# A Brief Roadmap into Uncertain Knowledge Representation via Probabilistic Description Logics
## 3 Representing Uncertain Knowledge
### Statistical Logic  
The second class in [PERSON]'s classification is statistical logic, where the uncertainty is distributed over the objects of the domain, but only one \"possible world\" is considered at a time. That is, rather than the situation of the world, the unknown refers to the properties of specific individuals [34; 35].  
Syntactically, probabilistic description logics based on the statistical logic semantics often look very similar to those with subjective probabilities: each axiom is associated with a probability degree. The difference only becomes apparent in the semantics. In this case, the semantics assign a probability distribution for each property (or a combination thereof) within the elements of the domain. For example, an axiom stating that a property \(A\) is a subproperty of the property \(B\) with probability \(p\) is interpreted as a _conditional probability_ expressing that the probability of observing \(B\) given that \(A\) was observed is \(p\). This generally complicates reasoning as it requires the development of new techniques for dealing with individuals and transferring the properties among them [36].  
The additional difficulties of dealing with statistical probabilistic logics become obvious when exploring the literature. Indeed, while probabilistic description logics based on subjective probabilities abound, and their properties have been deeply studied, the variants based on statistical probabilities are extremely limited. Moreover, their reasoning complexity tends to grow as well [37]. Hence, despite being very useful in many situations--indeed, providing an adequate form of uncertainty in many practical scenarios--these logics are largely unexplored.  
In knowledge representation, one is often interested in the practical and computational properties of the languages as a way to guarantee adequate answers within reasonable time bounds, and minimise the effort of implementing, optimising, and updating the systems. However, the choice of semantics is fundamental to obtain the right answers. One way to summarise the difference between subjective and statistical probabilities is that in the former, an axiom holds in all the individuals or not, depending on the world under consideration; while in the latter, the properties of the axiom hold for some individuals and not in others. Consider for example the statement \"a person is female with probability 0.5\". Under subjective probabilities, this statement is interpreted as the knowledge that in half of all possible worlds, _every person is female_. Under statistical probabilities, instead, it is interpreted as stating that half of all persons are female. Although the difference may look very subtle at first sight, a simple reasoning question may highlight the deep differences between both: if one takes two random individuals and asks what the probability is that one is female and the other is male (assuming that there is additional knowledge about gender in the knowledge base), a statistical probabilistic approach would yield the (intuitive) answer that this probability is 0.5; a subjective probabilistic approach would instead set this probability to 0, because in its semantics, either all individuals are male, or all are female (and thus, there cannot be one of one gender and one of another). This simple example can be used as a general test to understand which kind of semantics is adequate in a given application.