## Default Theory

One suggestion for representing and reasoning with commonsense knowledge was developed by Ray Reiter and is known as default theory. The idea is to reason in first order logic but to have available a set of default rules which are used only if an inference can not be obtained within the first order formulation. The general framework of this proposal is depicted below.

In this proposal there are two sets of expressions, D and W. W is a set of expressions in standard first order logic. D is a set of default rules expressed in a unique syntax. This syntax is illustrated in the figure to the right. Each rule is a rule of inference such that if the expressions above the line (the premises) can be established, then the expressions below the line (the conclusion) may be asserted.

The premises consist of two components. The first is a set of first order expressions and is referred to as the prerequisite. These expressions must be proven (in the standard deductive sense) to be true in order for the rule to be applicable. The next set is referred to as the consistency test. These expressions must be consistent with the current database. That is, it must be proven that the negation of the expressions does not follow from the current database. If the rule is proven to be applicable, then the expressions referred to as the consequent are added to the database.

Default theory is an attempt to formally capture the idea that a set of beliefs may permit certain conclusions to be drawn even though these conclusions are not logically implied by the current set of beliefs. Note, however, that a default rule only applies if the conclusion can not be deductively derived from the existing set of beliefs. And, a conclusion can be derived in this fashion only if there is a default rule that supports its derivation.

At the bottom of this figure is shown an example default rule that is used to capture the commonsense knowledge that: Typically an American adult owns a car. Note that if our beliefs allowed us to infer that some person, say John, owns or does not own a car then this rule would not be invoked. The rule is only accessed if we wish to know whether or not John owns a car and an answer can not be deduced from our current beliefs. This default rule is applicable if we can prove from our beliefs that John is an American and an adult, and believing that there is some car that is owned by John does no't lead to an inconsistency. If these two sets of premises are satisfied, then the rule states that we can conclude that John owns a car.