Logical Proportions – Further Investigations
Logical proportions may be viewed as a Boolean counterpart of the notion of numerical proportions. They relate two pairs of binary variables, say (a,b) and (c, d), by stating the conjunction of two equivalences between similarity or dissimilarity indicators pertaining to these pairs of variables. Two variables are regarded as similar if they are both true or both false, they are dissimilar if one is true while the other is false. Logical proportions include the logical expression of the analogical proportion as a particular case. Although the phrase ‘logical proportion’ dates back to an early proposal by Piaget sixty years ago, the general definition of logical proportions has been proposed only very recently. There are 120 distinct logical proportions that can be organized in different subfamilies according to the way similarity or dissimilarity indicators are put in relation. Besides, subsets of logical proportions satisfying noticeable requirements such as permutation properties, or code independency (stability when taking the negation of all literals) have been also identified. The paper pursues the investigation of logical proportions having remarkable properties. In particular, the proportions that are homogeneous in the sense that they involve the same indicators for evaluating the pairs (a,b) and (c, d) are proved to be the 12 symmetric proportions. Some of the results are rather easy to prove, while others may be checked through tedious enumeration procedures. For this reason a program that helps establishing results is presented. A series of remarkable and sometimes surprising results are presented, which help to characterize logical proportions, to understand their potential interest, and to grasp their cognitive appeal.
Keywordsanalogical proportion proportion propositional logic
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