New Methods in Qualitative Calculus

A qualitative calculus based on signs was first developed by economists to overcome one major difficulty: we often face a lack of quantitative data. A number of methods such as comparative statics were proposed to solve qualitative-model-based problems. A new interest has recently arisen in qualitative methods as applied to process control, control theory, and artificial intelligence (the author is himself involved in Qualitative Physics a subfield of AI). New approaches, such as order of magnitude reasoning, have extended the scope of "what is qualitative". Beyond some aspects specific to these fields for instance representing how humans reason on the behavior of a system is a major concern in AI this has led to new results and methods for dealing with qualitative models. These results are likely to be applicable within any framework, including economical modelling. This paper discusses these new theoretical aspects of qualitative calculus. We hope that it will provide a better insight into these mathematical-like topics and contribute to the emergence of a general theory as to "what is qualitative".