Didier Dubois and Henri Prade wrote the following quote in a 1988 book entitled Non-standard Logics for Automated Reasoning:

A degree of truth is not a degree of uncertainty about truth.

This is a very important quote, and gets right to the matter I’d like to highlight. It distinguishes those problems with truth which are answered by two distinct, but related theories.

To answer the question of “a degree of uncertainty about truth”, something known as “Probabilistic Logic” was created. This merged together classical forms of logic, with its propositions and predicates, with Bayesian (or Bayesian-style) probability theory. It puts probability theory in a subjective perspective, and assigns probabilities to rules and statements, without the need for a frequency-based possible-worlds probability calculation.

As Didier and Henri rightly point out however, this really should not be confused with “a degree of truth”. For a degree of truth, fuzzy theory (i.e. fuzzy set theory and fuzzy logic) holds the solution. Fuzzy theory allows an object or statement to have a degree of membership of a set or a particular scenario.

For example “John is tall”. Tall is a vague concept, and “John” has a degree of truth of belonging to the vague “tall” concept. This is how fuzzy set theory, and matches our human way of thinking about tallness.

From a probabilistic logic perspective we would need to ask “what is the probability that John is Tall?”, which is quite a different question.

Of course, this is an area which has, for some reason, been a thorn in the fuzzy theorists side. There are many more probability theorists in this world at the moment, than there are fuzzy theorists. Once you start working with probability theory, it is easy to apply it to everything, even if it doesn’t quite fit. There are also some strong believers of probability theory, often labelled “Bayesians”, which attempt to assert that fuzzy set theory and fuzzy logic is somehow weak because the models can be “made-up” by experts instead of generated through statistics. Many fuzzy theorists have argued back, saying that its flexible model is actually a strength and not a weakness.

My own (current) research draws heavily from fuzzy set theory, but it (i.e. my current research) also has an element of probability theory as it implements data mining algorithms such as association rule mining and sequence pattern mining, which have a statistical element. I’m keen to investigate some more areas of the overlap between fuzzy and probability theories, as I consider them both to have a place (as do most other fuzzy theorists in fact). Of particular interest is the relationship of Fuzzy Formal Concept Lattices and Credal Networks. (Credal Networks are (and I simplify here) Bayesian Networks with added imprecision).

Let me know your thoughts on the above, and whether you have any hints or tips on the above. Feel free to email me or post a message in the comments box on this blog post.