A Treatise on Probability, Ch 4

In this chapter, Keynes critiques the Principle of Indifference (abbreviated P.I.) which is his own name for what Bernoulli called ‘The Principle of Non-Sufficient Reason.’ The principle states that when several alternatives are possible—and yet there is no information whatsoever to prefer one over another—each alternative should be given equal probabilities. Suppose you are picking up a friend at the airport and don’t know which gate they will arrive at. What is the probability of arrival at gate A versus not-A? The P.I. calls for assigning P(A) a value of 1/2, or more generally 1/n where n is the number of known alternatives. Unfortunately, this principle quickly leads to paradoxes, which Keynes demonstrates in this chapter.  His goal is not to demolish the P.I., but instead refine it.

A Treatise on Probability, Chapter 3

Must all probabilities be expressed as quantities?
All this talk about ‘degrees of belief’ might lead one to wonder how to measure such degrees (assuming they can be measured). Are they counted or weighed? Keynes begins by surveying a few other opinions on the matter. One opinion is that a numerical comparison of two probabilities is possible, though in practice it it may be impossible.  Much like counting the number of cells in the human body…we assume that there is such a number even though we don't know it. Some things are weighed and others are counted, but this is merely a limitation on our knowledge.

Subjective probability assessments and the Bayesian method

I've been reading John Loftus for about 5 years now, a considerable sample size.  The chart below represents (rather facetiously) an assessment of topics, of which any one might come up as the main subject of a blog post. Now, let's assume my pie chart could have been created using one of two methods.  Both methods start with the same list of topics and then rank them, but each method suggests a different approach to ranking, and perhaps even a different interpretation of what the ranking means.  

1. How frequently the topic has been the main subject of a post

2. My degree of belief that the topic will be the main subject of the next post

Assuming we must rely on memory, an approach to #1 might be to rank each topic's probability in relation to the others on the list.  (Remember this is just a subjective ranking.)  Given this approach, we will  end up with a nice tidy list where the probabilities add up to 1, and each probability is relative to the list.

With #2, the approach could be different.  I think we're less inclined to make our probabilities coherent (add up to 1) when we think in terms of 'degree of belief', and perhaps for good reason.   When asked about degree of belief, we are apt to consider additional evidence beyond mere frequency.  For example, we might hold an additional belief that John gets more frustrated with Christians during the holidays.

But is this an unfair characterization?  My perception of the frequencies (in #1) might account for additional evidence (such as John's holiday disposition) whether or not I realize that it does.  For instance, the fact that his posts tend to gravitate towards "Christians are deluded!" during the holidays could very well be the reason I believe John is more frustrated with Christians during the holidays.  Should we characterize this "fact that his posts tend to.." as a property out there in the world, a degree of belief, or both?

How do we sort things like this out, assuming we have good reason to?  One promising approach to critical thinking about probability is called the Bayesian method.  This method starts with a process for ranking one's degrees of belief with respect to some set of propositions, and when new evidence comes to light those degrees of belief are updated (recalculated), and thus the change in our degree of belief in some proposition with respect to new evidence can be accounted for mathematically.  The hardest part about this method is simply figuring out where to start.  What is the probability of any given belief we already hold?  One way to answer this question is to ask another one, "Would you rather bet on belief x or bet on the chance that a fair coin toss will land heads?"  If you think your belief is more likely than heads, now you're a little closer to knowing your degree of belief (more than 50% sure).  After we get that nailed down, things fall into place as new evidence comes along.  Critics of the Bayesian method accuse it of being overly subjective, yet it isn't clear to me that other approaches to interpreting probability are exempt from subjectivity.  Regardless, each approach to probability might prove useful in different situations.

Bayesian theory can also lead to a humanistic vision for the future...an optimistic outlook that may have been previously marred by the sense that philosophy might never reach agreement on much of anything.

