1. A method for a processor to respond to a Boolean query using an inconsistent knowledge base, the method comprising: receiving the Boolean query including an original formula; determining if the inconsistent knowledge base non-standardly entails the original formula; when the inconsistent knowledge base non-standardly entails the original formula, selecting a bridge formula that is non-standardly entailed by the inconsistent knowledge base, can be resolved with the original formula, and does not contradict the original formula; constructing a but answer comprising the original formula, a but formula, and the bridge formula, the but formula being a negation of a resolvent of the original formula and the bridge formula; validating the but answer; and transmitting or displaying the but answer with at least the original formula and the but formula.