An Irrefutable Refutation of Gettier Case II

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creativesoul
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An Irrefutable Refutation of Gettier Case II

Post by creativesoul » Wed Sep 13, 2017 8:01 am

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.

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.

QED
Last edited by creativesoul on Wed Sep 13, 2017 8:12 am, edited 1 time in total.

creativesoul
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Re: An Irrefutable Refutation of Gettier Case II

Post by creativesoul » Wed Sep 13, 2017 8:04 am

There's quite a bit to consider here regarding the sheer scope of application that this refutation has for philosophy on a whole. Aside from the cottage industry that owes it's very existence to this particular case, there is more. Namely, anything and everything that has to do with deriving disjunction. Unless I'm missing something, in addition to refuting the Case II, this places the very notion of deriving disjunction under tremendous scrutiny, which in turn places para-consistent logic - in part at least - under the same.

True premisses and valid form cannot yield false conclusions.
False premisses and valid form cannot yield true conclusions.

Gettier's Case II has Smith working from a false premiss. For whatever reason, convention has been bewitched for half a century. I've shown that arriving at belief that:((p v q) is true) because (p)) requires more than one deduction. Gettier's formulation requires only one. One cannot arrive at belief that:((p v q) is true). It can't be done, unless that is; one wants to conflate being true with being called "true" as a result of being the product of a valid inference. Validity is insufficient for truth. Prior to ever getting to belief that:((p v q) is true), one must first go through belief that:((p v q) is true if...) and belief that:((p v q) is true because(insert belief statement(s) corresponding to the prior 'if')). That holds for any and all interpretations thereof.

Salva veritate

What I'm saying is that the following is not only my argument. It is not only an adequate account of Smith's thought/belief process, but it is also necessary for any and all disjunction. It's a formula that doubles as a solution. This can be tested by virtue of filling it out...

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'))(from p1,p3)

It shows us something about the difference between propositions and belief(s) as well. Case II was a thorn in the side of many folk due to the fact that it required us to say that Smith believed a proposition which contained a statement that Smith did not believe. My solution solves that.

It also sheds light upon what it actually takes to believe a disjunction, as compared/contrasted to just believing that it follows from some belief or other. Furthermore, it shows that Gettier's formulation is invalid.

That's huge.

Londoner
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Re: An Irrefutable Refutation of Gettier Case II

Post by Londoner » Wed Sep 13, 2017 9:13 am

For casual readers, I should perhaps say that I understand the Gettier problems were concerned with what we understand by the term 'knowledge'.

To have 'knowledge' the traditional (Platonic) idea was that I believe something, and the thing I believe is true. I must also believe it for a reason (my belief is justified, it is not just a guess). 'Knowledge' = JTB = justified, true, belief.

The Gettier examples are ones which fit all these criteria, except that although I have a reason, that reason is false. I think it is justified, so I can tick that box, but it isn't i.e. I am right, except not for the reason I think I am.

Is that still 'knowledge'? We would be inclined to say it wasn't, so do we need to add an additional condition for what we count as 'knowledge'?

So I do not think this is a question about logic, or truth in the logical sense, but about what any claim to 'know' something implies. In other words, it isn't about the possibility we may get things wrong, but about what I mean when I claim I 'know' something.

Again, for any casual reader who is mystified, I understand that the point about the 'Either Jones owns a Ford, or Brown is in Boston' is that I have a good reason to think Jones does own a Ford. And I do not think Brown is in Boston. So I think a statement linking the two with an either/or conjunction must also be true.

However, by chance Brown really is in Boston. And I am wrong about Jones owning a Ford. So my statement, with that either/or conjunction is still true, but not for the reason I believed it was.

