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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 (consequences may be derived only because of lack of evidence of the contrary), abductive reasoning (consequences are only deduced as most likely explanations), some important approaches to reasoning about knowledge (the ignorance of a consequence must be retracted when the consequence becomes known), and similarly, belief revision (new knowledge may contradict old beliefs).
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. The default assumption 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/defeasible reasoning/argument (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 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.
Reasoning about knowledge
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 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.
Proof-theoretic versus model-theoretic formalizations of non-monotonic logics
Proof-theoretic formalization of a non-monotonic logic begins with adoption of certain non-monotonic rules of inference, and then prescribes contexts in which these non-monotonic rules may be applied in admissible deductions. This typically is accomplished by means of fixed-point equations that relate the sets of premises and the sets of their non-monotonic conclusions. Defaults logics and autoepistemic logic are the most common examples of non-monotonic logics that have been formalized that way.
Model-theoretic formalization of a non-monotonic logic begins with restriction of the semantics of a suitable monotonic logic to some special models, for instance, to minimal models, and then derives the set of non-monotonic rules of inference, possibly with some restrictions in which contexts these rules may be applied, so that the resulting deductive system is sound and complete with respect to the restricted semantics. Unlike some proof-theoretic formalizations that suffered from well-known paradoxes and were often hard to evaluate with respect of their consistency with the intuitions they were supposed to capture, model-theoretic formalizations were paradox-free and left little, if any, room for confusion about what non-monotonic patterns of reasoning did they cover. Examples of proof-theoretic formalizations of non-monotonic reasoning, which revealed some undesirable or paradoxical properties or did not capture the desired intuitive comprehensions, that have been successfully (consistent with respective intuitive comprehensions a with no paradoxical properties, that is) formalized by model-theoretic means include first-order circumscription, closed-world assumption, and autoepistemic logic.
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