Full Idea
One might add to one's logic an 'uncountable quantifier', or a 'Chang quantifier', or a 'two-argument quantifier', or 'Shelah's quantifier', or 'branching quantifiers'.
Gist of Idea
There are at least five unorthodox quantifiers that could be used
Source
Leslie H. Tharp (Which Logic is the Right Logic? [1975], §3)
Book Reference
'Philosophy of Logic: an anthology', ed/tr. Jacquette,Dale [Blackwell 2002], p.39
A Reaction
[compressed - just listed for reference, if you collect quantifiers, like collecting butterflies]