El retículo de las lógicas de primer orden con cuantificadores cardinales

Associate to every class S of cardinals a quantifier Q^S so that Q^Sx𝜙(x) holds just in case the number of individuals satisfying 𝜙(x) is a cardinal belonging to S. This includes the well know cardinal quantifiers Q_∝. We give a simple combinatorial condition on the classes S and S' necessary and sufficient to have

Lωω (Q^S) ≤ Lωω (Q^S´).

A similar result is shown for logics generated by families of such quantifiers. Some applications follow; for example, it is shown that if nω denotes the set of multiples of the natural number n, then Lωω(Q^nω) ≤ Lωω (Q^mω) if and only if n divides m. Also, we construct infinite descending chains of logics.