A vexing question in current presupposition theory is what a sentence with a presupposition trigger presupposes when it is in the scope of different quantifiers. Four dominant theories in the area, namely, Universal Theories, Existential Theories, Similarity Theory, and Trivalent Logic Theory, make different predictions about presuppositions of sentences like "Each/Some/None of the boys knows he is lucky." The current paper argues that previous experimental studies are not sufficient to adjudicate among the four theories, and this work is designed to investigate this issue. The results of naturalness judgment tasks show that (i) a trigger type effect is attested once individual variation is taken into consideration; (ii) existential presuppositions are highly available from "none" sentences (pace Chemla 2009); (iii) no participant showed a pattern in which universal presuppositions projected from all types of quantifiers (pace Sudo et al. 2012). This paper proposes to modify the Trivalent Logic Theory (Fox, 2013) by weakening the assumed universal statement in the truth conditions of quantified sentences to an existential one, and extends the triggering source of the repair mechanism (A-operator) and eliminates the redundant Pragmatic Strengthening strategy. This study mainly lends support to the trivalent logic approach and the assumption that presuppositions can be collapsed with truth conditions. It also suggests that the lexical meaning of quantifiers at the presupposition level should be existential rather than universal, independent of quantifier type.