The traditional analysis of knowledge (TAK), as put forward by Gettier[i]:

S knows that p if and only if (iff)

1) p is true

2) S believes that p

3) S is justified in believing that p

I will argue that TAK is redundant because we can only ever know p with indubitable certainty through mathematics. As a result, for all practical purposes we should not define knowledge as something that passes TAK. I will propose knowledge constitutes p coupled with a probability, and that whether something is true or not does not necessitate knowledge. This definition includes mathematical principles that I would propose we know with a probability of 1, and anything that contravenes the law of non-contradiction would have a probability of 0. These would lead to a subjectively configured probability that would have an associated subjectively derived level of certainty. The important point is you only have knowledge if you are aware that what you know could be true, or false. In the end we will be left with 3 choices: keep subjective definitions as sufficient for knowledge, enforce weak objective definitions in a step towards universal knowledge, or enforce strong objective definitions which for all practical purposes will be un-implementable other than in theory.

Please note that I am not considering the sceptical viewpoint that even mathematical principles could not be known with indubitable certainty, and that to pick one of many examples, every time we count the 4 sides of a square we are tricked into not seeing the 5th side; for the purposes of this paper I will discount this problem although I believe it would fit within the model, just that absolutely nothing could be known with indubitable certainty and you would deduce from an imperfect basis.

With this in mind first I would like to highlight why we can only have knowledge of mathematics with indubitable certainty, whereas we cannot know any of the ‘p’s’ put forward in examples related to Gettier. Everything is only subjectively true unless we can state it in pure mathematics, and this is the objective truth we are striving for, that we want to be able to term knowledge by TAK. Even if we take p to be ‘my house is still standing where I left it’, we could never call this knowledge because for all we know it might not be true – there might have been an earthquake, a plane may have fallen out of the sky and flattened it, a truck could have driven into it - I’m sure you can think of a load of weird and wonderful Gettier-like examples of where this belief is not true.

The two important things that allow mathematics to be known with indubitable certainty are that it is objective and perfectly defined. For example 1 + 1 = 2 is objective and universal, when presented with one unit, and given another, everyone in the world will agree there are two units now. This is an admirable quality, and what allows this universal objectivity? Mathematics is perfectly defined; there is no argument as to whether 1 + 1 = 2, 200, or 2000; 1 + 1 = 2 by definition.

Therefore to know anything with indubitable certainty in the world one has to satisfy these mathematical criteria; we have to have an objective agreed universal definition. So why is this a problem?

Take a sheep. How do you define ‘sheep’? What is a ‘sheep’? Take two sheep, how do we know they are both sheep? Are they identical? Are they even the same breed, the same sex, the same shape or size, in fact, are they biologically identical, if not, how do they satisfy ‘sheep’? What if one of the sheep is dead and not alive, is it still a ‘sheep’? What if one is stuffed? Do we need to add a clause in that a sheep has to have blood circulating, with a certain amount of that blood being oxygenated? Does the sheep need to think in a certain way to be a sheep? Does it have to be having instantaneously and simultaneously the same perceptions as another sheep to be identical to it?

‘Sheep’ is just an abstract term we give to an object that sufficiently satisfies a massive list of criteria. Moreover I would argue that if you have the belief that you see a ‘sheep’, and you are justified in seeing a ‘sheep’, then it is true that you are seeing a ‘sheep’. I argue this because in calling the object a ‘sheep’, you are just saying that it satisfies enough of the properties that you look for in a sheep, and it doesn’t satisfy the qualities of any other object more so. As I tried to outline before, who can argue, it is not like we have an objectively agreed definition of a sheep.

Consequently, what you say is a sheep, is a sheep. However, what you say, even by your own meaning is not always the truth; until you can know everything about the sheep, you can not be sure exactly if it is a sheep. But more than this, you need to have an exact definition of what a sheep is, so that you can gradually become surer and surer that it is a sheep by gradually satisfying more and more of the definition as you learn more about it.

If we can have a universal objective definition of a sheep, then we could have objective knowledge that we are seeing a sheep. You have 2 choices, you either make this knowledge realistic and attainable and risk for example a fake sheep actually being included in this objective knowledge, or you make the objective definition of a sheep rigorous and accurate, so that it is essentially in mathematical principles However, with this we will only ever know if something is a sheep by probability, because we can never through our perceptions of an object know enough about it to satisfy the mathematical principles. Moreover we have to reduce the sheep to maths, and we can never reduce a sheep to maths – just look at it, is it a bunch of floating numbers and formulas to you? You would therefore have to be able to deduce a sheep from mathematical principles, which even if theoretically possible would be completely pointless because even if we now know what a mathematical, objective sheep is we cannot reduce the real sheep in front of us in to maths. Moreover it would be impossible for us to match our mathematical sheep with the perceived sheep.

