Friday, November 26, 2010

A Treatise on Probability, Chapter 1

In chapter one, Keynes lays out his terms.  He begins by distinguishing direct knowledge and knowledge attained by argument.  With respect to the latter, what does it mean to say that a proposition is probable?
"The terms certain and probable describe the various degrees of rational belief about a proposition which different amounts of knowledge authorize us to entertain."
So we have two sets of related propositions: for now, let's label them as evidence and conclusion(s).
"Between two sets of propositions, therefore, there exists a relation, in virtue of which, if we know the first, we can attach to the latter some degree of rational belief."
I'm interested to see how Keynes will show that such a relationship exists.  Furthermore, it seems we must know about this relation between evidence and conclusion before we can rationally affirm anything about the probability of the conclusion. 

Questions that come to mind
1. How do we quantify the degree to which evidence confers support on a conclusion? 
2. How do we know this relationship, and how does that knowledge give rise to a degree of rational belief? 
3. Is one's degree of rational belief commensurate with the degree to which evidence confers supports on the conclusion?

Keynes says there is a "purely logical" relationship between the evidence and the conclusion.  So while we may not have certainty about the truth of the conclusion, we may have certainty about the appropriate degree of  belief that one should rationally hold with respect to that provocative!  'Hmm so I'm not sure about x, but I'm perfectly certain about the degree to which I'm not sure about x.'  I couldn't put the book down at this point.

The first chapter drives home the point that the stated probability of a conclusion is relative to the evidence or "corpus of knowledge."  Add or subtract relevant evidence and the probability of the conclusion may change.  But doesn't this imply that rational belief is subjective, since not everyone considers the same evidence when assessing the probability of conclusions?  Not quite.

Subjective Sets of Propositions
Propositions can be given differing probabilities with respect to differing bodies of evidence.  Person A has evidence 'a', and person B has evidence 'b'.  When assessing some proposition p, A and B needn't have the same rational degree of belief in p, since these may be different:
1. The probability of p given a, or P(p/a)
2. The probability of p given b, or P(p/b)

As a second example: ask me tomorrow, and my degree of belief might have changed due to new evidence.  So it's all mushy relativistic psychology stuff right?  Wrong. 

Objective Relations Among Propositions
The evidential content may vary, but the degree to which evidence confers support on the conclusion is "purely logical."  This again raises the issue in question #3.  He has more or less used 'degree of rational belief' and 'degree to which y confers evidence on x' synonomously in the first chapter.  He does mention this towards the end:
"I do not believe that any of [the initial terms and definitions] accurately represent that particular logical relation which we have in our minds when we speak of the probability of an argument."
So perhaps my head scratching is warranted, and things will become clearer as we move along.  For now the overall idea seems clear.  Let x be a conclusion and y be some set of propositions given as premises in a probabalistic argument.  When Keynes says 'x is probable' he means 'x is probable given y', which is longhand for P(x/y).  Thus, when one has knowledge of P(x/y), they can have a rational degree of belief in x.  What I'm not sure of, is if P(x/y) is .75 then is my degree of rational belief in x also .75?  All this talk about having knowledge of P(x/y) raises some epistemic questions, which Keynes will address in chapter 2.


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