Diagramas lógicos de Marlo para el razonamiento visual y heterogéneoválidos en lógica matemática y aristotélica

Abstract

This doctoral thesis presents the logical diagrams developed by the author for didactic purposes, with explanations about its operation, solved exercises, some data on its effectiveness in the classroom and a historical comparison with other classical forms of visual representation. We present two kinds of complementary logical diagrams: On the one hand, the Marlo networks of expectations, which are tree diagrams. On the other hand, Marlo diagrams, which represent the relationships between variables in geometric figures such as circles, triangles, and squares that we call propositional models. In both cases, these diagrams are intended to be tools for teaching logic in an intuitive way. Throughout the thesis, the reasoning is considered "heterogeneous": the visual information is integrated with the formal and natural language, which facilitates a better understanding of the inference processes; Furthermore, human reasoning, by its nature, is always accompanied by a dose of uncertainty, but that does not make it irrational. That is why we combine logic with probability theory in these diagrams. Let us bear in mind that we are very often forced to make decisions without all the information, and therefore our diagrams are intended to help students reason based not only on what is true or false, but also on what is likely to be true or likely false. In the first place, the Ortega concept of vital reason is made explicit as a philosophical framework, so that the principles of logic remain as supports for the learning of bio-psycho-social organisms, allowing them to generate and communicate expectations adapted to their circumstances. Secondly, some definitions of logical diagrams are reviewed, emphasizing that they have no less ability than formal language to show in a homeomorphic way the relationships between variables or terms established in the propositions. Subsequently, the networks of expectations are presented as Bayesian structures that can be transferred to a spreadsheet. Thus, we process mathematically multiple nuances between uncertainty and the certainty that something will become true or false. The following shows how Marlo diagrams, thanks to the foundations of Aristotelian logic and the doctrine of predicate quantification, facilitate inferences with the same precision, but more economically, than other diagrams such as Venn's in which the rigor of the conclusion requires showing explicitly and exhaustively all the combinations of variables that define the universe of discourse. The journey through the comparative history of logical diagrams aims to place our proposal within a long tradition that has illustrious names. In this section we intend to recognize influences and antecedents, so that it is possible to fairly evaluate to what extent our contributions may or may not be original. The thesis ends with multiple exercises solved in the networks of expectations and in Marlo's diagrams.