"The model of reasoning that I am inclined to accept is that everyone starts where they start, and then conditionalizes their belief systems on the evidence. And then, maybe, however many generations it takes, we can reach a consensus." - Dr. Victor Reppert, in a discussion about his essay on the probability of miracles.

This looks very similar to scientific optimism.  A scientist might say, "given enough observations, humans will eventually develop a coherent theory of the physical world."  A Bayesian might say (eg., Dr. Reppert), "given enough evidence, humans will eventually develop a coherent (shared) set of beliefs about the world."

Notice how the Bayesian account didn't put physical in front of world?  Of course, a materialist will define evidence as purely physical (remember, we're talking about a Bayesian definition of evidence here).  But it isn't clear to me that evidence qua evidence must be physical in a Bayesian model.  Perhaps I'm wrong about that?  I'm not sure.  Dr. Reppert is a theist, so I can only assume that he believes humans will ultimately converge on a belief system that is...at minimum...not eliminatively [sic?] materialistic?  But yet science, by it's own lights, is headed towards reducing the world down to some ultimate materialistic explanation.  So why is Bayesianism headed in another direction?  Perhaps there is a catch regarding one's view of the mind here?

What if science eventually reduces the mind down to some strictly materialistic explanation?  Could a Bayesian continue to hope that humans might eventually converge on a system of beliefs that is compatible with (bare bones) Christianity or theism?  At face value, I don't see any reason why one can't include religious experience in their probability model.  However, I don't see the optimistic side of all this (as a Christian).  To me, it seems that one's belief/non-belief in God will determine which direction Bayesianism takes you.  But then again, how does one account for the man that doesn't believe in God and then encounters some set of experiences that persuade him otherwise (and vice versa for apostates)?  How does one account for John Loftus' Outsider Test for Faith, which seems to suggest that we should provisionally "reset" our prior probabilities (with respect to one's religion) back to a neutral position, and then have a look at the evidence relative to that.  Looks like I have much more thinking to do about this.

*Update*
Dr. Reppert has posted his reply over at dangerous idea.  And in case any readers are (inexcusably!) unaware of the Monty Python reference:

A Treatise on Probability, Chapter 2

Summary
We know many things simply by contemplating the objects of direct acquaintance (experience, understanding, and perception). We can can leverage this direct knowledge to arrive at indirect knowledge about other propositions. For example, perhaps we know the proposition “3 horses—red, green, and blue--crossed the finish line” and we wish to know something about the proposition “The red horse crossed the finish line.” What moves must we make to arrive at knowledge about the red horse crossing the finish line? We must know a secondary proposition that expresses the logical relationship between the two propositions. In this case, that proposition would be “p bears probability-relationship .33 with respect to proposition h.” Yet how did we come to know this secondary proposition? By directly perceiving a logical relation: in this case certain axioms of probability. I’m not sure if that example is perfect, but it’s the best I can do.

Is this a typo?

I'm reading through Probability: A Philosophical Introduction.  I have read this page about 10 times and have concluded there is either a typo in the book or a hole in my head. 

If A=>B, then if P(B)=1, P(A)=1

So if A entails B, then if the probability of B is 1, the probability of A is 1....huh?  Isn't that affirming the consequent?

A - John finished the race first place
B - John finished the race

John finished the race first place entails that John finished the race.

The probability (epistemic) that John finished the race is 1...we watched.  But suppose several horses crossed the line at the exact same time (as far as we can tell from the nose bleed section), so we have no evidence about first place other than the above entailment and the proposition "Either horse 1, 2, or 3 won first place."  So how in the world do we know that the probability of John finishing first place is 1?  Shouldn't our credence be .33 that John won first place while we wait for the announcement?  Actually... A=>B doesn't seem to be relevant at all.

*Update*
Confirmed as a typo by Dr. Mellor himself.  He informed me that the fallacy of affirming the consequent only applies to material implication and not entailment (logical implication).  The reason isn't quite clear to me, but there are more important things to tackle.  Suffice it to say that A->B is not exactly the same as A=>B.

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 conclusion...how 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.