Everything about Non-monotonic Logic totally explained
A
non-monotonic logic is a
formal logic whose
consequence relation isn't
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 can't reduce the set of what is known. A monotonic logic can't 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's later learned that the considered animal is a penguin. This example shows that a logic that models default reasoning shouldn't 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 shouldn't 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.
Reasoning about knowledge
If a logic includes formulae that mean that something isn't known, this logic shouldn't be monotonic. Indeed, learning something that was previously not known leads to the removal of the formula specifying that this piece of knowledge isn't 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.
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