# A Brief Roadmap into Uncertain Knowledge Representation via Probabilistic Description Logics
## 2 The Many Faces of Uncertainty  
Since our goal is to formally handle uncertain knowledge, we should start to clarify what uncertainty _is_, and how it can be quantified, combined, and more generally, manipulated. Slimming over the many and deeply interesting philosophical discussions on the topic, we consider the most etymological version of the term: uncertainty as a lack of certainty. In this respect, uncertain knowledge is _not_ a lack of knowledge (which classical knowledge representation languages effectively handle by means of so-called _open world semantics_) nor imprecise knowledge, which is the scope of fuzzy logic [11]. Instead, uncertainty refers to properties or events which either hold or not, but we cannot certainly know which is the case beforehand. Consider the typical example of a coin toss. Before the toss, we know that it will either land on _heads_ or _tails_, but we have no way of knowing a priori which will be the case; thus, we are uncertain of the result, up until the point when the coin is tossed. Note that uncertainty of a property may be an indirect consequence of other certain or uncertain properties, some of which may not be obviously stated. For instance, if one makes a bet on the coin toss, then it is known with certainty that _if_ the coin lands on heads, then they win $1, and they loose $1 otherwise. However, it is still uncertain which will be the actual case until the toss is made. The next obvious question is how can one represent and manage such uncertainty. When first encountering this question, most people immediately turn to the notion of _probability_. Indeed, probabilities and similarly percentages are taught to most of us at a relatively early stage, and we encounter them in almost all aspects of our daily lives; in fact, we use probabilistic terminology in our daily life interactions. This familiarity with the theory of probabilities is both a blessing and a curse. On the one hand, it greatly reduces the _entry cost_ of dealing with uncertainty in a formal setting, removing the wall of introducing a new theory along with its nomenclature and notation. On the other, the heavy baggage of probabilities includes many misunderstandings and erroneous intuitions that we have grown used to accept as true. Partly for this reason and in part for other issues that we will subsequently highlight, other uncertainty representations have been proposed; most notably, possibility theory [5] and evidence theory [7; 8] (often referred to also as the _Dempster-Shafer theory_). For the scope of this paper, as in most of the literature of uncertain knowledge representation, we give more weight to the advantages of the easiness of presentation over the likelihood of misunderstandings. Thus, we consider probability theory as the basis for representing and managing uncertainty in the context of knowledge representation. However, we still need to take into account the different interpretations of probability as uncertainty. Despite its unified name, and the use of probabilities for handling it, not all uncertainty is equal. [PERSON] [15] already hinted at it when describing the different types of logics of probability. Broadly speaking, [PERSON]'s classification considers two kinds of views on uncertainty: a _statistical_ one referring to a proportion of the population satisfying a property of interest, and a _subjective_ one dealing with beliefs about possible worlds. The difference lies in how the uncertainty is used within a derivation or reasoning process, but mirrors existing differences from the real-life use of probabilities. However, it is important to note that [PERSON]'s classification is orthogonal to the usual distinction between frequentist and Bayesian probabilities, which we will omit further mentioning anything about in this text. Statistical probabilities come into play when speaking about proportionality and a random selection of elements.