Probability, Truth and Flow Graph
Zdzisław Pawlak · Electronic Notes in Theoretical Computer Science · 2003
In 1913 Jan Łukasiewicz proposed to use logic as mathematical foundations of probability. He claims that probability is “purely logical concept” and that his approach frees probability from its obscure philosophical connotation. He recommends to replace the concept of probability by the concept of a truth value, which can be regarded as a degree of truth, i.e., a number between 0 and 1, of propositional functions (called in his work indefinite propositions). Further he shows that all laws of probability can be obtained from a properly built logical calculus. In this paper we show that the idea of Łukasiewicz can be also expressed differently. Instead of using truth values in place of probability, stipulated by Łukasiewicz, we propose, in this paper, using of deterministic flow analysis in flow networks (graphs). In the proposed setting, flow is governed by some probabilistic rules (e.g., Bayes’ rule), or by the corresponding logical rules, proposed by Łukasiewicz, though, the formulas have entirely deterministic meaning, and need neither probabilistic nor logical interpretation. They simply describe flow distribution in flow graphs. However, flow graphs introduced here are different to those proposed by Ford and Fulkerson, for optimal flow analysis, because they model rather flow distribution in a plumbing network, then the optimal flow. The flow graphs considered in this paper can be also used as a description of a decision algorithms, where branches of the graph are interpreted as decision rules. This feature causes that flow networks can be also used as a new tool for data analysis, and knowledge representation.