Belief, nor justification, nor inference confer truth. — unenlightened

Yes. Should have made it clearer that inference preserving truth is something like a precedent for what we expect inference to do with justification.

On belief and assertion, I defer to Moore's paradox: asserting that P appears to carry with it a non-cancelable implicature that I believe that P. You can modify your degree of belief with "I'm not sure but I think" and the like, but you can't set it to zero.

Naturally if your belief in the premises of an inference is less than 100%, your belief in the conclusion should be less than 100%. Being the conclusion of an inference doesn't add or subtract certainty. -- We're talking here about perfect entailment. If you only have "If P, then it's likely that Q", that's a whole 'nother deal.

When I first saw Gettier, I had a similar reaction as creative, and as you're hinting at here -- that the "if p" tags along. As it turns out, this is the aspect of modus ponens that Tarski highlights by calling it the "rule of detachment" -- you get to detach the conclusion from the argument for it.

And I think that's right, with the proviso that your degree of belief in the conclusion, or the degree to which belief is justified, will track your degree of belief in your premises, or the degree to which those beliefs are justified.

When you have time. Read the following carefully... It is where I'm at in all this. Still honing it. p3 needs left alone. It's not an argument per se, it's a report of the thought/belief process necessary for arriving at believing Q when Q is a disjunction arrived at from believing P following Gettier's formulation...

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

You wrote:

Well I just waded through all this, and I have to admit to some skimming. I'll make a few preliminary remarks, and see who wants to swallow them whole and who wants to bite their heads off.

1. (p v q) appears to say more than p, but actually says less. 'The glass contains water or the glass contains vodka' says less than 'The glass contains water'.

Indeed. Well put. That is precisely one point I've been making. Here's the thing...

Gettier, and evidently many others want to say that Smith can get to belief that:((p v q) is true). That is their aim. The target. The candidate. The problem is that one cannot arrive at that without going through belief that ((p v q) is true if... (insert belief about the truth conditions of this particular disjunction)), and ((p v q) is true because...(insert belief corresponding to the "if" in p3)).

Once one arrives here, there's nothing more to the belief, and certainly nothing less.Salva veritate

Given the formula in the beginning of his paper, Gettier does not - cannot - take account of what it takes to arrive at believing Q when Q is a disjunction deduced fro believing P. There's nothing about Smith's thought/belief regarding the truth conditions of the disjunction. Michael has left that fact sorely neglected.

My argument, is about what the thought/belief consists in/of. It seems that you've understood. No surprise given that you've been one of the few who've been able to follow my position regarding thought/belief. Thanks for joining in, by the way...



As you say...

2. Nobody in the real world forms arbitrary unrelated disjunctions of things they believe and things they have no belief about, or believe to be false, except for rhetorical purposes, or I'm a monkey's uncle.

3. I'm not a monkey's uncle.

Gettier's mistake is conflating knowledge of the rules of entailment/disjunction with belief. Believing that (g), (h), and (i) are entailed by (f) is not equivalent to believing the disjunctions.

There's nothing about Smith's thought/belief regarding the truth conditions of the disjunction. Michael has left that fact sorely neglected. — creativesoul

There is, and I haven't. One of the truth conditions of "Jones owns a Ford or Brown is in Barcelona" is Jones owning a Ford, which Smith believes.

Believing that (g), (h), and (i) are entailed by (f) is not equivalent to believing the disjunctions.

How many times do we have to go over this? Smith doesn't just believe that (g), (h), and (i) are entailed by (f). He also believes (f), and so also believes (g), (h), and (i). I explained this simple fact to you in the very first reply.

To repeat my earlier question, do you believe that this statement is true?

London is the capital city of England and/or I was born in Leeds.

I assume that you do, because I assume that you believe that London is the capital city of England. If you were being honest then you would admit to this. And with that, your argument against Gettier fails.

While it is true that (g), (h), and (i) are entailed by (f), and it is also true that Smith could accept/believe that all three are valid forms of disjunction. It is not true that Smith could believe anything at all about Brown's location. I mean, Gettier clearly states that Smith is totally ignorant about that. Thus, Smith - himself - would not form belief about Brown's location. One cannot know they are ignorant about Brown's location and simultaneously form and/or hold a belief about where Brown is located.

The mistake here is conflating knowledge of the rules of entailment/disjunction with belief. Believing that (g), (h), and (i) are entailed by (f) is not equivalent to believing the disjunctions. — creativesoul

A few possible counters:

• Believing a disjunctive statement is not the same as beliefs about the disjuncts.
• Knowledge that (g), (h), and (i) are entailed by (f), and belief in (f) will, upon reflection, result in belief in the disjunctions. I I know that I have two arms, and I know that two is an even number, then, upon reflection, I will come to believe that I have an even number of arms.

