April 9th, 2012

## Calculating the Sufficiency of Belief Probabilistically

Here is a real simple model that can be used to determine how we weight our beliefs.  As an evidentialist, I appropriate the degree of belief of commitment to a belief according to it’s evidence.  The question of sufficiency may be expressed probabilistically.  If my belief p is sufficient then it must be 0 < p ≤ 1 where p is > .5.  Expressing the value of p is difficult and may certainly have values that may be measured and compared but there are also instances where p must be assigned an arbitrary value that must be determined intuitively or against the aggregate whole of one’s current knowledge [if p is novel].  Additionally, if p is not equivalent to 1 then all future tensed propositions may only be expressed probabilistically.

Why these criteria for determining the value of p?  There may be instances when p may be assigned a definitive value.  Suppose that I am colorblind, to an extent, and red and purple are indistinguishable for me and that they are the same shade.  Suppose I have three marbles in my pocket that are similar in weight and texture (or otherwise may only be distinguished by color) and I have been told by a reliable source that there is one red marble and two purple marbles.  The probability of me pulling out the red marble is .333.  I pull out a marble and it is red.  Let p be the belief that I pulled a purple marble from my pocket.  With the background knowledge, k, that I have the inability to differentiate red from purple; I am justified in assigning a value of .333 to p even though I actually pulled out a red marble.  Would I then be justified?