Multiple-Valued Logic Interpretations of Analogical, Reverse Analogical, and Paralogical Proportions

Boolean logic interpretations have been recently proposed for analogical proportions (i.e. statements of the form \´\´$a$ is to $b$ as $c$ is to $d$"), and for two other related formal proportions named reverse analogy (\´\´what $a$ is to $b$ is the reverse of what $c$ is to $d$"), and paralogy (\´\´what $a$ and $b$ have in common $c$ and $d$ have it also"). These proportions relate items $a$, $b$, $c$, and $d$ in different ways, on the basis of their differences, or of their similarities. This paper investigates multiple-valued models for these proportions. These extensions may serve different purposes, such as taking into account graded features, or handling crisp ones in non binary attribute domains in a compact way. After summarizing the main results in the binary case, we discuss what multiple valued patterns make sense for the different proportions, starting with tri-valued interpretations, and then considering $[0,1]$-valued interpretations. It appears that Lukasiewicz implication-based interpretation fits well with the intended meaning of analogy and reverse analogy, while a minimum-based interpretation is more suitable for paralogy in case of graded features. Besides, the interest of an interpretation based on Post algebra in case of non binary attribute domains is briefly outlined. Similarities and differences with the standard Boolean setting are highlighted.