Now I agree that this example is a bit iffy but not because of the logic. (I think the problem you point out is a problem about logic, rather than a mistake in logic). The question here is still 'what do we mean by a claim of 'knowledge''? We need some sort of a definition that can be a bridge between the subject's mental state and external fact. This turns out to be difficult.

creativesoul
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Re: An Irrefutable Refutation of Gettier Case II

Post by creativesoul » Thu Sep 14, 2017 8:15 am

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 is precisely false. S does not arrive at belief that:((p v q) is true). The half a century mistake is granting that Smith believes that:((p v q) is true)) simply as a result of belief that:((p v q) follows from (p)). An astute reader will prudently note that Gettier sets out a thought/belief process above. Smith believes that:((p is true) and then deduces Q from P and accepts Q as a result of this deduction. That is to say nothing more than Smith accepts/believes that:((p v q) follows from (p).

It does not follow from that that Smith believes that ((p v q) is true).

It takes belief that:((p v q) is true if (p) or (q) is true) and belief that((p v q) is true because (p)).
Last edited by creativesoul on Thu Sep 14, 2017 8:29 am, edited 2 times in total.

creativesoul
Posts: 489
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Re: An Irrefutable Refutation of Gettier Case II

Post by creativesoul » Thu Sep 14, 2017 8:21 am

p1. ((p) is true)
p2. ((p v q) follows from (p))
p3. ((p v q) is true if...(insert belief statement(s) about what makes this particular disjunction true))
C1. ((p v q) is true because... (insert belief statement(s) corresponding to the "if" directly above))(from p1,p3)

That's what any and all belief towards ((p v q) is true) must go through. Gettier wants us to agree that believing Q only requires accepting that Q follows from P. He leaves out both p3 and C1. Smith believes that ((p v q) is true because (p)). Smith holds false belief. False belief, no matter how it is arrived at, is not a problem for true belief... justified or otherwise.

creativesoul
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Re: An Irrefutable Refutation of Gettier Case II

Post by creativesoul » Thu Sep 14, 2017 8:37 am

Londoner wrote:
Wed Sep 13, 2017 9:13 am

...I think the problem you point out is a problem about logic, rather than a mistake in logic...
I point out the inherent inadequacy of Gettier's formula. It only requires one deduction. Belief that:((p v q) follows from (p)) exhausts Gettier's formula.

I would agree. It neglects to draw the crucial distinction between kinds of Q's...

creativesoul
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Re: An Irrefutable Refutation of Gettier Case II

Post by creativesoul » Thu Sep 14, 2017 8:40 am

It also shows that entailment does not necessarily(always) preserve truth.

Londoner
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Re: An Irrefutable Refutation of Gettier Case II

Post by Londoner » Thu Sep 14, 2017 12:58 pm

Is the example quoted an 'exclusive or'? It might be an 'inclusive or', but the fact we can't tell is indicative of a problem. So, although we can translate the example into symbolic Ps and Qs, that does not preserve the meaning - or rather the lack of clarity - of the original.

Nobody would ever say: Either Jones owns a Ford, or Brown is in Boston, not unless the context was one in which it was understood there was some sort of material connection between those two facts. In the Gettier example, where the speaker is sure Jones owns a Ford, the second part would really be interpreted as meaning something more like 'or I'm a donkey'. So it doesn't follow from the first part. It doesn't assert a fact (which we can tell because we can substitute 'Barcelona' or 'Brest-Litovsk' for 'Boston'). It only emphasizes the speaker's state of mind, that they are quite sure Jones owns a Ford.

So I would say that the reason we cannot tell what sort of 'or' it is, is because it is neither, it is not really any sort of logical connective between two propositions, because there is no second proposition.

I think this is another example of why no normal language sentences can be translated into symbolic logic. And vice-versa.

creativesoul
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Re: An Irrefutable Refutation of Gettier Case II

Post by creativesoul » Fri Sep 15, 2017 2:48 am

p1. ((p) is true)
p2. ((p v q) follows from (p))
p3. ((p v q) is true if...(insert belief statement(s) about what makes this particular disjunction true))
C1. ((p v q) is true because... (insert belief statement(s) corresponding to the "if" directly above))(from p1,p3)

Any and all disjunction derived from belief using Gettier's formula will go through every step above in order to arrive at believing Q.

An insincere purveyor of disjunction cannot get through the above...

The Merrillian Lie-Trap

:mrgreen:

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