Non-monotonic logic
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A non-monotonic logic is a
formal logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premis ...
whose conclusion
relation Relation or relations may refer to: General uses *International relations, the study of interconnection of politics, economics, and law on a global level *Interpersonal relationship, association or acquaintance between two or more people *Public ...
is not
monotonic In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order ...
. In other words, non-monotonic logics are devised to capture and represent defeasible inferences (cf.
defeasible reasoning In philosophical logic, defeasible reasoning is a kind of reasoning that is rationally compelling, though not deductively valid. It usually occurs when a rule is given, but there may be specific exceptions to the rule, or subclasses that are su ...
), i.e., a kind of inference in which reasoners draw tentative conclusions, enabling reasoners to retract their conclusion(s) based on further evidence. Most studied formal logics have a monotonic entailment relation, meaning that adding a formula to a theory never produces a pruning of its set of conclusions. 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 (conclusions may be derived only because of lack of evidence of the contrary),
abductive reasoning Abductive reasoning (also called abduction,For example: abductive inference, or retroduction) is a form of logical inference formulated and advanced by American philosopher Charles Sanders Peirce beginning in the last third of the 19th century ...
(conclusions are only deduced as most likely explanations), some important approaches to reasoning about knowledge (the ignorance of a conclusion must be retracted when the conclusion becomes known), and similarly,
belief revision Belief revision is the process of changing beliefs to take into account a new piece of information. The logical formalization of belief revision is researched in philosophy, in databases, and in artificial intelligence for the design of rational age ...
(new knowledge may contradict old beliefs).


Abductive reasoning

Abductive reasoning Abductive reasoning (also called abduction,For example: abductive inference, or retroduction) is a form of logical inference formulated and advanced by American philosopher Charles Sanders Peirce beginning in the last third of the 19th century ...
is the process of deriving a sufficient explanation of the known facts. An abductive logic should not be monotonic because the likely explanations are not necessarily correct. For example, the 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

Belief revision Belief revision is the process of changing beliefs to take into account a new piece of information. The logical formalization of belief revision is researched in philosophy, in databases, and in artificial intelligence for the design of rational age ...
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 A paraconsistent logic is an attempt at a logical system to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing "inconsistency-tolerant" syste ...
, which tolerate inconsistency rather than attempting to remove it.


See also

*
Logic programming Logic programming is a programming paradigm which is largely based on formal logic. Any program written in a logic programming language is a set of sentences in logical form, expressing facts and rules about some problem domain. Major logic prog ...
*
Negation as failure Negation as failure (NAF, for short) is a non-monotonic inference rule in logic programming, used to derive \mathrm~p (i.e. that ~p is assumed not to hold) from failure to derive ~p. Note that \mathrm ~p can be different from the statement \neg p ...
* Stable model semantics *
Rational consequence relation In logic, a rational consequence relation is a non-monotonic consequence relation satisfying certain properties listed below. Properties A rational consequence relation \vdash satisfies: :; REF : Reflexivity \theta \vdash \theta and the so-cal ...


Notes


References

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External links

* * * {{Logic Belief revision Formal epistemology Logic Non-classical logic Reasoning