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My main motivation is in terms of epistemic logic. In epistemic logic X knows that p if p is true in all states that X sees as epistemically indistinguishable from the current state. In this sense we can think of knowledge purely as a product of the epistemic relation.

However this also means that knowledge can be attained with different epistemic relations. For example if X only sees states s,t,u as available, and p is true in all of them then X knows that p. In fact perhaps X's epistemic state has changed so she only sees s as available, then she still knows that p. Yet intuitively her sense of know in the first case is different from her sense of know in the second case. I guess because she got 'new information'.

So then i thought maybe that the 'content' of an epistemic sentence must change at each instance. However what if an individual knows that p at two different instances, whilst receiving no useful information (e.g. doesn't remove available states) between the two? Are there cases of knowledge different?

If I know that p because of a certain reason, and X knows that p for another, in what sense is our knowledge of p comparable?

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If I know that p because of a certain reason, and X knows that p for another, in what sense is our knowledge of p comparable?

It is comparable in respect of having as an identical component the knowledge that p. But this evidently not what concerns you. If we know the same thing in different ways, is our knowledge comparable ?

There are different routes of the same knowledge. Take, for instance, the Pythagorean theorem : (a) the square of the hypotenuse is equal to the sum of the squares of the other two sides. When I last looked into this, I counted well over 100 different proofs of this theorem. If I know (a) by proof 1 and you know it by proof 15, we still both know it. Our knowledge is not merely 'comparable', I'd be inclined to say, but the very same : we both know the same thing, namely that the square of the hypotenuse is equal to the sum of the squares of the other two sides.

However, let's slow down. You might want to argue that what we know is not separable from the grounds on which we know it. I know that Pythagoras' theorem is true but the full specification of my knowledge includes the grounds on which I know it. What I know is that because proof 1 is valid therefore the square of the hypotenuse is equal to the sum of the squares of the other two sides. What you know is that because proof 15 is valid therefore the square of the hypotenuse is equal to the sum of the squares of the other two sides. From this point of view, our knowledge is different.

To take this further. Suppose I know that the components of vectors can be used to add vectors together. You set me a problem and I use the formula I have learned and get the right result. In this sense I know (to invent an example) that the answer is (0, 50) + (50, 0) = (50, 50). But suppose all I know is how to use the formula; I have no idea of the math that supports it. If Professor IQ is set the same problem, s/he knows not only the formula but the a large body of algebra and trig that sits behind vector calculation. There is a case here for saying that because of the much greater understanding of the formula that Professor IQ has than I have, in knowing that the answer is (0, 50) + (50, 0) = (50, 50) we do not know the same thing. If in our knowledge we include the grounds of knowledge, our knowledge is not comparable.

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