Definitions

# Non-monotonic logic

A non-monotonic logic is a formal logic whose consequence relation is not monotonic. Most studied formal logics have a monotonic consequence relation, meaning that adding a formula to a theory never produces a reduction of its set of consequences. Intuitively, monotonicity indicates that learning a new piece of knowledge cannot reduce the set of what is known. A monotonic logic cannot handle various reasoning tasks such as reasoning by default (facts may be known only because of lack of evidence of the contrary), abductive reasoning (facts are only deduced as most likely explanations), reasoning about knowledge (the ignorance of a fact must be retracted when the fact becomes known), and belief revision (new knowledge may contradict old beliefs).

## Default reasoning

An example of a default assumption is that the typical bird flies. As a result, if a given animal is known to be a bird, and nothing else is known, it can be assumed to be able to fly. This fact must however be retracted if it is later learned that the considered animal is a penguin. This example shows that a logic that models default reasoning should not be monotonic. Logics formalizing default reasoning can be roughly divided in two categories: logics able to deal with arbitrary default assumptions (default logic, defeasible logic, and answer set programming) and logics that formalize the specific default assumption that facts that are not known to be true can be assumed false by default (closed world assumption and circumscription).

## Abductive reasoning

Abductive reasoning is the process of deriving the most likely explanations of the known facts. An abductive logic should not be monotonic because the most likely explanations are not necessarily correct. For example, the most likely explanation for seeing wet grass is that it rained; however, this explanation has to be retracted when learning that the real cause of the grass being wet was a sprinkler. Since the old explanation (it rained) is retracted because of the addition of a piece of knowledge (a sprinkler was active), any logic that models explanations is non-monotonic.

If a logic includes formulae that mean that something is not known, this logic should not be monotonic. Indeed, learning something that was previously not known leads to the removal of the formula specifying that this piece of knowledge is not known. This second change (a removal caused by an addition) violates the condition of monotonicity. A logic for reasoning about knowledge is the autoepistemic logic.

## Belief revision

Belief revision is the process of changing beliefs to accommodate a new belief that might be inconsistent with the old ones. In the assumption that the new belief is correct, some of the old ones have to be retracted in order to maintain consistency. This retraction in response to an addition of a new belief makes any logic for belief revision to be non-monotonic. The belief revision approach is alternative to paraconsistent logics, which tolerate inconsistency rather than attempting to remove it.

## References

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• G. Brewka (1991). Nonmonotonic Reasoning: Logical Foundations of Commonsense. Cambridge University Press.
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• M. Cadoli and M. Schaerf (1993) "A survey of complexity results for non-monotonic logics" Journal of Logic Programming 17: 127-60.
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• M. L. Ginsberg, ed. (1987) Readings in Nonmonotonic Reasoning. Los Altos CA: Morgan Kaufmann.
• Horty, J. F., 2001, "Nonmonotonic Logic," in Goble, Lou, ed., The Blackwell Guide to Philosophical Logic. Blackwell.
• W. Lukaszewicz (1990) Non-Monotonic Reasoning. Ellis-Horwood, Chichester, West Sussex, England.
• D. Makinson (2005) Bridges from Classical to Nonmonotonic Logic, College Publications.
• W. Marek and M. Truszczynski (1993) Nonmonotonic Logics: Context-Dependent Reasoning. Springer Verlag.