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In this paper we investigate the consequences and consistency of the downward L\"owenheim-Skolem theorem for extension of the first order logic by the Magidor-Malitz quantifier. We derive some combinatorial results and improve the known upper bound for the consistency of Chang's Conjecture at successor of singular cardinals.arXiv:1512.09299v4 fatcat:h2iqg67efje5jbolmmgaendvju