McCarthy used circumscription to figure out how to solve the frame problem. He added to first-order logic to minimize the extension of some predicates, where the extension of a predicate is the set of tuples of values that the predicate is factual. 

It was the first way that circumscription could be used. It is like the closed-world assumption, which says that anything that isn't known to be true is false. 

Objective

There are three missionaries and three cannibals on one side of a river. They need to get across in a boat that only holds two, with the additional constraint that the cannibals on either side of the river can never outnumber the missionaries. It is an original problem that McCarthy considered (as otherwise, the missionaries would be killed and, presumably, eaten). 

McCarthy's focus was not on determining a course of action to achieve the aim (a solution to which can be found in the article on the missionaries and cannibals problem) but on ruling out possibilities that are not brought up in the text. For example, there are better solutions than going half a mile south and crossing the river on the bridge because it needs to be mentioned in the problem statement. 

However, the existence of this bridge is not ruled out by the given issue statement. Instead, the lack of the bridge is a natural result of the assumption inherent in the problem statement that it includes all the information necessary to find a solution. 

Usage

Later, McCarthy used circumscription to put into words the unspoken principle of inertia: nothing changes unless explicitly said to do so. However, McCarthy's technique could produce incorrect outcomes, such as in the Yale shooting dilemma. Alternative approaches to the frame problem also correctly formalize the Yale shooting problem.

Predicate circumscription

McCarthy's first-order logic-based definition of circumscription stands as the gold standard. In first-order logic, predicates serve as the propositional logic (true or false) equivalent of variables. In addition, First-order logic's version of circumscription involves forcing predicates to be false whenever possible to acquire the formula's circumscription, a process known as the minimization of predicates.

Pointwise circumscription

Vladimir Lifschitz developed a kind of first-order circumscription known as pointwise circumscription. Pointwise and predicate circumscription coincide in the propositional case. Pointwise circumscription is justified because it minimizes the value of a predicate for each tuple of values individually rather than reducing the predicate's extension.

Domain and formula circumscription

Instead of extending predicates, McCarthy's earlier formulation of circumscription relies on limiting the scope of first-order models. In particular, a model is deemed inferior to another if it has a smaller domain and the models agree on evaluating the shared tuples of values. Predicate circumscription is a more basic variation of circumscription.

McCarthy later popularised the formalism of formula circumscription. In this application of circumscription, the extension of a formula rather than the extension of a predicate is minimized. 

Conclusion

Circumscription only sometimes handles contradictory information in the right way. Ray Reiter gave this example: If you throw a coin over a checkerboard, the coin could land on a black square, a white square, or both. But there are a lot of other places where the coin is not supposed to be. For example, the coin is not on the floor, fridge, or moon. 

Furthermore, Thomas Eiter, Georg Gottlob, and Yuri Gurevich presented the idea of theory curbing as a solution. The theory states that the model in which both coins are true is the one that circumscription fails to choose. It selects the least upper-bound models in addition to circumscription. The least upper bounds of all models are included until it is closed.