Probability is Logic

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 That is, the mechanism will certainly be invoked, and one of those N objects will certainly be withdrawn.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 premises): C = “The object withdrawn is black.”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’.

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.

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1. Hagfish Bagpipe

”We can’t help ourselves by thinking something is “going on” with probability questions.”

Adding illegitimate premises to logical statements is the Original Sin. God gave simple, explicit premises to Drs. Adam & Eve: you may eat from any tree except this one, if you eat from this one you die. Dr. Serpent added the illegitimate premise: eat it, you’ll become gods. Drs. Adam & Eve believed Dr. Serpent’s bad model, which had very poor predictive skill, resulting in the flawed decisions you see all around. A cautionary tale to all logicians, modelers, statisticians, and orchardmen.

2. Jan Van Betsuni

[Under (P’) || N’ = N + 1 ||] If “the mechanism” selects an infinite series of objects (replacing the selected object each time) ~ do we know with absolute (perfect) certainty that eventually the lone white object will be selected ~ ((NO)) ~ it doesn’t “have to be”

3. “You believe men landed on the moon?”
“YOU believe in the MOON?”

4. Leo

The discussion of the single white object occupies almost as much space as the discussion of the large number of black objects. In the economy of attention the white one gets as much as all the black ones. If instead of adhering to the logic, a heuristic shortcut is used, choosing the white object is about as likely as choosing a black one, at least based on the attention paid to each.

Maybe that explains the addition of extra hypotheses.

5. Milton Hathaway

“But if you have had {formal training in probability}, the whole thing suddenly becomes suspicious.”

Or if you are an experienced engineer. To be fair, though, an engineer is more likely to be dealing with the equivalent very large number of ensembles of black/white balls, and if the white ball shows up in even one of them, your bridge will fall down or your building will collapse. An experienced engineer will never, and indeed must never, accept the premises as given. The chicken is never spherical.

6. i Like to think of prob as a means of dealing with a lack of information. In reality the P(white ball) is either 1 or 0, but, because we don’t know which it is, we use an estimate – sometimes based on physics (causality), sometimes on past observed frequency, sometimes just a wild guess, and mostly some combo of those 3 mixed up with some other guesses as to how important the answer is. i..e. no info => p=0.5; but in your example we know n(black) and n(white) so we guess 1/n+1 largely because it doesn’t matter. If it became life/death, however, we’d go bonkers imagining scenarios other than true random choice and estimating their odds…

7. Pat Cusack

Wouldn’t the cautious engineer simply reach into the googlplex and remove the dangerous white ball from the ensemble?

8. Robin

And yet, the bottom line is that engineering codes use factors of safety against failure. A simplified example for, say, a concrete structure: strength capacity of design / stress from design loads > 1.5.

Is not the acceptance of these numerator and denominator values based upon Briggs’ simple logic?

I say this because I have seen two PhD level scientists, pretending to be Engineers, question the entire premise of these codes; ie that there are internal voids, flaws, defects, inadequate construction practice etc that one cannot see. We are the experts you know!

This really happened to my astonishment and disbelief. I tried to reason with them; tried to go through the records, tried to go through the codes, the assumptions behind the codes, tried to go through calculations side by side to show the folly of their conjectures, but it was no use.

These conjectures delayed the project for two years and ultimately cost the owner about \$100 million. This was just the first arbitration settlement. There are more cases winding their way through the courts.

But it’s possible that experiments may force you to update your premises.

Suppose we take P’ to be true, draw just 10 balls, and one of them is white. I’d argue that the probability of that happening is MUCH smaller than the probability some part of P’ not being true.

Or, what’s the probability that I’ll get seven hundred tails in a row tossing a fair coin? Much smaller than the probability that the coin I’m tossing is actually not a fair one.

Or (one of my favorite real life examples), what is the probability of seeing a 25-sigma event (given a normal distribution)? That’s what some financial engineers said was happening in 2008 once everything started falling apart. I’m fairly certain they didn’t actually see 25-sigma events, and the fact that their models told them that’s what’s happening is evidence that they were egregiously wrong.

10. Fr. John Rickert, FSSP

Reminds me of a joke — actually, not really a joke, maybe — one of my great math professors told us. An engineer, a physicist, and a mathematician are riding in a car in Scotland. They see a black sheep. The engineer says, “The sheep in Scotland are black!” The physicist replies, “Well, -some- of the sheep in Scotland are black.” The mathematician replies, “No. There exists at least one sheep in Scotland, at least one side of which is black.”

Book should be ready soon! Thanks for the “shout out.”

11. PhilH

Your argument flies out the window if the objects are swans.