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Default logic (or default theory) is a non-monotonic logic proposed by Ray Reiter. Default logic seeks to better represent human methods of reasoning. It behaves like standard logic when complete information is available, but lets us infer things in the presence of incomplete information. Where in standard logic knowing that something is a bird tells us nothing of whether or not it can fly (since some birds do not fly, and many things that are not birds also fly), we have an intuitive notion that birds fly. Default logic lets us infer, barring evidence to the contrary, that anything which is a bird also flies.
Formal Definition of a Default Theory
A default theory of logic consists of a pair <D, W> where D is a set of expressions in first-order predicate calculus, and W is a set of default inference rules called default rules. These are of the form
- <math>\frac{Prerequisite(X) : Consistent_1(X) \and ... Consistent_n(X)}{Conclusion(X)}<math>
That is, if we know something about X, and each of Consistentn does not conflict with whatever else we might know about X, then we may infer the conclusion.
Some Default Theories
Returning to the example of birds flying, we can define our default rules as
- <math>W = \left\{ \frac{Bird(X) : Flies(X)}{Flies(X)} \right\} <math>,
and our initial premises as D = { Bird(Condor), Bird(Penguin), ¬Flies(Penguin), Flies(Airplane) }.
(In our simple theory, our consistency requirements are identical to the conclusion. This is not always the case in more complicated theories.)
From these, we can safely infer that a condor flies, because a condor is a bird and we don't know that condors can't fly. However we cannot infer that penguins fly, because it is inconsistent with our knowledge that penguins can't fly. As in standard logic we cannot say anything about whether or not an airplane is a bird because we have no inference rules available to do that, default or otherwise.
The computer language Prolog uses default logic with its closed-world assumption, which says that if it does not know something to be true, then it is false. Note that this rule has no prerequisite.
- <math>W = \left\{ \frac{ : {\neg}X}{{\neg}X} \right\} <math>
References
- Schmidt, Charles F. Default Logic (http://www.rci.rutgers.edu/~cfs/472_html/Logic_KR/DefaultTheory.html). Retrieved August 10th. 2004.
- Ramsay, Allan (1999). Default Logic (http://www.ccl.umist.ac.uk/teaching/material/5005/node33.html). Retrieved August 10th. 2004.
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