non-monotonic logic a logic that fails to be monotonic, i.e., in proof-theoretic terms, fails to meet the condition that for all statements u1, . . . un,f,y, if ‘u1, . . . un Yf’, then, for any y, ‘u1, . . . un, y Y f’. (Equivalently, let Γ represent a collection of statements, u1 . . . un, and say that in monotonic logic, if ‘Γ Y f’, then, for any y, ‘Γ, y Y f’ and similarly in other cases.) A non-monotonic logic is any logic with the following property: For some Γ, f, and y, ‘ΓNML f’ but ‘Γ, y K!NML f’. This is a weak non-monotonic logic. In a strong non-monotonic logic, we might have, again for some Γ, f, y, where Γ is consistent and Γ 8 f is consistent: ‘Γ, y YNML > f’.
A primary motivation (among AI researchers) for non-monotonic logic or defeasible reasoning, which is so evident in commonsense reasoning, is to produce a machine representation for default reasoning or defeasible reasoning. The interest in defeasible reasoning readily spreads to epistemology, logic, and ethics.
The exigencies of practical affairs requires leaping to conclusions, going beyond available evidence, making assumptions. In doing so, we often err and must leap back from our conclusions, undo our assumptions, revise our beliefs. In the literature’s standard example, Tweety is a bird and all birds fly, except penguins and ostriches. Does Tweety fly? If pressed, we may need to form a belief about this matter. Upon discovering that Tweety is a penguin, we may have to retract our conclusion. Any representation of defeasible reasoning must capture the nonmonotonicity of this reasoning. Non-monotonic logic is an attempt to do this within logic itself – by adding rules of inference that do not preserve monotonicity. Although practical affairs require us to reason defeasibly, the best way to achieve non-monotonicity may not be to add non-monotonic rules of inference to standard logic. What one gives up in such systems may well not be worth the cost: loss of the deduction theorem and of a coherent notion of consistency. Therefore, the challenge of non-monotonic logic (or defeasible reasoning, generally) is to develop a rigorous way to represent the structure of non-monotonic reasoning without losing or abandoning the historically hard-won properties of monotonic (standard) logic. See also ARTIFICIAL INTELLIGENCE , DEFAULT LOGIC , DEFEASIBILIT. F.A.