It can be seen that we are left with 3 options. We could create strong objective universal definitions for everything it is possible to know in the world, this would be a perfect definition of something down to a molecular level if necessary. If you are able to know something well enough to sufficiently satisfy every single part of that definition, then you have knowledge and in every circumstance this knowledge would be true. Using this basis means that all of the examples put forward by Gettier do not count as knowledge as they are not true. From a practical perspective this would mean it is impossible to know anything in the world other than mathematical principles and their immediate deductions via perception.

The second option would be to enforce weak objective universal definitions. There would be a list of criteria for everything it is possible to know in the world and you would couple this with the knowledge that according to how much you know of this definition there is a relative level of certainty in the knowledge you have. In this situation, when presented with the exact same perceptions, everyone would arise with the same knowledge, even if that knowledge is false.

The option I am proposing however is that we enforce none of these and instead the only definition we enforce is that of knowledge: knowledge is not merely knowing p, it is knowing p with a level of certainty or more precisely a subjectively derived level of certainty based on probabilities. In this circumstance TAK is redundant. Knowledge is the product of a justified belief and a level of certainty:

S knows that p if and only if (iff)

1) S believes that p

2) S is justified in believing that p

3) S has a level of certainty of p, such that the perceptions triggering the certainty in p don’t trigger a greater certainty in anything other than p

It is evident something we know can turn out to be false; just look at all of the Gettier examples. This in my eyes is a practical, usable analysis of knowledge that forgives us that we can only be 100% sure of a few things.

This allows that if perceptions can satisfy a certain number of a given set of properties of a subjective definition of what ‘p’ should constitute, and it satisfies more properties of the chosen definition to match ‘x’ with than with any other definitions of any other objects, and it is a justified belief then we can have knowledge of ‘p’ as long as we maintain that it is not certain it is ‘p’.

To offer one workable example: Lars in New Zealand seeing a ‘sheep’ in a field. The problem is Lars has a true belief that there is a sheep in the field that is justified and not on false grounds but this is not knowledge. We obviously have a problem in the definition of ‘sheep’ as has been outlined previously but short of flatly rejecting that p is true, I will disregard this notion of truth. Therefore, Lars believes there is a ‘sheep’ in the field and he is justified because he sees something that is triggering ‘sheep’ in his mind, integrally, his perceptions trigger ‘sheep’ more than any other object in his mind, so by my analysis of knowledge Lars knows there is a sheep in the field, with a level of certainty adequate to his perceptions. The fact that it is a goat and that there is another sheep out of sight in the field is irrelevant.

This I hope also gives us a nice insight of why we have had such a hard time solving the Gettier problem problem. If we take p to be something we can know with indubitable certainty, like 1 + 1 = 2, then we can draw up a sound analysis of knowledge for it and we won’t find it Gettiered, because there is no possible example where 1 + 1 = 2, not one interpretation of 1 + 1 =2 can be Gettiered. However, as soon as we stray away from mathematical principles, for example into p being the proposition highlighted in any of the previous representations, as this proposition p can only be known with an element of certainty, even if it is 99.99999% certainty, there would be 1 case in each 1000000 where the knowledge was found to be untrue, and this example is one of those that is Gettiered. Of course, most problems put forward to counter analyses of knowledge are not this certain and are therefore a lot more easily Gettiered.

I have proposed a different account of knowledge and realise I have not dealt with perceived refutations or justified why this account is better – if I had attempted all that in this paper I would have done none of it justice.

[i] Gettier, Edmund L. Is Justified True Belief Knowledge? From Analysis 23 1963 sourced at http://www.ditext.com/gettier/gettier.html no later than 30/11/2007

## Friday, February 15, 2008

### Problems of Truth, Names & Certainty

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## 2 comments:

Of course, 1+1=2 is not a brilliant example, since in binary, 1+1=10, for example. You'd need to qualify it with "in the decimal system" every time, but that gets tedious. It might be worth adding a disclaimer at the top to explain that when you say 1+1=2, you in fact mean 1+1=2 in the standard decimal system of arithmetic.

I think your point merges with mine made throughout the essay... that we continuously make assumptions on the information laid out before us.

Given your comment as 'p' and applying it to my own formation:

S knows that p if and only if (iff)

1) S believes that p

2) S is justified in believing that p

3) S has a level of certainty of p, such that the perceptions triggering the certainty in p don’t trigger a greater certainty in anything other than p

(You would be S, obviously.)

If you trully believe P, then I would suggest that you contravene (2) by not being justified in believing p.

That would be to say given the context and information surrounding all the utterances of '1+1=2' it would be daft to suggest 'that the perceptions triggering the certainty in p don’t trigger a greater certainty in anything other than p'.

I hope you see my point!

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