We show that propositional logic and its extensions can support answer-set programming in the same way stable logic programming and disjunctive logic programming do. To this end, we introduce a logic based on the logic of propositional schemata and on a version of the Closed World Assumption. We call it the extended logic of propositional schemata with CWA (PS+, in symbols). An important feature of this logic is that it supports explicit modeling of constraints on cardinalities of sets. In the paper, we characterize the class of problems that can be solved by finite PS+ theories. We implement a programming system based on the logic PS+ and design and implement a solver for processing theories in PS+. We present encouraging performance results for our approach --- we show it to be competitive with smodels, a state-of-the-art answer-set programming system based on stable logic programming.