Nonmonotonicity

(a subtopic of Reasoning)

"Consider putting an axiom in a common sense database asserting that birds can fly. Clearly the axiom must be qualified in some way since penguins, dead birds and birds whose feet are encased in concrete can't fly. A careful construction of the axiom might succeed in including the exceptions of penguins and dead birds, but clearly we can think up as many additional exceptions like birds with their feet encased in concrete as we like. Formalized nonmonotonic reasoning [citations] provides a way of saying that a bird can fly unless there is an abnormal circumstance and reasoning that only the abnormal circumstances whose existence follows from the facts being taken into account will be considered." - John McCarthy

Nonmonotonicity. In John McCarthy's paper, Generality in Artificial Intelligence (1971-1987).

• Also see:
  • Applications of Circumscription to Formalizing Common Sense Knowledge. By John McCarthy. Artificial Intelligence (1986). "We present a new and more symmetric version of the circumscription method of nonmonotonic reasoning first described in (McCarthy 1980) and some applications to formalizing common sense knowledge."
  • Getting machines to think like us. Newsmaker interview with John McCarthy. By Jonathan Skillings. CNET News.com (July 3, 2006). " What are some of the big things that have been learned over the last 50 years that have helped shape research in artificial intelligence? McCarthy: Well, I suppose one of the big things was the recognition that computers would have to do nonmonotonic reasoning. Could you elaborate on that--on nonmonotonic reasoning? McCarthy: OK. In ordinary logical deduction, if you, say, have a sentence P that is deducible from a collection of sentences--call it A--and we have another collection of sentences B, which includes all the sentences of A, then it will still be deducible from B because the same proof will work. However, humans do reasoning in which that is not the case. Suppose I said, 'Yes, I will be home at 11 o'clock, but I won't be able to take your call.' Then the first part, "I will be home at 11 o'clock,"--you would conclude that I could take your call, but then if I added the 'but' phrase, then you would not draw that conclusion. So nonmonotonic reasoning is where you draw a conclusion, which may be a correct conclusion to draw, but it isn't guaranteed to be true because some added facts may prevent it. Now, that was around 1980, or a little bit before, that formalizing nonmonotonic reasoning began, and it's turned into a fairly big field now."

Non-Monotonic Logic. Entry in The Stanford Encyclopedia of Philosophy by G. Aldo Antonelli (March 27, 2006 revision), Edward N. Zalta (ed.). "The term 'non-monotonic logic' covers a family of formal frameworks devised to capture and represent defeasible inference, i.e., that kind of inference of everyday life in which reasoners draw conclusions tentatively, reserving the right to retract them in the light of further information. Such inferences are called 'non-monotonic' because the set of conclusions warranted on the basis of a given knowledge base does not increase (in fact, it can shrink) with the size of the knowledge base itself. This is in contrast to classical (first-order) logic, whose inferences, being deductively valid, can never be 'undone' by new information. ... One of the most significant developments both in logic and artificial intelligence is the emergence of a number of non-monotonic formalisms, which were devised expressly for the purpose of capturing defeasible reasoning in a mathematically precise manner. The fact that patterns of defeasible reasoning have been accounted for in such a rigorous fashion has wide-ranging consequences for our conceptual understanding of argumentation and inference. Pioneering work in the field of non-monotonic logics began with the realization that ordinary first-order logic is inadequate for the representation of defeasible reasoning accompanied by the effort to reproduce the success of FOL in the representation of mathematical, or formal, reasoning. Among the pioneers of the field in the late 1970's were (among others) J. McCarthy, D. McDermott & J. Doyle, and R. Reiter (see Ginsberg (1987) for a collection of early papers in the field and Gabbay et al (1994) for a more recent collection of excellent survey papers). In 1980, the Artificial Intelligence Journal published an issue (vol. 13, 1980) dedicated to these new formalisms, an event that has come to be regarded as the 'coming of age' of non-monotonic logic."

Nonmonotonic Logics. In MIT Encyclopedia of Cognitive Science (1999). Entry by Leora Morgenstern.

Non-monotonic and defeasible logics. By John Zeleznikow and Dan Hunter. "Where we have a set of axioms defined in the logical language, adding new axioms or facts may increase the set of true statements or may leave the set unchanged. However in no case would the addition of new statements reduce the number of true statements in the set of predicates. This feature is known as the monotonicity of the logic. It means that the logic is indefeasible, that there is no situation where the statement p Æ q can be defeated by further information. ... However in human reasoning, and particularly in law, we often find that the existence of both p Æ q and p does not necessarily entail q. We often have special exceptional cases defined either in statutes or resulting from cases interpreting statutes."