## Wednesday, July 10, 2024

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P: The total human population is an even number.

The probability of each of P and Not-P = 1/2.
The probability of (P & Not-P) = 0.

What you say is true, but misses the point.

Let the conjunctive proposition be *9/11 was an inside job & Hillary Clinton ran a child molestation ring out of a D. C. pizza parlor*

The least plausible conjunct is the second. The plausibility of the conjunction is the same as the the plausibility of the second conjunct. Or so says Rescher. But the probability of the conjunction is not the same, but must be less than, the probability of the least probable conjunct. For example, if the prob of the first conjunct is 1/2 and the prob of the second is 1/8, then the prob of the conjunction is 1/16.

Naive calculation of probabilities given two probabilities p1 and p2 would say that "or" adds probabilities p1 + p2 and "and" multiplies them p1 * p2, but most actual real world calculations must modify this because the p1 and p2 are usually not fully independent. For example, P and not-P are perfectly dependent, with no independence to do the "p1 * p2" calculation correctly as a simple product. See for example here: https://stats.stackexchange.com/questions/439150/range-of-probability-for-non-independent-events

In a world where 9/11 was an inside job it would seem more likely that other conspiracies are also real, so the probability of "9/11 was an inside job & Hillary Clinton ran a child molestation ring out of a D. C. pizza parlor" is actually greater than the simple product of the two conspiracies' probabilities (though still basically zero).

I see. You wanted a conjunction containing a "least plausible" conjunct. I didn't want to invite disputes over relative plausibility, as your example does.

If I may:

But the probability of the conjunction is not the same [as], but must be less than, the probability of the least probable conjunct.

If you're offering that as a principle, it's falsified by counterexamples. Let P be 1 = 1 and Q be Hillary Clinton ran a child molestation ring out of a D.C. pizza parlor. The probability of (P & Q) is not less than that of Q.

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