You can get into a real linguistic tangle with trying to talk about uncertainty, as Donald Rumsfeld discovered.  We say that Metarule is the language of uncertainty, what do we mean by this?

The standard scientific way to approach the world is to concern ourselves first with things we can measure and things we can be certain of. Classical logic and pure mathematics are all about certainties and absolute proof. Statistics is the main-stream method for dealing with uncertainty, and it attempts to apply pure maths and logical techniques to an uncertain world. Because statistics has been around a long time, we tend to trust it implicitly, just as we trust the findings of pure mathematics, but it may come as a surprise to find out that many of the suppositions of Statistics do not flow from first principles: there is a great big fat leap of faith required. The more classical statisticians have one main leap of faith, that the probability of something can be approximated by its frequency in trials. The less classical statisticians, known as the Bayesians, go further, in assuming you can apply probabilities to things whose frequency you can’t directly measure. See here for more details.  In both cases their justifications are that these techniques work, in some cases, such as thermodynamics, fantastically well, in other cases, well, not quite so well…

I mention this so that you can understand that there is no one divinely inspired algebra for uncertainty. 

When we talk about probabilities, we are in any event talking about only one aspect of uncertainty. Because doctors, sociologists, marketers and business people use probabilities so frequently we tend to act as  if they are the only available measure, and that the random chance of an event can be used to describe all forms of uncertainty.

Consider a court of law in a criminal case. Are the jurors who listen to a case considering just the probability of the defendant being guilty? I hope not, because the train of chances that brings a defendant to court and the circumstances of the crime are probably way outside the experience of the jurors; they’ve usually got little or no prior information about the likelihood of the events described in the courtroom.

Instead I hope they are considering the plausibility of the testimony and the arguments presented.

Similarly when engineers design a new nuclear power station, I hope it is the plausibility of the design going into meltdown that they are considering rather than the probability.

Lotfi Zadeh, the father of Fuzzy Logic has been ‘crying in the wilderness’ on this subject for a long time. He and his followers have created a vast body of work on an alternate measure of uncertainty, based on this idea of plausibility, that they call Possibility Theory.

Possibility and probability theories are not in competition. They consider different aspects of uncertainty. The only places where they clash are where Probability theory, as the older more established theory, has wandered outside its natural boundaries.

Metarule is concerned with plausibility more than probability. This is because Fuzzy logic supplies a natural way to expand classical logic and include possibility theory into that logical framework. Possibility theory can give us many useful insights into the real world, but it is also mathematically tractable.  This has many benefits.

If you want to simulate a real world system were the parameters are uncertain and have tolerances, for instance investment planning,  the standard method for the statistical world, Monte Carlo simulation requires thousands or millions of passes through the model to build up a picture of the outcomes. Fuzzy arithmetic requires one.

Although phrases like 'Fuzzy logic' and 'possibility theory' make the technologies used in Metarule sound mysterious, in fact their operation is very intuitive and closely models human reasoning.

I hope to revisit this subject in the near future.

Andy