2022-03-04: Probabilistic law of the excluded middle

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Probabilistic law of the excluded middle

I'm a modeler -- I'm interested in making sense of problems, and thus, my primary interest in tools is in how they help me resolve problems. One of being finite and prioritizing my interests as such is that I am not as deeply an expert in any of the things I would like to be. Most of the time, that's fine, but many interesting questions the I find are precisely interesting because they fall into these interesting (real or apparent) corner cases where the standard tools start to have difficulty.

And so it goes with the title of this post -- I wish I was more of an expert in probability to discuss this, but I have to start somewhere.

I've grown to feel that in mathematics, the law of the excluded middle, while a useful postulate, is one whose invocation needs to be explicit. Proofs by contradiction are useful "bounds", but not as useful as direct constructions, and everybodies' lives will be easier when we know that distinction up-front. And as a modeler, focusing studies on algorithms and constructions rather than contradictions leaves us better prepared to solve new problems.

I suspect the same issue is present in probability theory, but I'm not aware of how it is discussed in that context. Whatever it is, the idea has not seeped into the core curriculum far enough yet to inform my every-day practice. So, I'm raising it as a question.

What is the probabilistic equivalent of the law of excluded middles, what are its consequences, and would avoidance of the excluded middle change probabilistic reasoning and modeling?

I expect this is something that a few people understand deeply already, either explicitly or implicitly, but the rest of us need to understand it better too.