apply Gettier's reasoning to this proposition:

1. One or both of "Jones owns a Ford" and "Brown is in Barcelona" is true

A belief in 1 is justified if one is justified in believing that Jones owns a Ford, and a belief in 1 is true if Brown is in Barcelona. Smith has a justified true belief.

Why? — Michael

Because the disjunction is explicitly saying that one or other may be false. So it does not say p is true it says p might be false, but in case p is false, then q must be true. And it also says that q might be false, but in case q is false p must be true.

But since there is in fact no connection between p and q, there is no justification for saying it.

p1. ((p) is true)
p2. ((p v q) follows from (p))
p3. ((p v q) is true if...(insert belief statement(s) regarding what makes this particular disjunction true))
C1. ((p v q) is true because... (insert belief statement(s) corresponding to the prior 'if'))

Gettier is proposing a thought/belief process for arriving at believing Q when Q is a disjunction deduced from believing P. Case II fills that out, but it doesn't account for p3 and C1, which are necessary for a rational person to arrive at believing Q.

Gettier's reasoning is flawed because it's based upon a grossly inadequate (mis)understanding of what Smith's believing Q requires. Smith is purportedly rational. Smith's arrival at believing Q requires and is exhausted by C1. That holds good for every imaginable disjunction arrived at from belief that:((p) is true). There is never a problem of any kind.

Believing a disjunction - for a rational person - is nothing more and nothing less than knowing what it means which requires knowing what makes it true and believing that those conditions have been met.

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.

No rational person would think it's reasonable to believe A but unreasonable to believe A ∨ B. — Srap Tasmaner

Then I must be an unreasonable person, because I think that to reason thus: "Probably A, but if not A then definitely B" is cuckoo.

Because the disjunction is explicitly saying that one or other may be false. So it does not say p is true it says p might be false, but in case p is false, then q must be true. And it also says that q might be false, but in case q is false p must be true.

But since there is in fact no connection between p and q, there is no justification for saying it. — unenlightened

This doesn't seem like the correct interpretation of the disjunction at all.

Let's say that there's a group of kids, and I believe that one of them is mine. If someone were to ask me "is one of those kids yours?" I will answer "yes". Am I saying that if it isn't the one I think it is then it must be one of the others? Not at all. But I'm still saying that one of them is mine, and it's true that one of them is mine if in fact one of them is mine.

Now replace a group of kids with a group of sentences, with one of which I believe to be true.

"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.

Because A ∨ B ↔(¬A→B), I guess.

That means ∨-introduction comes to P→(¬P→Q) for any Q, which, duh.

Because A ∨ B ↔(¬A→B), I guess.

That means ∨-introduction comes to P→(¬P→Q) for any Q, which, duh. — Srap Tasmaner

I suppose that means that the term "or" suffers from the same sort of problem as the term "if ... then ..." (which unenlightened brought up earlier with the example of the rain dance). But I don't think this undermines Gettier's reasoning either way, so it seems a sort of misplaced criticism.

So to rephrase the issue, and continue with my earlier example, if I believe that my child is eating cake, and if I believe that three children are eating cake, then I believe that the statement "my child is one of the three eating cake" is true. This seems perfectly reasonable.