I don't think the Gettier problem is really new. I think it was mentioned as early as the Theatetus or however you spell it.

Also, there's something fishy about investigating commonsense beliefs by means of formal logic in a thought experiment where we imagine people constructing formal structures in order to deduce things. It seems like a weird scenario bound to produce nonsense.

To repeat my earlier question, do you believe that this statement is true?

London is the capital city of England and/or I was born in Leeds. — Michael

Michael, try my newly minted Anti-Gettier.

Given that it is the case that it rains every day, do you believe that "if I do the rain dance it will rain tomorrow"?

Logic insists it is true; common sense insists that if you believe in the efficacy of rain dances, you've gone badly wrong somewhere.

I seems to me that the logic of conditionals, conjunctions, disjunctions, fails to take any account of semantic content.

I'm wondering if all the difficulties can be resolved by adding a fourth criterion to the tradition: Knowledge is justified true meaningful belief. This would allow me to answer your question, "No. I believe London is the capital of England, and where you were born has nothing to do with it." And it would allow you to answer me, "No, I believe it will rain tomorrow whether you dance or not"

And these "no's" are denials of significance rather than of logic.

I will say my prayers before bedtime, and/or someone will die in the night. Justified true belief. Forming meaningless connectives is like dividing by zero; there ought to be a law against it.

Given that it is the case that it rains every day, do you believe that "if I do the rain dance it will rain tomorrow"? — unenlightened

Yes. To believe that this conditional is true is not (necessarily) to believe that the antecedent is the cause of the consequent.

I'm wondering if all the difficulties can be resolved by adding a fourth criterion to the tradition: Knowledge is justified true meaningful belief. — unenlightened

If you have to add to the JTB account of knowledge in light of Gettier's argument then you've agreed with Gettier that the JTB account of knowledge fails (or at least isn't sufficient).

Such JTB+X accounts of knowledge are a direct response to the apparent validity of the Gettier cases.

I don't mind agreeing with Gettier if he will allow that his argument and my agreement is meaningless. :)

Yes. To believe that this conditional is true is not (necessarily) to believe that the antecedent is the cause of the consequent. — Michael

Well is this not exactly what @creativesoul has been trying to articulate, that one can assent to the truth only by denying the meaning? When someone says 'Hands up or I shoot', the convention is that if you put your hands up, they don't shoot, and if you put your hands up and they shoot you anyway, you are entitled to be pissed off, and they might as well not have said anything.

Hey Pneum!

When you have time. Read the following carefully... It is where I'm at in all this. Still honing it. p3 needs left alone. It's not an argument per se, it's a report of the thought/belief process necessary for arriving at believing Q when Q is a disjunction arrived at from believing P following Gettier's formulation.

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

When someone says 'Hands up or I shoot', the convention is that if you put your hands up, they don't shoot, and if you put your hands up and they shoot you anyway, you are entitled to be pissed off, and they might as well not have said anything. — unenlightened

That's an implied exclusive or. The Gettier case is an explicit inclusive or.

And I don't actually understand how this relates to what I said to you. I was addressing your material conditional, whereas with creative we are talking about disjunctions.

I wrote:

Believing that (g), (h), and (i) are entailed by (f) is not equivalent to believing the disjunctions.

Michael replied:

How many times do we have to go over this? Smith doesn't just believe that (g), (h), and (i) are entailed by (f). He also believes (f), and so also believes (g), (h), and (i). I explained this simple fact to you in the very first reply.

You mean this???

It is if you believe that f is true.

Gratuitous assertions won't do at this juncture Michael. I've detailed exactly what believing any and all Q's requires when Q is a disjunction deduced from believing P. Show me which step is unnecessary.

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 only gets to p2. I've successfully argued for this without subsequent refutation or due attention.

I've clearly shown that p3 and C1 are left completely unaccounted for in Gettier's paper. Arriving at believing Q when Q is a disjunction deduced from believing P requires another deduction. That deduction is clearly accounted for in my formula, but is left neglected in Gettier's.

So, you could show which step in my formula is unnecessary for believing a disjunction. You could show how believing Q requires only one deduction from P. You could always just take the argument from the top. You could always just imagine any disjunction you like... fill in my formula... and look at the results. There is never a problem.

Fill it out...

You mean this???
"It is if you believe that f is true." — creativesoul

Yes. If I believe that (f) is true and if I believe that (g), (h), and (i) are entailed by (f) then I will believe that (g), (h), and (i) are true. That's straightforward rationality.

So contrary to your repeated strawmen, nobody here is saying that "believing that (g), (h), and (i) are entailed by (f) is equivalent to believing the disjunctions".

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

That's where you get.

It doesn't make Q true. It makes Q the conclusion of a valid inference.

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

That's where you get.

It doesn't make Q true. It makes Q the conclusion of a valid inference. — creativesoul

Yes it does. It's modus ponens. This is elementary logic.

