I was reading a draft of a book our own Fr John Rickert’s is working on. Visual Logic: Seeing Classical and Modern Logic: An Introduction, a neat little handbook showing how to demonstrate and prove logical statements using clever diagrams. His chapter on basics led me to another argument to show that probability is logic. The epistemological side of logic, I mean.
Let’s start a logical argument. There is no probability here, just old-school logic.
Suppose we have some enormous container in which are N objects, all black and all indistinguishable from each other (except as to location, of course). N is very large. Indeed, N is 10 to the googol (which is 10^100), all to the googol, all to the googol, and so on, ten to the googol times. All times 2. N is ridiculously large, but still finite.
There exists a mechanism that can select an object from this container. And this will certainly happen. That is, the mechanism will certainly be invoked, and one of those N objects will certainly be withdrawn.
Those are the premises, which we can call P (the whole shebang).
Here, then, is the proposition of interest we wish to examine with respect to those (and no other! premises): C = “The object withdrawn is black.”
Well, it’s obvious. C is true given P. That is, we deduce C given P. If we believe P, we must believe C.
That last sentence is epistemological, of course. And, ignoring academic solipsists who pretend nothing can be known with certainty (other than they have to teach three and not two classes this semester as they have been promised), the sentence is in no way controversial.
Very well, let’s change the premises very, very slightly, while keeping C, and seeing what happens.
We augment N by 1, this time adding a “white” object to the mix. Therefore we have N’ = N + 1. The new premises, in which no other changes have been made, we label P’.
What about C?
It is no longer certain, given P’. We cannot deduce C given P’. If we believe P’, we are not compelled to believe C.
However, C seems pretty likely, doesn’t it? We are almost at the full deduction of logic to C from P’, are we not?
We could say the probability C is true, given P—the original premises—was 1. And we can say the probability C is true given P’ is nearly true. Indeed, via simple arguments, we can say the probability C is true given P’ is N/(N+1) ~ 1.
And if that isn’t close enough to 1 for you, simply take the original N and take it to the power of googol. Do that as many times as you want, as long as you stop short of infinity.
Now if you have had little or no formal training in probability, you will accept all this with ease. So easily that this whole discussion might seem absurd.
But if you have had training, the whole thing suddenly becomes suspicious.
If you are like most, you can’t help yourself and you begin to add premises to P’. Like “What if the white one is on top or the container isn’t well mixed?” Well mixed? Or “What if the selecting mechanism isn’t fair?” Fair? What does “fair” mean? Or “What if the white one is different in other ways than just color?”
Or “So where if there’s a near infinite number of black. These shrink to almost equality with that one white!” Or if you’re woke: “What if the mechanism used to do the picking was under the control of a white supremacist? Why is he using and black and white anyway, the racist.”
All of a sudden, details like this come seeping, then even flooding in. There wasn’t even a hint of suspicion in the first case, with P and all black balls. But with P’, with that one dinky white nothing in a seething universe of black, these nagging details bother us.
In one respect, this is a good sign. It means you are thinking of cause, and cause is the most important thing in science. But our example doesn’t have to be science, or even empirical. They can be leprechauns instead of objects, wholly fictional imaginary or otherwise impossible objects. Or it can be empirical but not objects (did you think there were balls?). Te container can be a computer, and the object bits. Or the computer can be a leprechaun computer. None of these affect the logic in the least. Nor the probability.
But even with these swaps, the doubt that all is well with the example increases upon us.
Yet, of course, we have no right—no logical right, that is—to insert these other premises. They are not part of P’. We have only P’.
There is no information—as in no: a hard, strict adamantine no—in P’ about mixing, or fairness, or anything else. P’ is perfectly compatible with the white object being anywhere in the container. And have no information how the mechanism picks.
We are not entitled to read that information into P’. That’s cheating. We’d never do that with P. Or to any other problem we think is strict logic. But we do when it comes to probability. We can’t help ourselves by think something is “going on” with probability questions.
Buy my new book and learn to argue against the regime: Everything You Believe Is Wrong.