What does 'nonmonotonic' mean?

In classical logic, adding premises never removes conclusions: the set of theorems only grows, which is called monotonic. Reasoning is nonmonotonic when new information can cause earlier conclusions to be withdrawn, as happens with default assumptions that turn out to have exceptions.

Why can't classical logic handle commonsense defaults?

Classical logic is monotonic, so a default like 'birds typically fly' cannot be stated as a strict rule without becoming false for exceptions like penguins. Nonmonotonic logics let such defaults hold tentatively and be overridden when specific contrary facts are known.

Nonmonotonic Reasoning

Nonmonotonic reasoning formalizes inference in which adding new information can retract previously drawn conclusions, capturing the defeasible, default-laden character of commonsense thought.

Definition

A reasoning system is nonmonotonic when the set of conclusions does not grow monotonically with the premises: learning a new fact can invalidate an earlier inference, as when discovering that a particular bird is a penguin retracts the default conclusion that it can fly.

Scope

This topic covers logics and methods for reasoning with incomplete information and defaults, where conclusions are tentative and may be withdrawn: default logic, circumscription, the closed-world assumption, negation as failure, and the broader idea of defeasible inference. It addresses why classical logic's monotonicity is inadequate for commonsense reasoning and how nonmonotonic formalisms restore the ability to jump to and later revise conclusions. Probabilistic approaches to uncertainty are treated under reasoning under uncertainty.

Core questions

Why is classical logic monotonic, and why does commonsense reasoning require nonmonotonicity?
How do default rules license tentative conclusions in the absence of contrary evidence?
How does circumscription minimize the extension of abnormality predicates to capture defaults?
How are conflicts between defaults and exceptions resolved when new information arrives?

Key concepts

monotonicity vs. nonmonotonicity
default rules and extensions
circumscription
closed-world assumption
negation as failure
defeasible inference
abnormality predicates
belief revision

Key theories

Default logic: Reiter's default logic augments classical logic with default rules of the form 'if A holds and it is consistent to assume B, then conclude B,' producing sets of conclusions (extensions) that license commonsense defaults while remaining retractable.
Circumscription: McCarthy's circumscription is a nonmonotonic inference that minimizes the extension of selected (abnormality) predicates, formalizing the assumption that things are as normal as the known facts permit and thereby justifying default conclusions.
Closed-world assumption and negation as failure: Treating the absence of a fact from a knowledge base as evidence of its falsehood (closed-world assumption, realized operationally as negation as failure) is a widely used nonmonotonic mechanism, central to logic programming and databases.

Clinical relevance

Nonmonotonic mechanisms appear wherever systems must reason with incomplete information and sensible defaults: logic programming and deductive databases (via negation as failure), rule-based expert systems with exceptions, answer set programming for combinatorial reasoning, and formal models of commonsense and legal reasoning.

History

A cluster of foundational papers in the 1980 special issue of Artificial Intelligence, including Reiter's default logic and McCarthy's circumscription, launched the formal study of nonmonotonic reasoning in response to the frame problem and the inadequacy of classical logic for commonsense. The field later fed into answer set programming and belief revision.

Key figures

Raymond Reiter
John McCarthy
Drew McDermott
Jon Doyle
Robert C. Moore

Seminal works

reiter1980
mccarthy1980

Frequently asked questions

What does 'nonmonotonic' mean?: In classical logic, adding premises never removes conclusions: the set of theorems only grows, which is called monotonic. Reasoning is nonmonotonic when new information can cause earlier conclusions to be withdrawn, as happens with default assumptions that turn out to have exceptions.
Why can't classical logic handle commonsense defaults?: Classical logic is monotonic, so a default like 'birds typically fly' cannot be stated as a strict rule without becoming false for exceptions like penguins. Nonmonotonic logics let such defaults hold tentatively and be overridden when specific contrary facts are known.