A computational problem is defined as a set of problem instances with specified properties. An algorithm solves a problem if it computes the correct answer to every problem instance in that set.
The use of a problem instance as an expository example is to represent its problem class and to provide some idea of how the algorithm works. A single problem instance can always be addressed by special pleading but the test of an algorithm is whether it handles the whole set of problem instances.
The program I wrote uses an extended topological variant of Peirce’s Alpha Graphs as its main data structure for representing propositions and it uses a general purpose algorithm that finds the complete set of satisfying logical interpretations for any proposition given on input. This is tantamount to using propositional calculus as a very simple form of declarative programming language.