On the amount of information resulting from empirical and theoretical knowledge. (English) Zbl 1076.62006

Vajda, Igor; Veselý, Arnošt; Zvárová, Jana

Rev. Mat. Complut. 18, No. 2, 275-286 (2005).

Summary: We present a mathematical model allowing to formally define the concepts of empirical and theoretical knowledge. The model consists of a finite set \(\mathcal P\) of predicates and a probability space \((\Omega,{\mathcal S}, P)\) over a finite set \(\Omega\) called ontology which consists of objects \(\omega\) for which the predicates \(\pi\in{\mathcal P}\) are either valid \((\pi(\omega) =1)\) or not valid \((\pi(\omega) = 0)\). Since this is a first step in this area, our approach is as simple as possible, but still nontrivial, as it is demonstrated by examples. A more realistic approach would be more complicated, based on a fuzzy logic where the predicates \(\pi\in{\mathcal P}\) are valid on the objects \(\omega\in\Omega\) to some degree \((0\leq\pi(\omega)\leq 1)\).
We use the classical information divergence to introduce the amount of information in empirical and theoretical knowledge. By an example is demonstrated that information in theoretical knowledge is an extension of the “semantic information” introduced formerly by Y. Bar Hillel and R. Carnap [Anoutline of a theory of semantic information. Tech. Rep. (1964)] as an alternative to the information of Shannon [see C. E. Shannon and W. Weaver, The mathematical theory of communication. (1949; Zbl 0041.25804)].

MSC:

62B10	Statistical aspects of information-theoretic topics
94A17	Measures of information, entropy
60A99	Foundations of probability theory
94D05	Fuzzy sets and logic (in connection with information, communication, or circuits theory)
62A01	Foundations and philosophical topics in statistics

Citations:

Zbl 0041.25804