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I would like to describe a reasoning with quantifier "almost always". For example, if probability of $Z$ is above $95$%, I would like to say that $Z$ is "almost always true".

Is there a modal or fuzzy logic with such a quantifier? If not, is it even possible to define a logic with this quantifier? May be, it would contradict some assumptions about quantifiers in a meaningful logic?

Alejandro Bergasa Alonso
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  • I have seen "almost everywhere" in measure theory. If you have a measure, then a statement P is true on the measure space "almost everywhere" if it is true everywhere except a null set (which is usually a subset of a measure zero set), so actually this is not completely irrelevant since you have a probability measure (iirc this is usually a measure which has value one on the space). – Stephen Landsittel Jan 16 '21 at 15:44
  • But "almost always Z" (95% probability) means Z may be false on a subset with non-zero (5%) measure. So, this is something else, I think. Besides, "almost everywhere" is not a quantifier in a logic (?) As far as I know, there are no defined logical operators with it. –  Jan 16 '21 at 15:51
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    There are well-developed theories of generalized quantifiers. – Rob Arthan Jan 16 '21 at 15:52
  • You could also be interested in modal logic? One could imagine a modality "almost surely". But it wouldn't be a quantifier. – Dabouliplop Jan 16 '21 at 15:57
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    I would think that "above 95%" would be a very bad interpretation of "almost always true" and that no meaningful logic could be based on such an interpretation. (I mean that whatever meaningful logical framework you come up with that has an "almost always true" modality, then "with probability at least 95%" will not yield a realization of this modality.) – Captain Lama Jan 16 '21 at 16:01
  • To Idéophage: Yes, I am very interested. Could you, please, hint at why it is "modality" and not "quantifier"? Can you point me to an answer? –  Jan 16 '21 at 16:02
  • To Captain Lama: this is what I want to know. Why is this? –  Jan 16 '21 at 16:04
  • Marina, you can type "@CaptainLama" instead of "To Captain Lama", so that the other user has a notification and sees your comment. About the "modality versus quantifier" thing, I said that because you could either ask for a quantifier that says "for almost all x, blabla", or you could ask for a unary modality that says "it is almost always the case that blabla". I don't really know how to explain more than that. About what CaptainLama is saying, I guess it's because if you iterate several times "above 95%", you can get something which cannot be called "almost surely" anymore. But I don't know. – Dabouliplop Jan 16 '21 at 21:45
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    The problem with 95% is that the intersection of two events that occur at a 95% each might have less than 95% probability by itself. That means that $\forall^{95}x(\varphi\land\psi)$ does not imply $\forall^{95}x\varphi\land\forall^{95}x\psi$. If that is not something you expect from a universal quantifier, that's fine. – Asaf Karagila Jan 16 '21 at 22:43
  • @ Asaf Karagila I understand now that it can not be quantifier. The logic I found (see the answer) uses "more than $a$ percent" as modal operator, not quantifier. Would you have objections against it? –  Jan 16 '21 at 22:51

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This logic exists https://www.researchgate.net/publication/221551396_The_Modal_Logic_of_Probability

I just found it. They define modal operators L_a, meaning probability is above a. I guess, modal operator is something used in propositional logic, and quantifiers are used with predicates with variables.