192 11 Stages of Cognitive Development Finally, another limiting aspect of Piaget’s model is that it did not recognize any stages beyond formal operations, and included no provisions for exploring this possibility. A mumber of researchers [Bic88, Arl75, CRK82, Rie73, Mar01| have described one or more postformal stages. Commons and colleagues have also proposed a task-based model which provides a framework for explaining stage discrepancies across tasks and for generating new stages based on classification of observed logical behaviors. [IK1<90] promotes a statistical conception of stage, which provides a good bridge between task-based and stage-based models of development, as statistical modeling allows for stages to be roughly defined and analyzed based on collections of task behaviors. [CRK&82] postulates the existence of a postformal stage by observing elevated levels of abstrac- tion which, they argue, are not manifested in formal thought. [CTS* 98] observes a postformal stage when subjects become capable of analyzing and coordinating complex logical systems with each other, creating metatheoretical supersystems. In our model, with the reflexive stage of development, we expand this definition of metasystemic thinking to include the ability to consciously refine one’s own mental states and formalisms of thinking. Such self-reflexive re- finement is necessary for learning which would allow a mind to analytically devise entirely new structures and methodologies for both formal and postformal thinking. In spite of these various critiques and limitations, however, we have found Piaget’s ideas very useful, and in Section 11.4 we will explore ways of defining them rigorously in the specific context of CogPrime’s declarative knowledge store and probabilistic logic engine. 11.3.1 Perry’s Stages Also relevant is William Perry’s [Per70, Per81] theory of the stages (“positions” in his terminol- ogy) of intellectual and ethical development, which constitutes a model of i