Is there a "standard" form for a) a generalized existence quantor, assuring that there are at least n objects existing, b) a generalized forall quantor, allowing that n objects violate its assurance? Of course any such statement can be written with $\forall,\exists$. In any case I never saw a (say) $\forall^N$ in math literature yet, and googling didn't brought up anything (as usual, I probably just used the wrong search terms...)
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See Generalized Quantifiers – Mauro ALLEGRANZA Mar 31 '20 at 10:09
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See also my answers to Why haven't mathematicians come up with an efficient way of writing “sufficiently”, e.g. “for $n$ sufficiently large” (a "stack exchange" version of this 21 December 2004 sci.math post) and Relations between the "$\forall$" quantifier and integration. – Dave L. Renfro Mar 31 '20 at 11:49
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(waves finger) Original research :-) (I.e. I can't suppose the casual reader of my work is familiar with the notation.) But it didn't surprise me the concept already exists. – Hauke Reddmann Apr 02 '20 at 08:29