Writing complex queries in SQL is a challenge for users. Prior work has developed several techniques to ease query specification but none of these techniques are applicable to a particularly difficult class of queries: quantified queries. Our hypothesis is that users prefer to specify quantified queries interactively by trial-and-error. We identify two impediments to this form of interactive trial-and-error query specification in SQL: (i) changing quantifiers often requires global syntactical query restructuring, and (ii) the absence of non-answers from SQL's results makes verifying query correctness difficult. We remedy these issues with DataPlay, a query tool with an underlying graphical query language, a unique data model and a graphical interface. DataPlay provides two interaction features that support trial-and-error query specification. First, DataPlay allows users to directly manipulate a graphical query by changing quantifiers and modifying dependencies between constraints. Users receive real-time feedback in the form of updated answers and non-answers. Second, DataPlay can auto-correct a user's query, based on user feedback about which tuples to keep or drop from the answers and non-answers. We evaluated the effectiveness of each interaction feature with a user study and we found that direct query manipulation is more effective than auto-correction for simple queries but auto-correction is more effective than direct query manipulation for more complex queries.