As stated in W11 Slide #36, the Principle of Indifference says:

• When faced with n > 1 possibilities that are mutually exclusive and jointly exhaustive, and you have no evidence about their relative likelihoods, a probability equal to 1/n for each possibility.

I also gave two examples in the lecture:

• Example 1:
• Paul graduated from either JPJC, ACJC, or SAJC (and exactly one of them), but you don’t know which and you don’t know the likelihood of him going to any one of them.
Assign a probability equal to 1/3 to the possibility that he went to SAJC.
• Example 2:
• You have one lucky draw coupon, and you know that one coupon from a pile (numbered 000,000,000 to 999,999,999) will be picked. Each coupon give you one chance of willing. Assign a probability equal to 1/1,000,000,000 to the outcome where your coupon is the winning coupon.

Now, the “probability” we are talking about is not something about the world–about where Paul actually went to school, or which coupon is the actual winning coupon. Rather, we are talking about the level of confidence you should have, if you are being rational, in one of the options being true, given that you don’t know yet.

This “level of confidence”, or your credence in a proposition, is also expressed as a probability (hence, between 0 and 1)–“0”, where one is absolutely confident that the proposition is false, and “1”, where one is absolutely confident that the proposition is true. So if one has a credence of 0.5 in a proposition, then one is as confident that it is true as that it is false.

The probabilities involved in the Principle of Indifference are basically Credences (or sometimes also called Subjective Probabilities); or more accurately, the Principle tells you what credences you should have, if you want to be rational. (Yes, it’s an Epistemic Norm.) That’s what’s going on when we say that you should “assign” a certain probability to an outcome.

But what’s the motivation for the Principle of Indifference in the first place?

The thought goes something like this: Presumably, how confident you are that a certain proposition is true should be proportionate to the degree to which that proposition is supported by the evidence available to you. If the evidence is strong, then you should also be more confident that the proposition in question is true; else, not so much.

But now, consider two mutually exclusive propositions–and you have literally no reason to think that one is more likely to be true than the other. Given the earlier thought that your confidence in a proposition should be proportionate to the degree to which that proposition is supported by the evidence available to you–it follows that whatever else you should do, you shouldn’t have different degrees of confidence in the two propositions–you should have the same degree of confidence in the two propositions, right? Once you impose the additional constrain that the options are jointly exhaustive, then you also know that your subjective probability for each should add up to 1. With all these pieces in place, you have the Principle of Indifference as it was introduced in Slide #36.

But as I later briefly mentioned (Slide #51), there is a well known problem facing the Principle of Indifference: There are often multiple ways to carve up the probability space into different sets of mutually exclusive and jointly exhaustive options. Imagine the following scenario that one of the students who stayed back to chat came up with:

You have a hundred marbles in a bag–99 blue and 1 red, all equal size and weight, same texture, i.e., indistinguishable by touch. You reach into the bag and grab exactly one. You can’t see what you grabbed. What’s the probability that it’s a red marble?

Option 1: There are 100 marbles. So grabbing each marble (one at a time) is mutually exclusive from grabbing another. And the 100 possibilities are jointly exhaustive. I have no reason to believe that the one I’m grabbing is a particular marble. So the probability that the one I grabbed is red = 1/100 (or 0.01).

Option 2: Each marble is red or blue, but not both. So grabbing a red marble is mutually exclusive from grabbing a blue one. And the two possibilities are jointly exhaustive–there are only red or blue marbles in the bag. I have no reason to believe that the one I’m grabbing is a red marble rather than a blue one, and vice versa. So the probability that the one I grabbed is red = 1/2 (or 0.5).

If the criteria we have is that the options have to be mutually exclusive and jointly exhaustive options—and that’s all we have to work with–then Option 1 and 2 are equally proper applications of the Principle of Indifference. What this means is that applying the Principle of Indifference exactly as taught in #36, can lead to inconsistent results. The important point is that this outcome comes about exactly because both options correctly applied the Principle exactly as it was taught in #36.

Examples of the above nature (and other related one) can be easily multiplied. Collectively, they present what is called the “Partition Problem” for the Principle of Indifference. For this reason, some philosophers reject the Principle of Indifference outright. Some of them say that no probabilities should be assigned in the absence of evidence, while others they say that any assignment would be equally rational. In contrast, the defenders of the Principle believe that it just needs to be properly reinforced. (In case you haven’t noticed, this is an example of a disagreement over epistemic standards!)

Michael Huemer–the one who gave us our Political Authority topic earlier this semester–is one of those who think that the Principle just needs to be reinforced. This is what he says:

Not everything in the world is equally fundamental; some things are explained by other things. The things that do the explaining are more fundamental than, or explanatorily prior to, the things that get explained. Some examples:

(i) The properties and relations of an object’s parts generally explain the features of the whole. The table is solid because its molecules have a stable configuration with relatively fixed distances from each other, not vice versa. (Note: explanatory priority is a necessary but not a sufficient condition for explanation. Microscopic features in general, including those that are not relevant to explaining solidity, are prior to such macroscopic features as solidity.)

(ii) Earlier events explain later events, and causes explain their eﬀects. The Treaty of Versailles helps to explain World War II, whereas World War II could play no role in explaining Versailles, since World War II hadn’t yet happened at the time the Versailles Treaty was signed.

(iii) General facts are explained by the more specific facts that make them true. An apple is colorful in virtue of being red, not vice versa. Michael Jordan is tall in virtue of being six feet six inches, not vice versa.

This relation of explanatory priority must be taken account of in assigning probabilities: if one set of possible facts is explanatorily prior to another, then the initial probabilities we assign to the more fundamental possibilities must constrain the probabilities of the less fundamental possibilities, not vice versa. Given two variables, X and Y, if X explains Y, then the initial probability distribution for Y must be derived from that for X (or something even more fundamental). Here, by “initial probabilities”, I mean probabilities prior to relevant evidence. Thus, if we are applying the Principle of Indiﬀerence, we should apply it at the more fundamental level.

Michael Huemer, Paradox Lost: Logical Solutions to Ten Puzzles of Philosophy, 174-175

(Chapter 8 of Huemer’s book is about the Principle of Indifference. The book as a whole makes for a nice read, by the way.)

If we take on board Huemer’s “Explanatory Priority Proviso”, then it will turn out that Option 1, rather than Option 2, properly applied the Principle of Indifference. This is because–plausibly–Option 1 makes use of a more fine-grain way of categorizing the possibility space–not just “red vs blue marble”, but “red marble, vs. blue marble #1, vs. blue marble #2, etc…” And that more fine-grain fact is explanatorily more basic–the fact that there are 99 blue marbles in the bag (a more specific facts) explains how there are blue marbles in the bag at all. It’s a neat proposal, but it is not universally accepted as of the last I checked.