Handling Analogical Proportions in Classical Logic and Fuzzy Logics Settings
ECSQARU '09 Proceedings of the 10th European Conference on Symbolic and Quantitative Approaches to Reasoning with Uncertainty
Journal of Artificial Intelligence Research
Learning by analogy: a classification rule for binary and nominal data
IJCAI'07 Proceedings of the 20th international joint conference on Artifical intelligence
Analogy, paralogy and reverse analogy: postulates and inferences
KI'09 Proceedings of the 32nd annual German conference on Advances in artificial intelligence
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Given a 4-tuple of Boolean variables (a, b, c, d), logical proportions are modeled by a pair of equivalences relating similarity indicators (a∧band ¬a∧ ¬b), or dissimilarity indicators (a∧ ¬band ¬a∧b) pertaining to the pair (a, b) to the ones associated with the pair (c,d). There are 120 distinct logical proportions. One of them models analogical proportions which correspond to statements of the form "ais tobascis tod". The paper inventories the whole set of logical proportions by dividing it into 5 subfamilies according to what their logical proportions express, and then identifies the proportions that satisfy noticeable properties such as full identity (the pair of equivalences defining the proportion hold as true for the 4-tuple (a, a, a, a)), symmetry (if the proportion holds for (a, b, c, d), it also holds for (c, d, a, b)), or code independency (if the proportion holds for (a, b, c, d), it also holds for (¬a, ¬b, ¬c, ¬d)). Finally, the paper provides a discussion of the potential interest of logical proportions, which clearly have a cognitive appeal.