1. p
2. p ⊨ p ∨ q
3. p ∨ q

Yes. If I believe that (f) is true and if I believe that (g), (h), and (i) are entailed by (f) then I will believe that (g), (h), and (i) are true. That's straightforward rationality.

So contrary to your repeated strawmen, nobody here is saying that "believing that (g), (h), and (i) are entailed by (f) is equivalent to believing the disjunctions".

Are you sure you want to argue this?

X-)

Yes.

If I believe that 1.(f) is true and if I believe that 2.(g), (h), and (i) are entailed by (f) then I will believe that 3.(g), (h), and (i) are true. That's straightforward rationality.

So contrary to your repeated strawmen, nobody here is saying that "believing that (g), (h), and (i) are entailed by (f) is equivalent to believing the disjunctions.

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

That's where you get.

3 in the quote above doesn't follow from p2 unless by "true" you mean being the result of a valid inference.

If I believe that 1.(f) is true and if I believe that 2.(g), (h), and (i) are entailed by (f) then I will believe that 3.(g), (h), and (i) are true. That's straightforward rationality.

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

That's where you get.

3 in the quote above doesn't follow from p2 unless by "true" you mean being the result of a valid inference. — creativesoul

If 1 is true and if 2 is true then 3 is true. Modus ponens. Again, elementary logic.

3 in the quote above doesn't follow from p2 — creativesoul

I'm not saying it does. It follows from 2 and 1. It's a syllogism with a major and minor premise.

Perhaps I should spell it out like this for you:

1. p
2. p ⊨ p ∨ q
3. p ∨ q (from 1 and 2)

Being the result of a valid inference isn't equivalent to being true.

Being the result of a valid inference isn't equivalent to being true. — creativesoul

I'm not saying that it is. I'm saying that if Smith is rational and if he recognises that the argument is valid and if he believes that the premises are true then he will believe that the conclusion is true.

In Gettier's case, the argument is valid, Smith is rational, and Smith believes that the premises are true. So Smith believes that the conclusion is true.

Then why keep calling (g), (h), and (i) "true"?

If 1 is true and if 2 is true then 3 is true. Modus ponens. Again, elementary logic.

Modus ponens doesn't help your report of Smith's belief. 1 is false and you know it. So, it is not the case that if 1 is true and 2 is true then 3 is true. 1 is false and yet you want 3 to be true. Your argument requires it.

Again elementary logic.

Your report upon Smith's thought/belief process is not equivalent to Smith's thought/belief process.

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 only gets to p2. I've successfully argued for this without subsequent refutation or due attention.

I've clearly shown that p3 and C1 are left completely unaccounted for in Gettier's paper. Arriving at believing Q when Q is a disjunction deduced from believing P requires another deduction. That deduction is clearly accounted for in my formula, but is left neglected in Gettier's.

So, you could show which step in my formula is unnecessary for believing a disjunction. You could show how believing Q requires only one deduction from P. You could always just take the argument from the top. You could always just imagine any disjunction you like... fill in my formula... and look at the results. There is never a problem.

Fill it out...

Then why keep calling (g), (h), and (i) "true"? — creativesoul

We're talking about what Smith believes to be true. If he believes that the premises of a valid argument are true then he will believe that the conclusion is true.

Modus ponens doesn't help your report of Smith's belief. 1 is false and you know it. So, it is not the case that if 1 is true and 2 is true then 3 is true. 1 is false and yet you want 3 to be true. Your argument requires it.

Again elementary logic. — creativesoul

We're talking about belief at the moment. You're trying to argue that Smith doesn't believe that 3 is true, despite believing that 1 and 2 are true. This is wrong. Smith is a rational person. He believes that the argument is valid and that the premises are true. Therefore he believes that the conclusion is true.

I'm arguing that your notion(and Gettier's) regarding what counts as believing Q, when Q is a disjunction deduced from believing P is utterly inadequate.

Is that clear enough?

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

I know what you're arguing. I'm explaining that you're wrong. If Smith believes that P is true and if Smith believes that P entails Q (and if Smith is rational) then Smith will believe that Q is true. Whether or not Q is a disjunction doesn't make a difference. Modus ponens is valid, and so the rational person who recognises this and who believes that the premises are true will also believe that the conclusion is true.

I'll repeat (and add to) my previous explanation yet again:

1. Smith believes that p
2. Smith believes that p ⊨ p ∨ q
3. From 1 and 2, Smith believes that p ∨ q

4. Smith's belief that p is justified by r
5. p ⊨ p ∨ q
6. From 4 and 5, Smith's belief that p ∨ q is justified by r
7. p is false and q is true
8. q ⊨ p ∨ q
9. From 7 and 8, p ∨ q is true
10. From 6 and 9, Smith has a justified true belief