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Monotonicity of entailment is a property of many logical systems such that if a sentence follows deductively from a given set of sentences then it also follows deductively from any superset of those sentences.
A non-monotonic logic is a formal logic whose entailment relation is not monotonic.In other words, non-monotonic logics are devised to capture and represent defeasible inferences, i.e., a kind of inference in which reasoners draw tentative conclusions, enabling reasoners to retract their conclusion(s) based on further evidence. [1]
The term monotonic transformation (or monotone transformation) may also cause confusion because it refers to a transformation by a strictly increasing function. This is the case in economics with respect to the ordinal properties of a utility function being preserved across a monotonic transform (see also monotone preferences ). [ 5 ]
In the case of a completely monotonic function, the function and its derivatives must be alternately non-negative and non-positive in its domain of definition which would imply that function and its derivatives are alternately monotonically increasing and monotonically decreasing functions.
Defeasible logic is a non-monotonic logic proposed by Donald Nute to formalize defeasible reasoning. In defeasible logic, there are three different types of propositions: strict rules specify that a fact is always a consequence of another; defeasible rules specify that a fact is typically a consequence of another; undercutting defeaters
In general, a sigmoid function is monotonic, and has a first derivative which is bell shaped. Conversely, the integral of any continuous, non-negative, bell-shaped function (with one local maximum and no local minimum, unless degenerate) will be sigmoidal.
The following result is a generalisation of the monotone convergence of non negative sums theorem above to the measure theoretic setting. It is a cornerstone of measure and integration theory with many applications and has Fatou's lemma and the dominated convergence theorem as direct consequence.
Default logic is a non-monotonic logic proposed by Raymond Reiter to formalize reasoning with default assumptions.. Default logic can express facts like “by default, something is true”; by contrast, standard logic can only express that something is true or that something is false.