I posted a similar question on math stack exchange with the same title, but I didn't get a helpful response. I am trying to develop a logical language where one can express variable numerical quantifiers. In first order logic, one can express statements like "there exist at most 10 x such that Px" and "there exist exactly 25 x such that Px". But I am trying to develop a language where one can express statements like "there exist at least 3n+1 x such that Px", where the n is itself a variable that can be quantified over. Everytime I come up with a definition, or develop a theory, I look on the Internet and find that someone else has already worked it out better. So, my question is, is there a paper or book where someone develops this kind of logical language?



Variable numerical quantifiers
There is an old notion of a generalized quantifier introduced by Andrzej Mostowski in late '50s:


A. Mostowski, "On a generalization of quantifiers", Fundamenta Mathematicae 44, 1957.


The paper is actually well-written, but you may also enjoy reading the following introduction:


J. A. Väänänen, "Generalized Quantifiers, an Introduction", European Summer School in Logic, Language and Information, 1997.




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As pointed out by Emil and Andreas in the comments, logics extended by some classes of non-standard quantifiers are interesting from the perspective of finite model theory and descriptive complexity --- here are, for example, some results on second-order generalized quantifiers (however, take into account, that I do not follow recent work on the subject):


A. Andersson, "On second-order generalized quantifiers and finite structures", Annals of Pure and Applied Logic, Volume 115, Issues 1–3, 2002.


Another context where you can find the notion of a generalized quantifier is the study of semantics for natural languages:


S. Peters, D. Westerståhl, "Quantifiers in Language and Logic", Clarendon Press, 2006.


