12. The non-transitory, computer-readable medium of claim 11 , wherein the selecting the bridge formula comprises: forming the bridge formula as a conjunction of a step number of different clauses in the inconsistent knowledge base; determining if the original formula and the bridge formula can be resolved; when the original formula and the bridge formula cannot be resolved so the bridge formula is rejected: determining if a termination condition is met; when the termination condition is not met, incrementing the step number and repeating above steps starting from the forming the bridge formula; when the termination condition is met, ending the method; when the original formula and the bridge formula can be resolved: determining if the original formula and the bridge formula are contradictory; when the original formula and the bridge formula are contradictory so the bridge formula is rejected: determining if the termination condition is met; when the termination condition is not met, incrementing the step number and repeating above steps starting from the forming the bridge formula; when the termination condition is met, ending the method; when the original formula and the bridge formula are not contradictory, using the bridge formula for the constructing and the validating.
11. A non-transitory, computer-readable medium encoded with computer-executable instructions for a processor to implement an inconsistent reasoner that responds to a Boolean query using an inconsistent knowledge base, the instructions 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.